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.

翻译：本文针对非线性Cosserat弹性体，研究了多尺度方法。具体而言，我们采用广义多尺度有限元法（GMsFEM）求解具有应变限制特性的各向同性Cosserat问题（确保在高应力下线性化应变有界）。此类应变限制Cosserat模型可应用于固体和生物纤维领域。然而，具有自然旋转自由度、非线性本构关系、高对比度及异质性的Cosserat介质，其解可能产生具有挑战性的多尺度特征，因此需要借助多尺度方法进行升尺度计算。为此，我们在该问题中采用离线型和基于残差的在线型（自适应或均匀）GMsFEM方法，并通过Picard迭代处理非线性。通过二维数值实验（涵盖小/大应变限制参数下的含孔介质、复合材料及随机异质介质），结果表明该方法具有收敛性、高效性和鲁棒性。进一步，数值结果证明该方法能提供良好的精度：在线型GMsFEM比离线型获得更精确的解，且在线自适应策略在自由度更少的情况下保持与均匀策略相当的精度。