Hussein Nassar

Non-reciprocal wave propagation in modulated elastic metamaterials

Time-reversal symmetry for elastic wave propagation breaks down in a resonant mass-in-mass lattice whose inner-stiffness is weakly modulated in space and in time in a wave-like fashion. Specifically, one-way wave transmission, conversion and amplification as well as unidirectional wave blocking are demonstrated analytically through an asymptotic analysis based on coupled mode theory and numerically thanks to a series of simulations in harmonic and transient regimes. High-amplitude modulations are then explored in the homogenization limit where a non-standard effective mass operator is recovered and shown to take negative values over unusually large frequency bands. These modulated metamaterials, which exhibit either non-reciprocal behaviours or non-standard effective mass operators, offer promise for applications in the field of elastic wave control in general and in one-way conversion/amplification in particular.

Fundamentals of 1D and 2D lattice-based mechanical topological insulators (Conference Presentation)

Spring-mass lattices constitute an accessible model for the understanding of various physical phenomena. Here, they are used to probe fundamental aspects of mechanical topological insulators. First, in gapped one-dimensional 2-periodic lattices, a simple interpretation of Zack’s phase and of the associated integer winding number is provided based on the stiffness coupling two consecutive masses. Nearest neighbor and non-nearest neighbor interactions are explored so as to generate more diverse winding numbers. Lattices with different winding numbers are shown to be topologically distinct. In that case, the difference in winding numbers is interpreted as a count of edge modes localized at the interface between the two topologically distinct lattices. The existence of these edge modes is verified through numerical modal analysis and through homogenization-type asymptotic analysis. The study is extended to two-dimensional systems. Although a visualization of the winding number, also known as a Chern number in this context, is harder, various aspects remain unchanged. Most importantly, an interface separating two topologically distinct gapped lattices will carry a number of edge modes. Last, robustness and immunity to back scattering of localized interface modes against defects is assessed for different systems.

Space-time modulations of phononic crystals (Conference Presentation)

When a set of resonators is attached to a master structure, a bandgap opens in the vicinity of the resonance frequency. Then, using piezoelectric circuitry for instance, the spring constant coupling the resonators to the structure can be tuned thus allowing to actively control the resonance frequency and subsequently the position of the bandgap. In this study, we investigate the consequences of dynamically changing the resonance frequency of a resonant metamaterial on its dispersion diagram. In particular, the resonance frequency is modulated periodically in space and in time at a uniform speed in a wave-like fashion and at low frequencies of the same order of magnitude of the resonance frequency itself. A two-scale asymptotic homogenization approach shows that the modulated resonant metamaterial effectively behave as another resonant metamaterial with a different set of resonance frequencies. Changing the modulation speed reveals interesting effective dynamics whereby the bandgaps of the original metamaterial split, move, condense and merge to form new band structures. The results are illustrated and exemplified through the analytical study of a onedimensional elastic medium coupled with a continuous distribution of spring-mass oscillators resonating at low frequencies. The conclusions point towards possible applications in breaking time-reversal symmetry, active wave control and filtering.