The law of reciprocity in acoustics and elastodynamics codifies a relation of symmetry between action and reaction in fluids and solids. In its simplest form, it states that the frequency-response functions between any two material points remain the same after swapping source and receiver, regardless of the presence of inhomogeneities and losses. As such, reciprocity has enabled numerous applications that make use of acoustic and elastic wave propagation. A recent change in paradigm has prompted us to see reciprocity under a new light: as an obstruction to the realization of wave-bearing media in which the source and receiver are not interchangeable. Such materials may enable the creation of devices such as acoustic one-way mirrors, isolators and topological insulators. Here, we review how reciprocity breaks down in materials with momentum bias, structured space-dependent and time-dependent constitutive properties, and constitutive nonlinearity, and report on recent advances in the modelling and fabrication of these materials, as well as on experiments demonstrating nonreciprocal acoustic and elastic wave propagation therein. The success of these efforts holds promise to enable robust, unidirectional acoustic and elastic wave-steering capabilities that exceed what is currently possible in conventional materials, metamaterials or phononic crystals. Nonreciprocal acoustic and elastic wave propagation may enable the creation of devices such as acoustic one-way mirrors, isolators and topological insulators. This Review presents advances in the creation of materials that break reciprocity and realize robust, unidirectional acoustic and elastic wave steering.

/pdf/nonreciprocity-in-acoustic-and-elastic-materials-4hinbae3h3.pdf

Nonreciprocity in acoustic and elastic materials

We investigate non-Hermitian elastic lattices characterized by non-local feedback control interactions. In one-dimensional lattices, we show that the proportional control interactions produce complex dispersion relations characterized by gain and loss in opposite propagation directions. Depending on the non-local nature of the control interactions, the resulting non-reciprocity occurs in multiple frequency bands characterized by opposite non-reciprocal behavior. The dispersion topology is also investigated with focus on winding numbers and non-Hermitian skin effect, which manifests itself through bulk modes localized at the boundaries of finite lattices. In two-dimensional lattices, non-reciprocity is associated with directional dependent wave amplification. Moreover, the non-Hermitian skin effect manifests as modes localized at the boundaries of finite lattice strips, whose combined effect in two directions leads to the presence of bulk modes localized at the corners of finite two-dimensional lattices. Our results describe fundamental properties of non-Hermitian elastic lattices, and open new possibilities for the design of metamaterials with novel functionalities related to selective wave filtering, amplification and localization. The results also suggest that feedback interactions may be a useful strategy to investigate topological phases of non-Hermitian systems.

Dynamics and topology of non-Hermitian elastic lattices with non-local feedback control interactions

In this paper, we report a new type of bandgap phenomenon called “amplitude-induced bandgap”, which is induced by amplitude in nonlinear metamaterials. Generally, bandgap phenomena in elastic metamaterials or phononic crystals have been achieved by the artificial unit cells which prevent wave propagation with Bragg scattering or internal resonance. However, the amplitude-induced bandgap originates from the amplitude and nonlinearity unlike previously discovered bandgap phenomena. Accordingly, the amplitude-induced bandgap is shown to exhibit various peculiar characteristics that have not been previously observed. Here, an analytic investigation about the amplitude-induced bandgap is conducted based on the perturbation approach in a nonlinear softening monoatomic chain. Also, various peculiar characteristics of the amplitude-induced bandgap are studied in detail. Numerical simulations using finite element analysis are carried out to validate the amplitude induced bandgap. In addition, the transmission characteristics of nonlinear bandgap are also analytically studied based on the recursive method and numerically verified through the simulations.

Amplitude-induced bandgap: New type of bandgap for nonlinear elastic metamaterials

Vibration properties and optimized design of a nonlinear acoustic metamaterial beam

Wave propagation through nonlinear acoustic metamaterials has generated numerous scientific interests for their enormous potential in practical applications these years. This study focuses on the effects of nonlinearity on the band properties of diatomic mass-in-mass chain with active control. By applying the Lindestedt–Poincare (L–P) perturbation method, analytical dispersion relations of the linear and nonlinear diatomic mass-in-mass system have been established and investigated by numerical simulation. Different from the monatomic mass-in-mass chain, this two mass-in-mass units forming a unit cell of the periodic structure results in four branches of the dispersion relation. The effects of nonlinearity on the band gaps of the system have been exhaustively illustrated. By only tuning the nonlinear constitutive relation parameter of the spring, the fourth branch and the third gap are found to be more sensitive compared to the other branches and gaps. It is concluded that closing and re-opening of the band-folding-induced gap in this nonlinear system is still possible. Here, a piezoelectric spring model is applied to the diatomic mass-in-mass to make the system available for wider applications. With the negative proportional control, a new stop band is generated which can be also captured in the monatomic nonlinear system. The new results here will help better analyze the band gap properties in nonlinear mechanical metamaterials and emphasize the great potentials of the topological analysis of such a nonlinear local resonance system that induces band-folding-induced band gaps.

Active control for acoustic wave propagation in nonlinear diatomic acoustic metamaterials

In linear time-invariant systems acoustic reciprocity holds by the Onsager-Casimir principle of microscopic reversibility, and it can be broken only by odd external biases, nonlinearities, or time-dependent properties. Recently it was shown that one-dimensional lattices composed of a finite number of identical nonlinear cells with internal scale hierarchy and asymmetry exhibit nonreciprocity both locally and globally. Considering a single cell composed of a large scale nonlinearly coupled to a small scale, local dynamic nonreciprocity corresponds to vibration energy transfer from the large to the small scale, but absence of energy transfer (and localization) from the small to the large scale. This has been recently proven both theoretically and experimentally. Then, considering the entire lattice, global acoustic nonreciprocity has been recently proven theoretically, corresponding to preferential energy transfer within the lattice under transient excitation applied at one of its boundaries, and absence of similar energy transfer (and localization) when the excitation is applied at its other boundary. This work provides experimental validation of the global acoustic nonreciprocity with a one-dimensional asymmetric lattice composed of three cells, with each cell incorporating nonlinearly coupled large and small scales. Due to the intentional asymmetry of the lattice, low impulsive excitations applied to one of its boundaries result in wave transmission through the lattice, whereas when the same excitations are applied to the other end, they lead in energy localization at the boundary and absence of wave transmission. This global nonreciprocity depends critically on energy (i.e., the intensity of the applied impulses), and reduced-order models recover the nonreciprocal acoustics and clarify the nonlinear mechanism generating nonreciprocity in this system.

/pdf/acoustic-nonreciprocity-in-a-lattice-incorporating-14m93itfam.pdf

Acoustic nonreciprocity in a lattice incorporating nonlinearity, asymmetry, and internal scale hierarchy: Experimental study.

In dynamical and acoustical systems, breaking reciprocity is achievable by employing external biases, spatial temporal variations of material properties, or nonlinearities. In this present work, we propose, theoretically analyze, and experimentally demonstrate the first passive broad-band mechanical diode, by adding an asymmetric local nonlinear interface to an otherwise linear waveguide. It is shown that for a broad range of input energies and frequencies, three different types of nonreciprocal behavior are obtained, (i) different transmitted energies depending on the direction of wave propagation, (ii) transmission of acoustic waves allowed in one direction but not in the reverse direction, and (iii) arrest of propagating waves at the nonlinear interface only in one (preferred) direction. A unique feature of the proposed acoustic device compared with previous designs is its capability to achieve passive nonreciprocity without altering or distorting the frequency content of the sending signal. The proposed system can pave the way for designing passive acoustic or thermal diodes, and adaptable nonreciprocal wave transmission devices of enhanced robustness that are tunable with, and passively adaptive to energy.

https://link.aps.org/accepted/10.1103/PhysRevB.99.214305

Broadband passive nonlinear acoustic diode

/pdf/higher-order-dispersion-stability-and-waveform-invariance-in-2nhphcd2c4.pdf

Higher-Order Dispersion, Stability, and Waveform Invariance in Nonlinear Monoatomic and Diatomic Systems

This paper presents a multiple-scales analysis approach capable of capturing internally resonant wave interactions in weakly nonlinear lattices and metamaterials. Example systems considered include a diatomic chain and a locally resonant metamaterial-type lattice. At a number of regions in the band structure, both the frequency and wave number of one nonlinear plane wave may relate to another in a near-commensurate manner (such as in a 2:1 or 3:1 ratio) resulting in an internal resonance mechanism. As shown herein, nonlinear interactions in the lattice couple these waves and enable energy exchange. Near such internal resonances, previously derived higher-order dispersion corrections for single plane wave propagation may break down, leading to singularities in the predicted nonlinear dispersion relationships. Using the presented multiple-scales approach and the two example systems, this paper examines internal resonance occurring (i) within the same branch and (ii) between different branches of the band structure, resolving the aforementioned singularity issue while capturing energy exchange. The multiple-scales evolution equations, together with a local stability analysis, uncover multiple stable fixed points associated with periodic energy exchange between internally resonant propagating modes. Response results generated using direct numerical simulation verify the perturbation-based predictions for amplitude-dependent dispersion corrections and slow-scale energy exchange; importantly, these comparisons verify the exchange frequency predicted by the multiple-scales approach.

Matthew D. Fronk

Papers

Acoustic nonreciprocity in a lattice incorporating nonlinearity, asymmetry, and internal scale hierarchy: Experimental study.

Broadband passive nonlinear acoustic diode

Higher-Order Dispersion, Stability, and Waveform Invariance in Nonlinear Monoatomic and Diatomic Systems

Internally resonant wave energy exchange in weakly nonlinear lattices and metamaterials

Direction-dependent invariant waveforms and stability in two-dimensional, weakly nonlinear lattices