Multiscale modelling of non-centrosymmetric nested lattice-like materials via high-frequency continualization

This study proposes an enhanced continualization framework through which non-centrosymmetric generalized micropolar continua are systematically identified from the underlying discrete microstructure, enabling an accurate description of periodic architected materials beyond standard elasticity. Within this setting, a hierarchical non-centrosymmetric lattice is mapped onto integral-type micropolar continuum models whose dispersion-band structure coincides with that of the underlying discrete Lagrangian system, and which are asymptotically approximated by non-local, gradient-type generalized micropolar continua of increasing order endowed with non-local inertia terms. Focusing in particular on first-order micropolar continua (i.e., Cosserat-type solids), the overall constitutive tensors are identified through a variational projection-based procedure that maps discrete kinematic descriptors onto continuum kinematic fields, ensuring full energetic equivalence between the discrete and continuum counterparts. The framework is applied to two prototypical non-centrosymmetric lattice-like microstructures: a square lattice with non-uniform ligament cross-sections, and a nested block lattice material with architected interfaces. In both cases, the reference models are formulated within a discrete Lagrangian setting and exhibit non-vanishing stress-curvature (or equivalently, couple-stress-strain) coupling terms as a direct consequence of non-centrosymmetry. Bloch-Floquet dispersion analyses are employed to assess the ability of the derived micropolar generalized continua to reproduce the spectral properties of the original discrete systems. The results highlight the role of non-centrosymmetry and shear-bending interactions in generating anisotropic and non-reciprocal wave propagation phenomena, and demonstrate that the proposed framework accurately captures the dispersion behaviour of non-centrosymmetric lattices. More generally, the approach provides a systematic pathway for the construction of higher-order, hierarchical, and non-local generalized continuum models with controlled spectral fidelity.