Session: 01-02-03: Phononic Crystals and Metamaterials

Paper Number: 150635

150635 - Quantification of Brillouin Zones in Non-Reciprocal Willis Monatomic Lattices 

Recent research in vibration and wave propagation has been largely motivated by the exploration of new wave phenomena, with elastic periodic lattices attracting significant interest due to their unique dynamics. The periodic nature of these lattices allows for analysis using the Bloch theorem, leading to a dispersion relation that relates the wavenumber and excitation frequency. This relationship is often depicted graphically as a dispersion diagram (or band structure), which repeats periodically along the wavenumber axis. Consequently, a segment of this relation, known as the Brillouin zone, often suffices to capture the entire dispersion behavior.

Elastodynamic reciprocity mandates that an elastic wave's behavior remains unchanged if the source and receiver locations are swapped. Such a behavior is captured by the symmetry of the Brillouin zone about the zero wavenumber. Introducing momentum bias within elastic media breaks such reciprocity (and the symmetry of the Brillouin zone), which proven vital for realizing new engineering applications (e.g., one-way wave guiding). However, non-reciprocity raises concerns about the adequacy of the traditional Brillouin-zone definitions to capture such a phenomenon. As a consequence, it becomes necessary to examine wavenumbers beyond the conventional Brillouin zones, as non-reciprocity renders these traditional definitions obsolete, particularly in the case of two-dimensional media.

This presentation aims to introduce a theory on quantifying Brillouin zones for non-reciprocal Willis monatomic lattices (WMLs), synthesized from the wave equation governing a moving elastic rod. The proposed theory suggests that the Brillouin-zone’s width remains constant in non-reciprocal WMLs and its boundaries shift by an analytically quantifiable amount. Numerical simulations are performed by means of spatiotemporal fast-Fourier transform of the time response of a WML with a finite number of unit cells to validate the proposed theory, substantiating that the forward-going and backward-going wavenumber ranges adjust to maintain a constant Brillouin zone width of 2π, regardless of the shift.

An extension to a two-dimensional, square-lattice counterpart of WMLs is also established, and their dynamics are derived from a modified form of the two-dimensional wave equation with a momentum bias applied in the two wave-propagation directions. Results show similar shifting of the Brillouin zone, yet its boundaries are deformed whilst maintaining a constant area (identical to its reciprocal counterpart) under all modulation strengths. This newly defined Brillouin zone is also verified numerically via the spatiotemporal fast-Fourier transform of the time response of a finite square WML, and the results are projected on directivity plots to allow for comparison with analytically obtained ones.

The mathematical framework developed here serves as a foundation for further investigations into Brillouin-zone quantification in various types of modulated media exhibiting wave non-reciprocity.

Presenting Author: Hasan Al Ba'ba'a Union College

Presenting Author Biography: Dr. Al Ba’ba’a received his Ph.D. from the State University of New York at Buffalo (UB) in 2019, M.S. from University of Wisconsin – Milwaukee (UWM) in 2015, and B.S. from Hashemite University in 2010, all specializing in mechanical engineering. Following his Ph.D. graduation, Dr. Al Ba’ba’a held positions at multiple universities, such as part-time lecturer at Eastern Michigan University, Research Fellow in the University of Michigan, and Postdoctoral Associate at the University of Southern California and UB. Being a recipient of a competitive teaching fellowship during his Ph.D. studies, Dr. Al Ba’ba’a became very interested in leading an academic career that balances teaching and scholarship, which eventually led him to joining Union College in the Fall of 2022, where he is currently working as a visiting assistant professor of mechanical engineering.

Areas of research that Dr. Al Ba’ba’a is interested in are structural dynamics, wave propagation, and control theory, all with the aim of controlling and reducing mechanical vibrations in dynamical systems. Particularly, he investigates various classes of optimally engineered materials, such as elastic metamaterials and phononic crystals, to establish mathematical models for better understanding of their dynamical characteristics and consequently invent new classes of them. Dr. Al Ba’ba’a received multiple awards during his academic career, including the Chancellor’s Graduate Student Award by the College of Engineering and Applied Sciences at UWM, and the Dean’s Graduate Achievement Award at UB. In addition, his novel phononic pendula design for reducing vibrations in overhead cranes awarded second place in the Silent Hoist and Crane Co. Materials Handling Prize.

Authors:

Hasan Al Ba'ba'a Union College