Modulation instability in nonlinear flexible mechanical metamaterials.

In this paper, we study modulation instabilities (MI) in a one-dimensional chain configuration of a flexible mechanical metamaterial (flexMM). Using the lumped element approach, flexMMs can be modeled by a coupled system of discrete equations for the longitudinal displacements and rotations of the rigid mass units. In the long wavelength regime, and applying the multiple-scales method we derive an effective nonlinear Schrödinger equation for slowly varying envelope rotational waves. We are then able to establish a map of the occurrence of MI to the parameters of the metamaterials and the wave numbers. We also highlight the key role of the rotation-displacement coupling between the two degrees of freedom in the manifestation of MI. All analytical findings are confirmed by numerical simulations of the full discrete and nonlinear lump problem. These results provide interesting design guidelines for nonlinear metamaterials offering either stability to high amplitude waves, or conversely being good candidates to observe instabilities.

Laser ultrasonics in a multilayer structure: Semi-analytic model and simulated examples.

Laser-generated elastic waves have been the subject of numerous experimental, theoretical, and numerical studies to describe the opto-acoustic generation process, involving electromagnetic, thermal, and elastic fields and their couplings in matter. Among the numerical methods for solving this multiphysical problem, the semi-analytic approach is one of the most relevant for obtaining fast and accurate results, when analytic solutions exist. In this paper, a multilayer model is proposed to successively solve electromagnetic, thermal, and elastodynamic problems. The optical penetration of the laser line source, as well as thermal conduction and convection, are accounted for. Optical, thermal, and mechanical coupling conditions are considered between the upper and lower media of the multilayer. The simulation of laser-generated ultrasounds in multilayer structures is of interest for the development of nondestructive evaluation methods of complex structures, such as bonded assemblies in aeronautics [as discussed in Hodé et al., J. Acoust. Soc. Am. 150, 2076 (2021)]. The developed Python code is provided for free at https://doi.org/10.5281/zenodo.4301720.

Laser ultrasonics in a multilayer structure: Plane wave synthesis and inverse problem for nondestructive evaluation of adhesive bondings.

A laser ultrasonic method is proposed for the nondestructive evaluation of bonded assemblies based on the analysis of elastic plane waves reflected from the bonding interface. Plane waves are numerically synthesized from experimentally detected cylindrical waves. Several angles of incidence with respect to the bonding interface are achieved by varying the delay in the synthesis step. An inverse problem using these plane waves is then solved to identify the normal and transverse interfacial stiffnesses that model the mechanical coupling between two bonded media. The semi-analytic model developed and detailed in Hodé et al. [J. Acoust. Soc. Am. 150, 2065 (2021)] is used to create the database that contains simulated laser-generated ultrasounds required to solve the inverse problem. The developed method is first validated with semi-analytic simulated input data where Gaussian noise has been added. Next, the method is applied using signals acquired on an aluminum alloy plate and on assemblies (with and without adhesion defects) made of two aluminum alloy plates bonded by an aeronautical structural epoxy adhesive film. Differences between the identified values of interfacial stiffnesses distinguish the three samples and obtain quantitative values to characterize the adhesive bonding.

Elastic Vector Solitons in Soft Architected Materials.

We demonstrate experimentally, numerically, and analytically that soft architected materials can support the propagation of elastic vector solitons. More specifically, we focus on structures comprising a network of squares connected by thin and highly deformable ligaments and investigate the propagation of planar nonlinear elastic waves. We find that for sufficiently large amplitudes two components-one translational and one rotational-are coupled together and copropagate without dispersion. Our results not only show that soft architected materials offer a new and rich platform to study the propagation of nonlinear waves, but also open avenues for the design of a new generation of smart systems that take advantage of nonlinearities to control and manipulate the propagation of large amplitude vibrations.

Surface waves in granular phononic crystals.

The existence of surface elastic waves at a mechanically free surface of granular phononic crystals is studied. The granular phononic crystals are made of spherical particles distributed periodically on a simple cubic lattice. It is assumed that the particles are interacting by means of normal, shear, and bending contact rigidities. First, Rayleigh-type surface acoustic waves, where the displacement of the particles takes place in the sagittal plane while the particles possess one rotational and two translational degrees of freedom, are analyzed. Second, shear-horizontal-type waves, where the displacement of the particles is normal to the sagittal plane while the particles possess one translational and two rotational degrees of freedom are studied. The existence of zero-group-velocity surface acoustic waves of Rayleigh type is theoretically predicted and interpreted. A comparison with surface waves predicted by the reduced Cosserat theory is performed, and some limitations of the latter are established.

Perfect and broadband acoustic absorption by critically coupled sub-wavelength resonators.

Perfect absorption is an interdisciplinary topic with a large number of applications, the challenge of which consists of broadening its inherently narrow frequency-band performance. We experimentally and analytically report perfect and broadband absorption for audible sound, by the mechanism of critical coupling, with a sub-wavelength multi-resonant scatterer (SMRS) made of a plate-resonator/closed waveguide structure. In order to introduce the role of the key parameters, we first present the case of a single resonant scatterer (SRS) made of a Helmholtz resonator/closed waveguide structure. In both cases the controlled balance between the energy leakage of the several resonances and the inherent losses of the system leads to perfect absorption peaks. In the case of the SMRS we show that systems with large inherent losses can be critically coupled using resonances with large leakage. In particular, we show that in the SMRS system, with a thickness of λ/12 and diameter of λ/7, several perfect absorption peaks overlap to produce absorption bigger than 93% for frequencies that extend over a factor of 2 in audible frequencies. The reported concepts and methodology provide guidelines for the design of broadband perfect absorbers which could contribute to solve the major issue of noise reduction.

Nonlinear Hysteretic Torsional Waves.

We theoretically study and experimentally report the propagation of nonlinear hysteretic torsional pulses in a vertical granular chain made of cm-scale, self-hanged magnetic beads. As predicted by contact mechanics, the torsional coupling between two beads is found to be nonlinear hysteretic. This results in a nonlinear pulse distortion essentially different from the distortion predicted by classical nonlinearities and in a complex dynamic response depending on the history of the wave particle angular velocity. Both are consistent with the predictions of purely hysteretic nonlinear elasticity and the Preisach-Mayergoyz hysteresis model, providing the opportunity to study the phenomenon of nonlinear dynamic hysteresis in the absence of other types of material nonlinearities. The proposed configuration reveals a plethora of interesting phenomena including giant amplitude-dependent attenuation, short-term memory, as well as dispersive properties. Thus, it could find interesting applications in nonlinear wave control devices such as strong amplitude-dependent filters.

Ultrasonic evaluation of the morphological characteristics of metallic powders in the context of mechanical alloying.

An ultrasonic method is proposed to characterize the morphological (geometrical) aspects of powders through the elastic modulus dependence of their packing on the factors of polydispersity, coordination number and particle shape. During the mechanical alloying process, the variation in geometrical characteristics of powders provides critical information. Ultrasonic parameters are shown to be sensitive not only to the average contact number per bead (i.e. the coordination number) but also to characteristics of the bead size distribution, when given the same sample preparation and confining pressure. These parameters, in turn, are sensitive to both the granular medium polydispersity and particle shapes. A non-monotonic behavior of the ultrasonic velocity (and of the derived compressional wave modulus) is observed throughout the alloying process, which thus offers possibilities for powder structure monitoring.

Enhancing the absorption properties of acoustic porous plates by periodically embedding Helmholtz resonators.

This paper studies the acoustical properties of hard-backed porous layers with periodically embedded air filled Helmholtz resonators. It is demonstrated that some enhancements in the acoustic absorption coefficient can be achieved in the viscous and inertial regimes at wavelengths much larger than the layer thickness. This enhancement is attributed to the excitation of two specific modes: Helmholtz resonance in the viscous regime and a trapped mode in the inertial regime. The enhancement in the absorption that is attributed to the Helmholtz resonance can be further improved when a small amount of porous material is removed from the resonator necks. In this way the frequency range in which these porous materials exhibit high values of the absorption coefficient can be extended by using Helmholtz resonators with a range of carefully tuned neck lengths.

Using simple shape three-dimensional rigid inclusions to enhance porous layer absorption.

The absorption properties of a metaporous material made of non-resonant simple shape three-dimensional rigid inclusions (cube, cylinder, sphere, cone, and ring torus) embedded in a rigidly backed rigid-frame porous material are studied. A nearly total absorption can be obtained for a frequency lower than the quarter-wavelength resonance frequency due to the excitation of a trapped mode. To be correctly excited, this mode requires a filling fraction larger in three-dimensions than in two-dimensions for purely convex (cube, cylinder, sphere, and cone) shapes. At long wavelengths compared to the spatial period, a cube is found to be the best purely convex inclusion shape to embed in a cubic unit cell, while the embedment of a sphere or a cone cannot lead to an optimal absorption for some porous material properties and dimensions of the unit cell. At a fixed position of purely convex shape inclusion barycenter, the absorption coefficient only depends on the filling fraction and does not depend on the shape below the Bragg frequency arising from the interaction between the inclusion and its image with respect to the rigid backing. The influence of the incidence angle and of the material properties, namely, the flow resistivity is also shown. The results of the modeling are validated experimentally in the case of cubic and cylindrical inclusions.

Directional asymmetry of the nonlinear wave phenomena in a three-dimensional granular phononic crystal under gravity.

We report the experimental observation of the gravity-induced asymmetry for the nonlinear transformation of acoustic waves in a noncohesive granular phononic crystal. Because of the gravity, the contact precompression increases with depth inducing space variations of not only the linear and nonlinear elastic moduli but also of the acoustic wave dissipation. We show experimentally and explain theoretically that, in contrast to symmetric propagation of linear waves, the amplitude of the nonlinearly self-demodulated wave depends on whether the propagation of the waves is in the direction of the gravity or in the opposite direction. Among the observed nonlinear processes, we report frequency mixing of the two transverse-rotational modes belonging to the optical band of vibrations and propagating with negative phase velocities, which results in the excitation of a longitudinal wave belonging to the acoustic band of vibrations and propagating with positive phase velocity. We show that the measurements of the gravity-induced asymmetry in the nonlinear acoustic phenomena can be used to compare the in-depth distributions of the contact nonlinearity and of acoustic absorption.

Slow relaxation and aging phenomena at the nanoscale in granular materials.

Granular matter exhibits a rich variety of dynamic behaviors, for which the role of thermal fluctuations is usually ignored. Here we show that thermal fluctuations can pronouncedly affect contacting nanoscale asperities at grain interfaces and brightly manifest themselves through the influence on nonlinear-acoustic effects. The proposed mechanism based on intrinsic bistability of nanoscale contacts comprises a wealth of slow-dynamics regimes including slow relaxations and aging as universal properties of a wide class of systems with metastable states.

Localized transversal-rotational modes in linear chains of equal masses.

The propagation and localization of transversal-rotational waves in a two-dimensional granular chain of equal masses are analyzed in this study. The masses are infinitely long cylinders possessing one translational and one rotational degree of freedom. Two dispersive propagating modes are predicted in this granular crystal. By considering the semi-infinite chain with a boundary condition applied at its beginning, the analytical study demonstrates the existence of localized modes, each mode composed of two evanescent modes. Their existence, position (either in the gap between the propagating modes or in the gap above the upper propagating mode), and structure of spatial localization are analyzed as a function of the relative strength of the shear and bending interparticle interactions and for different boundary conditions. This demonstrates the existence of a localized mode in a semi-infinite monatomic chain when transversal-rotational waves are considered, while it is well known that these types of modes do not exist when longitudinal waves are considered.

Second-harmonic generation for dispersive elastic waves in a discrete granular chain.

The propagation of nonlinear compressional waves in a one-dimensional granular chain driven at one end by a harmonic excitation is studied. The chain is described by a Fermi-Pasta-Ulam (FPU) lattice model with quadratic nonlinearity (α-FPU model), valid for strong initial compression of the chain by an external static force. A successive approximations method is used to obtain the analytical expressions for the amplitudes of the static displacement field and of the fundamental and second harmonics propagating through the lattice. Both propagating and evanescent second harmonics are shown to influence the nonlinear propagation characteristics of the fundamental frequency. The propagating regime is characterized by a periodic energy transfer between first and second harmonics, resulting from dispersion, which disappears when the second harmonic becomes evanescent.

Sustainable sonic crystal made of resonating bamboo rods.

The acoustic transmission coefficient of a resonant sonic crystal made of hollow bamboo rods is studied experimentally and theoretically. The plane wave expansion and multiple scattering theory (MST) are used to predict the bandgap in transmission coefficient of a non-resonant sonic crystal composed of rods without holes. The predicted results are validated against experimental data for the acoustic transmission coefficient. It is shown that a sonic crystal made from a natural material with some irregularities can exhibit a clear transmission bandgap. Then, the hollow bamboo rods are drilled between each node to create an array of Helmholtz resonators. It is shown that the presence of Helmholtz resonators leads to an additional bandgap in the low-frequency part of the transmission coefficient. The MST is modified in order to account for the resonance effect of the holes in the drilled bamboo rods. This resonant multiple scattering theory is validated experimentally and could be further used for the description and optimization of more complex resonant sonic crystals.

Absorption of sound by porous layers with embedded periodic arrays of resonant inclusions.

The aim of this work is to design a layer of porous material with a high value of the absorption coefficient in a wide range of frequencies. It is shown that low frequency performance can be significantly improved by embedding periodically arranged resonant inclusions (slotted cylinders) into the porous matrix. The dissipation of the acoustic energy in a porous material due to viscous and thermal losses inside the pores is enhanced by the low frequency resonances of the inclusions and energy trapping between the inclusion and the rigid backing. A parametric study is performed in order to determine the influence of the geometry and the arrangement of the inclusions embedded in a porous layer on the absorption coefficient. The experiments confirm that low frequency absorption coefficient of a composite material is significantly higher than that of the porous layer without the inclusions.

Linear and nonlinear Biot waves in a noncohesive granular medium slab: transfer function, self-action, second harmonic generation.

Experimental results are reported on second harmonic generation and self-action in a noncohesive granular medium supporting wave energy propagation both in the solid frame and in the saturating fluid. The acoustic transfer function of the probed granular slab can be separated into two main frequency regions: a low frequency region where the wave propagation is controlled by the solid skeleton elastic properties, and a higher frequency region where the behavior is dominantly due to the air saturating the beads. Experimental results agree well with a recently developed nonlinear Biot wave model applied to granular media. The linear transfer function, second harmonic generation, and self-action effect are studied as a function of bead diameter, compaction step, excitation amplitude, and frequency. This parametric study allows one to isolate different propagation regimes involving a range of described and interpreted linear and nonlinear processes that are encountered in granular media experiments. In particular, a theoretical interpretation is proposed for the observed strong self-action effect.

Experimental evidence of rotational elastic waves in granular phononic crystals.

A generalized theory of elasticity, taking into account the rotational degrees of freedom of point bodies constituting a continuum, was proposed at the beginning of the twentieth century by the Cosserat brothers. We report the experimental observation of coupled rotational-translational modes in a noncohesive granular phononic crystal. While absent in the classical theory of elasticity, these elastic wave modes are predicted by the Cosserat theory. However the Cosserat theory fails to predict correctly the dispersion of the elastic modes in granular crystals even in the long-wavelength limit.

Elastic waves in noncohesive frictionless granular crystals.

An ordered structure of noncohesive spherical beads constitutes a phononic crystal. This type of media combines the properties of wave propagation in phononic crystals (dispersion due to the geometrical periodicity) with the properties of wave propagation in granular media (nonlinearities, rotational degree of freedom) and gives the opportunity to have interesting features as tunable frequency band gaps for example. In this work, the acoustic bulk modes of a hexagonal close packed (hcp) structure of beads, considered as rigid masses connected by springs, are theoretically evaluated and their associated resonance frequencies are compared to experimental results. When friction is neglected, the elastic interaction between the beads are reduced to a normal spring interaction given by the Hertz theory. According to this theory, the rigidity of the contact depends on its static loading. The theory predicts the existence of elastic transverse and longitudinal acoustical-type modes and transverse and longitudinal optical-type modes. The acoustic transfer function of a hcp crystal slab built with stainless steel beads is measured and its resonance frequencies are compared to the theoretical predictions. Despite some differences between theory and experiments, which could come for instance from the disordered character of the contact loads, the developed theory and the experimental results show relatively good agreement.

Dispersion of elastic waves in three-dimensional noncohesive granular phononic crystals: properties of rotational modes.

The dispersion relations of bulk modes propagating within a hexagonal close-packed structure of noncohesive monodisperse spherical elastic beads are derived. The contacts are modeled by two springs with stiffnesses given by the Hertz-Mindlin theory, one for normal interactions and one for transverse interactions. The existence of the transverse interaction requires to take into account the rotational degrees of freedom of the beads in the analysis. This leads to the prediction of translational modes and, due to the rotational degrees of freedom, of rotational modes and coupled rotational and translational modes. The study of the dispersion relations in a direction of high symmetry allows to identify the different modes and the influence of the rotational degrees of freedom on the bulk mode propagation. The evaluated dispersion relations provide guidelines for the experimental observation of rotational modes. Opportunities for controlling the dispersion laws of the modes by an external loading on the granular structure are discussed.