Modeling boundary effects in finite-sized mechanical metamaterials via the reduced relaxed micromorphic model and the method of interface forces

In recent decades, metamaterials have attracted significant attention due to their exotic properties and promising applications. These properties usually arise from their periodic and engineered microstructures. We are particularly interested in mechanical metamaterials exhibiting band-gaps: frequency ranges where wave propagation is prohibited, enabling applications like vibration isolation. Band-gaps appear in the dispersion curves derived via Bloch-Floquet analysis, which assumes infinite periodicity. However, real metamaterials are finite, and their boundaries -dependent on the choice of unit cell ``cut”- introduce boundary effects absent in standard Bloch-Floquet analysis. To model large and/or complex metamaterials efficiently, enriched continuum theories are often used. In this thesis, we employ the reduced relaxed micromorphic model (RRMM), which effectively captures dispersion and band-gaps while reducing computational cost. Yet, like most homogenised models, it lacks information necessary to model boundary effects stemming from the choice of different unit cell ``cut”, when finite-sized specimens are constructed. To address this, we introduce the method of interface forces, enabling the RRMM to account for boundary effects in finite-sized metamaterials. We demonstrate its necessity and effectiveness for specific benchmark tests and propose an ansatz for the interface force expressions at Cauchy/RRM interfaces. Additionally, we distinguish between general boundary effects, edge effects, and boundary effects that propagate in the bulk and we explore the origin and invariance of specific edge effects’. Moreover we introduce edge tests and explore their potential for edge effect prediction. Finally, we discuss the implications of the choice of unit cell ``cut” on the transmissibility of metamaterials and potential applications.