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Advanced Materials and Smart Structures

Relatore: Prof. Francesco Braghin

Tutor: Prof. Stefano Beretta

Università di Provenienza: Politecnico di Milano - Ingegneria Meccanica

Titolo della Tesi: Topological and Nonreciprocal Wave Propagation in Stiffness-modulated Materials

Topological and Nonreciprocal Wave Propagation in Stiffness-modulated Materials


This PhD work consists in the design and experimental analysis of wave propagation in stiffness-modulated structures with space and space-time varying elastic parameters, in the attempt to:

  1. achieve a mechanical analogous of anelectric diode, in which elastic waves canpropagate only along one direction (i.e.from left to right but not vice-versa);
  2. move the vibrational energy betweendistinct points of a structure.


The opportunity to manipulate elastic wave propagation through modulated materials has drawn growing interest within the research community over the last years. In this context, nonreciprocal transmission of elastic signals is one of the most promising wave phenomena that have been elusive from the experimental perspective. In this thesis, theoretical, numerical and experimental analysis are performed on suitably designed space-time modulated structures, achieving an elastic analogue of an electric diode [1-3]. The second part of this PHD work is focused on robust transport of elastic waves through topologically non-trivial structures, which rely on the exploitation of a synthetic dimension, emerging from a relevant higher-order parameter space and mapped in a physical domain (spatial or temporal). It is demonstrated that, thanks to the system topology, the wave propagation behavior is suitable for the implementation of mechanical delay lines, waveguiding and wave de-multiplexing [4].


Schematic of the experimental setup employed to reproduce a space-time varying stiffness. The modulation profile is achieved though piezo patches with attached negative capacitance shunts, that are able to locally decrease the effective Young’s modulus of the patch.(Fig.1, Fig.2).

Experimentally measured dispersion relation (Fig.4) and measured wavefield for a narrowband excitation spectrum centered at 10.5 kHz (Fig.1, Fig.5). The switching frequency fm sign determines the propagation direction of the modulation which, in turn determines the allowed wave propagation direction.

Schematic of a stiffness modulated structure periodic along the x-direction and smoothly modulated along a second spatial (Fig.6) or temporal (Fig.7) dimension.

Natural frequencies of a finite strip of beam upon varying the phase parameter Φ (Fig.8) and corresponding topological modes.

Resulting wavefield for a topological waveguide characterized by a left-to-right transition of the edge state in space (Fig.10) and in time (Fig.11).


This PhD work embodies theoretical and experimental achievements, which constitute novelty with respect to the state of the art in the field. Specifically, in the first part of this work a 1kHz-wide one-way wave propagation is achieved in a space-time modulated beam, spanning a frequency range within 8-11 kHz. In the second part, instead, a theoretical framework to study adiabatic edge-to-edge transitions is proposed and experimentally validated, opening up new opportunities for engineering applications involving vibrations, such as structural health monitoring, energy harvesting and for the implementation of communication circuits based on elastic wave propagation.


[1] E. Riva, J. Marconi, G. Cazzulani, F. Braghin, Generalized plane wave expansion method for non-reciprocal discretely modulated waveguides, Journal of Sound and Vibration, 449, 172-181,(2019).
[2] J. Marconi, E. Riva, M. Di Ronco, G. Cazzulani, F. Braghin, M. Ruzzene, Experimental Observation of Nonreciprocal Band Gaps in a Space-Time-Modulated Beam Using a Shunted Piezoelectric Array, Physical Review Applied, 13(3), 031001, (2020).
[3] E. Riva, M. Di Ronco, A. Elabd, G. Cazzulani, F. Braghin, Non-reciprocal wave propagation in discretely modulated spatiotemporal plates, Journal of Sound and Vibration, 115186, 2020.
[4] E. Riva, M. Rosa, M. Ruzzene, Edge states and topological pumping in stiffness-modulated elastic plates, Physical Review B, 101(9), 094307, 2020.