Multiscale Modeling of Micropolar Media

Classical (Cauchy) continuum mechanics fails to capture size effects observed experimentally in materials with complex hierarchical microstructures — a well-documented phenomenon in bone tissue, where small samples exhibit significantly higher stiffness than classical models predict. Micropolar (Cosserat) continuum theory addresses this by introducing independent rotational degrees of freedom and couple-stress fields at the macroscale, effectively encoding a characteristic length scale related to the microstructure.

However, defining micropolar material parameters from microstructural data is non-trivial, even for simple neo-Hookean materials. This project develops a multiscale computational homogenization framework that bypasses the need for phenomenological micropolar constitutive laws at the RVE level by connecting:

• a Cauchy microstructure (described by classical displacement fields, amenable to standard constitutive models), to
• a Micropolar macroscale (Cosserat continuum with translations, rotations, forces, and couple-stresses),

via the Method of Multiscale Virtual Power (MMVP), extended to the Cauchy-to-Micropolar transition.

The development plan includes:

• Deriving the macroscopic equilibrium equations and homogenization operators so that the internal work at the macroscale — including the couple-stress term — is equivalent to the average deformation energy in the Cauchy RVE, satisfying the extended Hill-Mandel condition;
• Developing RVE boundary conditions that homogenize couple-stress and moment tensors from a purely displacement-based microscopic description;
• Implementing the FE² algorithm in FEniCSx, using elements with rotational degrees of freedom at the macroscale;
• Characterizing the size effect: parametric studies on how RVE volume variation influences the micropolar elastic constants and the model’s intrinsic characteristic length;
• Applying the framework to realistic bone tissue microstructures (osteons) to demonstrate that the proposed formulation captures the experimentally observed stiffness increase at small scales, where classical mechanics underestimates the structural response.

This line is supported by published results on micropolar computational homogenization (Thiesen et al., International Journal of Solids and Structures, 2026) and ongoing developments in the FEniCSx-based FE² platform.

Image: diagram of compact bone microstructure showing osteons, Haversian canals, periosteum, perforating (Volkmann’s) canals, and trabecular bone — the hierarchical architecture whose size-dependent mechanical response motivates the Micropolar macroscale description. From: OpenStax Anatomy and Physiology 2e, Fig. 6.12 (openstax.org), CC BY 4.0.