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Nassar H, Chen H, Norris AN, Haberman MR, Huang GL. Non-reciprocal wave propagation in modulated elastic metamaterials. Proc Math Phys Eng Sci 2017; 473:20170188. [PMID: 28690416 DOI: 10.1098/rspa.2017.0188] [Citation(s) in RCA: 68] [Impact Index Per Article: 9.7] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/16/2017] [Accepted: 06/01/2017] [Indexed: 11/12/2022] Open
Abstract
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.
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Affiliation(s)
- H Nassar
- Department of Mechanical and Aerospace Engineering, University of Missouri, Columbia, MO 65211, USA
| | - H Chen
- Department of Mechanical and Aerospace Engineering, University of Missouri, Columbia, MO 65211, USA
| | - A N Norris
- Department of Mechanical and Aerospace Engineering, Rutgers University, Piscataway, NJ 08854-8058, USA
| | - M R Haberman
- Department of Mechanical Engineering and Applied Research Laboratories, The University of Texas at Austin, Austin, TX 78712, USA
| | - G L Huang
- Department of Mechanical and Aerospace Engineering, University of Missouri, Columbia, MO 65211, USA
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Abstract
Transformation elasticity, by analogy with transformation acoustics and optics, converts material domains without altering wave properties, thereby enabling cloaking and related effects. By noting the similarity between transformation elasticity and the theory of incremental motion superimposed on finite pre-strain, it is shown that the constitutive parameters of transformation elasticity correspond to the density and moduli of small-on-large theory. The formal equivalence indicates that transformation elasticity can be achieved by selecting a particular finite (hyperelastic) strain energy function, which for isotropic elasticity is semilinear strain energy. The associated elastic transformation is restricted by the requirement of statically equilibrated pre-stress. This constraint can be cast as tr
F
= constant, where
F
is the deformation gradient, subject to symmetry constraints, and its consequences are explored both analytically and through numerical examples of cloaking of anti-plane and in-plane wave motion.
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Affiliation(s)
- A. N. Norris
- Mechanical and Aerospace Engineering, Rutgers University, Piscataway, NJ 08854-8058, USA
| | - W. J. Parnell
- School of Mathematics, Alan Turing Building, University of Manchester, Manchester M13 9PL, UK
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Norris AN, Shuvalov AL, Kutsenko AA. Analytical formulation of three-dimensional dynamic homogenization for periodic elastic systems. Proc Math Phys Eng Sci 2012. [DOI: 10.1098/rspa.2011.0698] [Citation(s) in RCA: 81] [Impact Index Per Article: 6.8] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022] Open
Abstract
Homogenization of the equations of motion for a three-dimensional periodic elastic system is considered. Expressions are obtained for the fully dynamic effective material parameters governing the spatially averaged fields by using the plane wave expansion method. The effective equations are of Willis form with coupling between momentum and stress and tensorial inertia. The formulation demonstrates that the Willis equations of elastodynamics are closed under homogenization. The effective material parameters are obtained for arbitrary frequency and wavenumber combinations, including but not restricted to Bloch wave branches for wave propagation in the periodic medium. Numerical examples for a one-dimensional system illustrate the frequency dependence of the parameters on Bloch wave branches and provide a comparison with an alternative dynamic effective medium theory, which also reduces to Willis form but with different effective moduli.
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Affiliation(s)
- A. N. Norris
- Mechanical and Aerospace Engineering, Rutgers University, Piscataway, NJ 08854-8058, USA
| | - A. L. Shuvalov
- Laboratoire de Mécanique Physique, Université de Bordeaux, CNRS, UMR 5469, Talence 33405, France
| | - A. A. Kutsenko
- Laboratoire de Mécanique Physique, Université de Bordeaux, CNRS, UMR 5469, Talence 33405, France
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Kutsenko AA, Shuvalov AL, Norris AN. Evaluation of the effective speed of sound in phononic crystals by the monodromy matrix method (L). J Acoust Soc Am 2011; 130:3553-3557. [PMID: 22225010 DOI: 10.1121/1.3654032] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/31/2023]
Abstract
A scheme for evaluating the effective quasistatic speed of sound c in two- and three-dimensional periodic materials is reported. The approach uses a monodromy-matrix operator to enable direct integration in one of the coordinates and exponentially fast convergence in others. As a result, the solution for c has a more closed form than previous formulas. It significantly improves the efficiency and accuracy of evaluating c for high-contrast composites as demonstrated by a two-dimensional scalar-wave example with extreme behavior.
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Affiliation(s)
- A A Kutsenko
- Université de Bordeaux, Institut de Mécanique et d'Ingénierie de Bordeaux, UMR 5295, Talence 33405, France
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Norris AN, Shuvalov AL. Elastodynamics of radially inhomogeneous spherically anisotropic elastic materials in the Stroh formalism. Proc Math Phys Eng Sci 2011. [DOI: 10.1098/rspa.2011.0463] [Citation(s) in RCA: 20] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022] Open
Abstract
A method for solving elastodynamic problems in radially inhomogeneous elastic materials with spherical anisotropy is presented, i.e. materials having
c
ijkl
=
c
ijkl
(
r
) in a spherical coordinate system {
r
,
θ
,
ϕ
}. The time-harmonic displacement field
u
(
r
,
θ
,
ϕ
) is expanded in a separation of variables form with dependence on
θ
,
ϕ
described by vector spherical harmonics with
r
-dependent amplitudes. It is proved that such separation of variables solution is generally possible only if the spherical anisotropy is restricted to transverse isotropy (TI) with the principal axis in the radial direction, in which case the amplitudes are determined by a first-order ordinary differential system. Restricted forms of the displacement field, such as
u
(
r
,
θ
), admit this type of separation of variables solution for certain lower material symmetries. These results extend the Stroh formalism of elastodynamics in rectangular and cylindrical systems to spherical coordinates.
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Affiliation(s)
- A. N. Norris
- Mechanical and Aerospace Engineering, Rutgers University, Piscataway, NJ 08854-8058, USA
| | - A. L. Shuvalov
- Université de Bordeaux, Institut de Mécanique et d'Ingénierie de Bordeaux, UMR 5295, Talence 33405, France
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Meziane A, Norris AN, Shuvalov AL. Nonlinear shear wave interaction at a frictional interface: energy dissipation and generation of harmonics. J Acoust Soc Am 2011; 130:1820-1828. [PMID: 21973335 DOI: 10.1121/1.3628663] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/31/2023]
Abstract
Analytical and numerical modeling of the nonlinear interaction of shear wave with a frictional interface is presented. The system studied is composed of two homogeneous and isotropic elastic solids, brought into frictional contact by remote normal compression. A shear wave, either time harmonic or a narrow band pulse, is incident normal to the interface and propagates through the contact. Two friction laws are considered and the influence on interface behavior is investigated: Coulomb's law with a constant friction coefficient and a slip-weakening friction law which involves static and dynamic friction coefficients. The relationship between the nonlinear harmonics and the dissipated energy, and the dependence on the contact dynamics (friction law, sliding, and tangential stress) and on the normal contact stress are examined in detail. The analytical and numerical results indicate universal type laws for the amplitude of the higher harmonics and for the dissipated energy, properly non-dimensionalized in terms of the pre-stress, the friction coefficient and the incident amplitude. The results suggest that measurements of higher harmonics can be used to quantify friction and dissipation effects of a sliding interface.
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Affiliation(s)
- A Meziane
- Institut de Mécanique et d'Ingénierie de Bordeaux-I2M-UMR CNRS 5295, Université de Bordeaux, 351, Cours de la Libération Talence 33405 Cedex, France.
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Shuvalov AL, Kutsenko AA, Norris AN, Poncelet O. Effective Willis constitutive equations for periodically stratified anisotropic elastic media. Proc Math Phys Eng Sci 2011. [DOI: 10.1098/rspa.2010.0389] [Citation(s) in RCA: 36] [Impact Index Per Article: 2.8] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022] Open
Abstract
A method to derive homogeneous effective constitutive equations for periodically layered elastic media is proposed. The crucial and novel idea underlying the procedure is that the coefficients of the dynamic effective medium can be associated with the matrix logarithm of the propagator over a unit period. The effective homogeneous equations are shown to have the structure of a Willis material, characterized by anisotropic inertia and coupling between momentum and strain, in addition to effective elastic constants. Expressions are presented for the Willis material parameters which are formally valid at any frequency and horizontal wavenumber as long as the matrix logarithm is well defined. The general theory is exemplified for scalar SH motion. Low frequency, long wavelength expansions of the effective material parameters are also developed using a Magnus series, and explicit estimates are derived for the rate of convergence.
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Affiliation(s)
- A. L. Shuvalov
- Université de Bordeaux, CNRS, UMR 5469, Laboratoire de Mécanique Physique, Talence 33405, France
| | - A. A. Kutsenko
- Université de Bordeaux, CNRS, UMR 5469, Laboratoire de Mécanique Physique, Talence 33405, France
| | - A. N. Norris
- Université de Bordeaux, CNRS, UMR 5469, Laboratoire de Mécanique Physique, Talence 33405, France
- Mechanical and Aerospace Engineering, Rutgers University, Piscataway, NJ 08854-8058, USA
| | - O. Poncelet
- Université de Bordeaux, CNRS, UMR 5469, Laboratoire de Mécanique Physique, Talence 33405, France
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Norris AN, Krylov VV, Abrahams ID. Flexural edge waves and comments on "A new bending wave solution for the classical plate equation". J Acoust Soc Am 2000; 107:1781-1785. [PMID: 10738833 DOI: 10.1121/1.428457] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [What about the content of this article? (0)] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
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Affiliation(s)
- A. V. Osipov
- Institute of Radiophysics, The St Petersburg State University, Uljanovskaja 1, Petrodvorets 198904, Russia
| | - A. N. Norris
- Department of Mechanical and Aerospace Engineering, Rutgers University, Piscataway, NJ 08855‐0909, USA
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Ye L, Cody G, Zhou M, Sheng P, Norris AN. Observation of bending wave localization and quasi mobility edge in two dimensions. Phys Rev Lett 1992; 69:3080-3083. [PMID: 10046720 DOI: 10.1103/physrevlett.69.3080] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [What about the content of this article? (0)] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
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