Modeling Wave Propagation in a Finite-Size Metamaterial through a Reduced Relaxed Micromorphic model: a study on Unit Cell “Cut”-related Boundary Effects
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Over the past few decades, there has been significant expansion in the research focused on identifying and characterizing metamaterials with exotic properties. Metamaterials are usually constructed by a periodic repetition of a unit cell in space and therefore they possess an effective microstructure. It is then easy to understand that modeling these materials is challenging due to the heavy computational cost required to simulate large domains of finite size. To overcome this limitation, we propose the use of an equivalent enriched model, specifically a reduced relaxed micromorphic model [1,2], which is based on a variational formulation that automatically guarantees the model's well-posedness and provides appropriate boundary conditions, crucial for properly modeling finite size samples. We conduct simulations to test the ability of this model to mimic the response of a finite-size microstructured material across a wide range of frequencies. We investigate the effect of choosing a different but equivalent unit cell “cut” on the metamaterial's behavior. In this way we stress that an enriched model should be able to capture boundary effects related to the unit cell “cut” if the metamaterial under investigation is of finite size, but also that microstructured simulations with different unit cell “cuts” should be run before any real application, as dispersion curves coming from standard Bloch-Floquet analysis on the unit cell concern only infinite metamaterials and therefore they convey no information on boundary effects that take place in specimens of finite size [3]. The metamaterial we study possesses a bandgap in the acoustic range, making it suitable for soundproof applications. REFERENCES [1] F. Demore, G. Rizzi, M. Collet, P. Neff, and A. Madeo. Unfolding engineering metamaterials design: Relaxed micromorphic modeling of large-scale acoustic meta-structures. Journal of the Mechanics and Physics of Solids, 168:104995, 2022. [2] G. Rizzi, P. Neff, and A. Madeo. Metamaterial shields for inner protection and outer tuning through a relaxed micromorphic approach. Philosophical Transactions of the Royal Society A, 380(2231):20210400, 2022. [3] P. Demetriou, G. Rizzi, and A. Madeo. Reduced relaxed micromorphic modeling of harmonically loaded metamaterial plates: investigating boundary effects in finite-size structures. Archive of Applied Mechanics, 2023.