  For nonlinear Cosserat elasticity, we consider multiscale methods in this
paper. In particular, we explore the generalized multiscale finite element
method (GMsFEM) to solve an isotropic Cosserat problem with strain-limiting
property (ensuring bounded linearized strains even under high stresses). Such
strain-limiting Cosserat model can find potential applications in solids and
biological fibers. However, Cosserat media with naturally rotational degrees of
freedom, nonlinear constitutive relations, high contrast, and heterogeneities
may produce challenging multiscale characteristics in the solution, and
upscaling by multiscale methods is necessary. Therefore, we utilize the offline
and residual-based online (adaptive or uniform) GMsFEM in this context while
handling the nonlinearity by Picard iteration. Through various two-dimensional
experiments (for perforated, composite, and stochastically heterogeneous media
with small and big strain-limiting parameters), our numerical results show the
approaches' convergence, efficiency, and robustness. In addition, these results
demonstrate that such approaches provide good accuracy, the online GMsFEM gives
more accurate solutions than the offline one, and the online adaptive strategy
has similar accuracy to the uniform one but with fewer degrees of freedom.
