Dynamics and Homogenization of Periodic Metamaterials (DHPM)

ABSTRACT

Periodic metamaterials can be realized in the form of lattices of beams deformable both axially and flexurally. These architected materials can exhibit uncommon mechanical properties for both static and dynamic applications, such as auxetic behaviour and elastic wave manipulation and control. The course aims to provide the fundamentals of the mechanics of periodic metamaterials. The student will learn (i) the most effective methods to analyze their static and dynamic behaviors; and (ii) the static and dynamic homogenization techniques to describe their macroscopic behavior.

COURSE CONTENT

• Dynamics of periodic beam lattices: unit cell, direct and reciprocal bases, governing equations, Bloch theorem, quasi-periodic conditions, dispersion relation, dynamic anisotropy, dispersion surfaces and waveforms
• Homogenization of periodic beam lattices: static homogenization, dynamic homogenization, lattice acoustic tensor
• Prestressed beam lattices: macro and micro instabilities, strain localization and shear bands in the lattice and the equivalent continuum, structured interfaces for wave channeling and manipulation

TEACHING ACTIVITIES

The teaching activity includes theoretical lessons, applications and exercises. The exam consists of a written test with multiple-choice questions on the topics of the course.

FURTHER READINGS

1. Estrin, Y., Bréchet, Y., Dunlop, J., Fratzl, P. (Eds.), 2019. Architectured Materials in Nature and Engineering: Archimats, Springer Series in Materials Science. Springer International Publishing, Cham. https://doi.org/10.1007/978-3-030-11942-3
2. Bigoni, D., 2012. Nonlinear Solid Mechanics: Bifurcation Theory and Material Instability, 1st ed. Cambridge University Press. https://doi.org/10.1017/CBO9781139178938
3. Fleck, N.A., Deshpande, V.S., Ashby, M.F., 2010. Micro-architectured materials: past, present and future. Proc. R. Soc. A. 466, 2495–2516. https://doi.org/10.1098/rspa.2010.0215
4. Bordiga, G., Cabras, L., Bigoni, D., Piccolroaz, A. (2019) Free and forced wave propagation in a Rayleigh-beam grid: Flat bands, Dirac cones, and vibration localization vs isotropization. International Journal of Solids and Structures, 161, 64-81. https://doi.org/10.1016/j.ijsolstr.2018.11.007
5. Bordiga, G., Cabras, L., Piccolroaz, A., Bigoni, D. (2019) Prestress tuning of negative refraction and wave channeling from flexural sources. Applied Physics Letters, 114, 041901. https://doi.org/10.1063/1.5084258
6. Bordiga, G., Cabras, L., Piccolroaz, A., Bigoni, D. (2021) Dynamics of prestressed elastic lattices: Homogenization, instabilities, and strain localization. Journal of the Mechanics and Physics of Solids, 146, 104198. https://doi.org/10.1016/j.jmps.2020.104198
7. Bordiga, G., Piccolroaz, A., Bigoni, D. (2022) A way to hypo-elastic artificial materials without a strain potential and displaying flutter instability. Journal of the Mechanics and Physics of Solids, 158, 104665. https://doi.org/10.1016/j.jmps.2021.104665
8. Bordiga, G., Bigoni, D., Piccolroaz, A. (2022) Tensile material instabilities in elastic beam lattices lead to a closed stability domain. Philosophical Transactions of the Royal Society A, 380, 20210388. https://doi.org/10.1098/rsta.2021.0388
9. Cabras, L., Bigoni, D., Piccolroaz, A. (2024) Dynamics of elastic lattices with sliding constraints. Proceedings of the Royal Society A, 480, 20230579. https://doi.org/10.1098/rspa.2023.0579
10. Viviani, L., Bigoni, D., Piccolroaz, A. (2024) Homogenization of elastic grids containing rigid elements. Mechanics of Materials, 191, 104933. https://doi.org/10.1016/j.mechmat.2024.104933