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Demiquel A, Achilleos V, Theocharis G, Tournat V. Modulation instability in nonlinear flexible mechanical metamaterials. Phys Rev E 2023; 107:054212. [PMID: 37329058 DOI: 10.1103/physreve.107.054212] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/16/2022] [Accepted: 04/20/2023] [Indexed: 06/18/2023]
Abstract
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.
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Affiliation(s)
- A Demiquel
- Laboratoire d'Acoustique de l'Université du Mans (LAUM), UMR 6613, Institut d'Acoustique - Graduate School (IA-GS), CNRS, Le Mans Université, France
| | - V Achilleos
- Laboratoire d'Acoustique de l'Université du Mans (LAUM), UMR 6613, Institut d'Acoustique - Graduate School (IA-GS), CNRS, Le Mans Université, France
| | - G Theocharis
- Laboratoire d'Acoustique de l'Université du Mans (LAUM), UMR 6613, Institut d'Acoustique - Graduate School (IA-GS), CNRS, Le Mans Université, France
| | - V Tournat
- Laboratoire d'Acoustique de l'Université du Mans (LAUM), UMR 6613, Institut d'Acoustique - Graduate School (IA-GS), CNRS, Le Mans Université, France
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2
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Ngapasare A, Theocharis G, Richoux O, Skokos C, Achilleos V. Wave-packet spreading in disordered soft architected structures. CHAOS (WOODBURY, N.Y.) 2022; 32:053116. [PMID: 35649992 DOI: 10.1063/5.0089055] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/22/2022] [Accepted: 04/20/2022] [Indexed: 06/15/2023]
Abstract
We study the dynamical and chaotic behavior of a disordered one-dimensional elastic mechanical lattice, which supports translational and rotational waves. The model used in this work is motivated by the recent experimental results of Deng et al. [Nat. Commun. 9, 1 (2018)]. This lattice is characterized by strong geometrical nonlinearities and the coupling of two degrees-of-freedom (DoFs) per site. Although the linear limit of the structure consists of a linear Fermi-Pasta-Ulam-Tsingou lattice and a linear Klein-Gordon (KG) lattice whose DoFs are uncoupled, by using single site initial excitations on the rotational DoF, we evoke the nonlinear coupling between the system's translational and rotational DoFs. Our results reveal that such coupling induces rich wave-packet spreading behavior in the presence of strong disorder. In the weakly nonlinear regime, we observe energy spreading only due to the coupling of the two DoFs (per site), which is in contrast to what is known for KG lattices with a single DoF per lattice site, where the spreading occurs due to chaoticity. Additionally, for strong nonlinearities, we show that initially localized wave-packets attain near ballistic behavior in contrast to other known models. We also reveal persistent chaos during energy spreading, although its strength decreases in time as quantified by the evolution of the system's finite-time maximum Lyapunov exponent. Our results show that flexible, disordered, and strongly nonlinear lattices are a viable platform to study energy transport in combination with multiple DoFs (per site), also present an alternative way to control energy spreading in heterogeneous media.
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Affiliation(s)
- A Ngapasare
- Nonlinear Dynamics and Chaos Group, Department of Mathematics and Applied Mathematics, University of Cape Town, Rondebosch 7701, South Africa
| | - G Theocharis
- Laboratoire d'Acoustique de l'Université du Mans (LAUM), UMR 6613, Institut d'Acoustique-Graduate School (IA-GS), CNRS, Le Mans Université, Le Mans, France
| | - O Richoux
- Laboratoire d'Acoustique de l'Université du Mans (LAUM), UMR 6613, Institut d'Acoustique-Graduate School (IA-GS), CNRS, Le Mans Université, Le Mans, France
| | - Ch Skokos
- Nonlinear Dynamics and Chaos Group, Department of Mathematics and Applied Mathematics, University of Cape Town, Rondebosch 7701, South Africa
| | - V Achilleos
- Laboratoire d'Acoustique de l'Université du Mans (LAUM), UMR 6613, Institut d'Acoustique-Graduate School (IA-GS), CNRS, Le Mans Université, Le Mans, France
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Miyazawa Y, Chong C, Kevrekidis PG, Yang J. Rogue and solitary waves in coupled phononic crystals. Phys Rev E 2022; 105:034202. [PMID: 35428101 DOI: 10.1103/physreve.105.034202] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/29/2021] [Accepted: 02/04/2022] [Indexed: 06/14/2023]
Abstract
In this work we present an analytical and numerical study of rogue and solitary waves in a coupled one-dimensional nonlinear lattice that involves both axial and rotational degrees of freedom. Using a multiple-scale analysis, we derive a system of coupled nonlinear Schrödinger-type equations in order to approximate solitary waves and rogue waves of the coupled lattice model. Numerical simulations are found to agree with the analytical approximations. We also consider generic initialization data in the form of a Gaussian profile and observe that they can result in the spontaneous formation of rogue-wave-like patterns in the lattice. The solitary and rogue waves in the lattice demonstrate both energy isolation and exchange between the axial and rotational degrees of freedom of the system. This suggests that the studied coupled lattice has the potential to be an efficient energy isolation, transfer, and focusing medium.
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Affiliation(s)
- Y Miyazawa
- Department of Aeronautics and Astronautics, University of Washington, Seattle, Washington 98195, USA
| | - C Chong
- Department of Mathematics, Bowdoin College, Brunswick, Maine 04011, USA
| | - P G Kevrekidis
- Department of Mathematics and Statistics, University of Massachusetts Amherst, Amherst, Massachusetts 01003-4515, USA
| | - J Yang
- Department of Aeronautics and Astronautics, University of Washington, Seattle, Washington 98195, USA
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Saitoh K, Mizuno H. Sound damping in soft particle packings: the interplay between configurational disorder and inelasticity. SOFT MATTER 2021; 17:4204-4212. [PMID: 33881038 DOI: 10.1039/d0sm02018d] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/12/2023]
Abstract
We numerically investigate sound damping in disordered two-dimensional soft particle packings. We simulate evolution of standing waves of particle displacements and analyze time correlation functions of particle velocities and power spectra. We control the strength of inelastic interactions between the particles in contact to show how the inelasticity affects anomalous sound characteristics of disordered systems: Increasing the strength of inelastic interactions, we find that (i) sound softening vanishes and (ii) attenuation coefficients exhibit a transition from the Rayleigh law to quadratic growth. We also report (iii) how the Ioffe-Regel limit frequencies depend on the strength of inelasticity as useful information for experiments and applications of the sound in disordered media. Our findings suggest that sound damping in soft particle packings is determined by the interplay between elastic heterogeneities and inelasticity.
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Affiliation(s)
- Kuniyasu Saitoh
- Department of Physics, Faculty of Science, Kyoto Sangyo University, Motoyama, Kamigamo, Kita-ku, Kyoto 603-8555, Japan.
| | - Hideyuki Mizuno
- Graduate School of Arts and Sciences, The University of Tokyo, Tokyo 153-8902, Japan
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Zhang Q, Umnova O, Venegas R. Nonlinear dynamics of coupled transverse-rotational waves in granular chains. Phys Rev E 2020; 100:062206. [PMID: 31962390 DOI: 10.1103/physreve.100.062206] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/09/2019] [Indexed: 11/07/2022]
Abstract
The nonlinear dynamics of coupled waves in one-dimensional granular chains with and without a substrate is theoretically studied accounting for quadratic nonlinearity. The multiple time scale method is used to derive the nonlinear dispersion relations for infinite granular chains and to obtain the wave solutions for semi-infinite systems. It is shown that the sum frequency and difference frequency components of the coupled transverse-rotational waves are generated due to their nonlinear interactions with the longitudinal wave. Nonlinear resonances are not present in the chain with no substrate where these frequency components have low amplitudes and exhibit beating oscillations. In the chain positioned on a substrate two types of nonlinear resonances are predicted. At resonance, the fundamental frequency wave amplitudes decrease and the generated frequency component amplitudes increase along the chain, accompanied by the oscillations due to the wave number asynchronism. The results confirm the possibility of a highly efficient energy transfer between the waves of different frequencies, which could find applications in the design of acoustic devices for energy transfer and energy rectification.
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Affiliation(s)
- Qicheng Zhang
- Acoustic Science and Technology Laboratory, Harbin Engineering University, Harbin 150001, China.,Key Laboratory of Marine Information Acquisition and Security (Harbin Engineering University), Ministry of Industry and Information Technology, Harbin 150001, China.,College of Underwater Acoustic Engineering, Harbin Engineering University, Harbin 150001, China
| | - Olga Umnova
- Acoustic Research Centre, University of Salford, Salford M5 4WT, England, United Kingdom
| | - Rodolfo Venegas
- University Austral of Chile, Institute of Acoustics, P.O. Box 567, Valdivia, Chile
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Molina MI, English LQ, Chang MH, Kevrekidis PG. Linear impurity modes in an electrical lattice: Theory and experiment. Phys Rev E 2020; 100:062114. [PMID: 31962490 DOI: 10.1103/physreve.100.062114] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/01/2019] [Indexed: 11/07/2022]
Abstract
We examine theoretically and experimentally the localized electrical modes existing in a bi-inductive electrical lattice containing a bulk or a surface capacitive impurity. By means of the formalism of lattice Green's functions, we are able to obtain closed-form expressions for the frequencies of the impurity (bound-state) eigenmodes and for their associated spatial profiles. This affords us a systematic understanding of how these mode properties change as a function of the system parameters. We test these analytical results against experimental measurements, in both the bulk and surface cases, and find very good agreement. Last, we turn to a series of quench experiments, where either a parameter of the lattice or the lattice geometry itself is rapidly switched between two values or configurations. In all cases, we are able to naturally explain the results of such quench experiments from the larger analytical picture that emerges as a result of the detailed characterization of the impurity-mode solution branches.
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Affiliation(s)
- M I Molina
- Departamento de Física, Facultad de Ciencias, Universidad de Chile, Santiago, Chile
| | - L Q English
- Department of Physics and Astronomy, Dickinson College, Carlisle, Pennsylvania 17013, USA
| | - Ming-Hua Chang
- Department of Physics and Astronomy, Dickinson College, Carlisle, Pennsylvania 17013, USA
| | - P G Kevrekidis
- Department of Mathematics and Statistics, University of Massachusetts, Amherst, Massachusetts 01003-4515, USA.,Mathematical Institute, University of Oxford, Oxford OX2 6GG, UK
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Saitoh K, Shrivastava RK, Luding S. Rotational sound in disordered granular materials. Phys Rev E 2019; 99:012906. [PMID: 30780228 DOI: 10.1103/physreve.99.012906] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/28/2018] [Indexed: 06/09/2023]
Abstract
We numerically study the evolution of elastic standing waves in disordered disk systems with a focus on the dispersion relations of rotational sound. As on a lattice, the rotational mode exhibits an optical-like dispersion relation in the high frequency regime, representing a shoulder in the vibrational density of states and fast oscillations of the autocorrelations of rotational velocities. If tangential stiffness between the disks is large enough, a lattice-based model perfectly describes the dispersion relation of the rotational mode. If it is comparable to or smaller than the normal stiffness, the model fails for short wavelengths. However, the dispersion relation then follows the model prediction for the transverse mode, implying that the fast oscillations of disks' rotations switch to acousticlike behavior. We evidence such a transition from rotational to transverse modes by analyzing their respective participation of different degrees of freedom to the eigenvectors.
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Affiliation(s)
- Kuniyasu Saitoh
- Research Alliance Center for Mathematical Sciences, Tohoku University, 2-1-1 Katahira, Aoba-ku, Sendai 980-8577, Japan
- WPI-Advanced Institute for Materials Research, Tohoku University, 2-1-1 Katahira, Aoba-ku, Sendai 980-8577, Japan
| | - Rohit K Shrivastava
- Faculty of Engineering Technology, MESA+, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands
| | - Stefan Luding
- Faculty of Engineering Technology, MESA+, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands
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Zheng LY, Pichard H, Tournat V, Theocharis G, Gusev V. Zero-frequency and slow elastic modes in phononic monolayer granular membranes. ULTRASONICS 2016; 69:201-214. [PMID: 26607105 DOI: 10.1016/j.ultras.2015.11.005] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/21/2015] [Accepted: 11/03/2015] [Indexed: 06/05/2023]
Abstract
We theoretically study the dispersion properties of elastic waves in hexagonal and honeycomb monolayer granular membranes with either out-of-plane or in-plane particle motion. The particles interact predominantly via normal and transverse contact rigidities. When rotational degrees of freedom are taken into account, the bending and torsional rigidities of the intergrain contacts can control some of the phononic modes. The existence of zero-frequency modes, zero-group-velocity modes and their transformation into slow propagating phononic modes due to weak bending and torsional intergrain interactions are investigated. We also study the formation and manipulation of Dirac cones and multiple degenerated modes. This could motivate variety of potential applications in elastic waves control by manipulating the contact rigidities in granular phononic crystals.
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Affiliation(s)
- Li-Yang Zheng
- LAUM, UMR-CNRS 6613, Université du Maine, Av. O. Messiaen, 72085 Le Mans, France.
| | - Hélène Pichard
- LAUM, UMR-CNRS 6613, Université du Maine, Av. O. Messiaen, 72085 Le Mans, France
| | - Vincent Tournat
- LAUM, UMR-CNRS 6613, Université du Maine, Av. O. Messiaen, 72085 Le Mans, France
| | - Georgios Theocharis
- LAUM, UMR-CNRS 6613, Université du Maine, Av. O. Messiaen, 72085 Le Mans, France
| | - Vitalyi Gusev
- LAUM, UMR-CNRS 6613, Université du Maine, Av. O. Messiaen, 72085 Le Mans, France.
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Pichard H, Duclos A, Groby JP, Tournat V, Zheng L, Gusev VE. Surface waves in granular phononic crystals. Phys Rev E 2016; 93:023008. [PMID: 26986406 DOI: 10.1103/physreve.93.023008] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/21/2015] [Indexed: 06/05/2023]
Abstract
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.
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Affiliation(s)
- H Pichard
- LAUM, UMR-CNRS 6613, Université du Maine, Av. O. Messiaen, 72085 Le Mans, France
| | - A Duclos
- LAUM, UMR-CNRS 6613, Université du Maine, Av. O. Messiaen, 72085 Le Mans, France
| | - J-P Groby
- LAUM, UMR-CNRS 6613, Université du Maine, Av. O. Messiaen, 72085 Le Mans, France
| | - V Tournat
- LAUM, UMR-CNRS 6613, Université du Maine, Av. O. Messiaen, 72085 Le Mans, France
| | - L Zheng
- LAUM, UMR-CNRS 6613, Université du Maine, Av. O. Messiaen, 72085 Le Mans, France
| | - V E Gusev
- LAUM, UMR-CNRS 6613, Université du Maine, Av. O. Messiaen, 72085 Le Mans, France
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Merkel A, Tournat V, Gusev V. Directional asymmetry of the nonlinear wave phenomena in a three-dimensional granular phononic crystal under gravity. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 90:023206. [PMID: 25215842 DOI: 10.1103/physreve.90.023206] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/25/2014] [Indexed: 06/03/2023]
Abstract
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.
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Affiliation(s)
- A Merkel
- LUNAM Université, Université du Maine, CNRS, LAUM UMR 6613, Avenue Olivier Messiaen, 72085 Le Mans, France
| | - V Tournat
- LUNAM Université, Université du Maine, CNRS, LAUM UMR 6613, Avenue Olivier Messiaen, 72085 Le Mans, France
| | - V Gusev
- LUNAM Université, Université du Maine, CNRS, LAUM UMR 6613, Avenue Olivier Messiaen, 72085 Le Mans, France
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