A boundary-inertia-aware relaxed micromorphic model for finite-sized metamaterials under loss of scale separation

Homogenized continuum models are widely used to describe wave propagation and band-gap behavior in mechanical metamaterials without explicitly resolving their microstructure. Their validity, however, typically relies on the classical separation of scales assumption, according to which the wavelength of the propagating disturbance is much larger than the characteristic size of the unit cell. In finite-sized metamaterial samples and at higher frequencies, this assumption progressively breaks down, and the dynamic response becomes strongly influenced by the way the microstructure is truncated at the external boundaries. In this work we introduce a concept in the homogenized description of mechanical metamaterials: the inertial contribution of macroscopic interfaces. We show that different truncations of the same lattice generate boundaries with distinct mass distributions, which lead to measurable differences in the dynamic response of finite-sized specimens. To capture this complex mechanism in a homogenized framework, we extend the relaxed micromorphic model by introducing a kinetic surface energy defined on the boundary of the considered body. This generates an additional inertial term in the boundary conditions that can be seen as the homogenized counterpart of the interface inertia produced by the truncation of the microstructure. As a result, the homogenized model can now distinguish between finite-sized specimens that share identical bulk properties but differ only in the configuration of their interfaces. The proposed formulation preserves the variational structure of the relaxed micromorphic model while enabling the continuum to reproduce boundary-dependent responses observed in fully resolved simulations, particularly in frequency regimes where scale separation no longer holds. This result opens the door to predictive homogenized models, enabling the upscaling towards larger-scale structures assembled from finite metamaterial building blocks, a step that would be computationally infeasible using fully resolved microstructured models.