Skip to main content
SpringerLink
Log in

Spectral Properties of Neumann-Poincaré Operator and Anomalous Localized Resonance in Elasticity Beyond Quasi-Static Limit

  • Published:
Journal of Elasticity Aims and scope Submit manuscript

Abstract

This paper is concerned with the polariton resonances and their application for cloaking due to anomalous localized resonance (CALR) for the elastic system within finite frequency regime beyond the quasi-static approximation. We first derive the complete spectral system of the Neumann-Poincaré operator associated with the elastic system in \(\mathbb{R}^{3}\) within the finite frequency regime. Based on the obtained spectral results, we construct a broad class of elastic configurations that can induce polariton resonances beyond the quasi-static limit. As an application, the invisibility cloaking effect is achieved through constructing a class of core-shell-matrix metamaterial structures provided the source is located inside a critical radius. Moreover, if the source is located outside the critical radius, it is proved that there is no resonance.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

References

  1. Ammari, H., Ciraolo, G., Kang, H., Lee, H., Milton, G.W.: Spectral theory of a Neumann-Poincaré-type operator and analysis of cloaking due to anomalous localized resonance II. Contemporary Math. 615, 1–14 (2014)

    Article  Google Scholar 

  2. Ammari, H., Ciraolo, G., Kang, H., Lee, H., Milton, G.W.: Anomalous localized resonance using a folded geometry in three dimensions. Proc. R. Soc. A 469, 20130048 (2013)

    Article  ADS  Google Scholar 

  3. Ammari, H., Ciraolo, G., Kang, H., Lee, H., Milton, G.W.: Spectral theory of a Neumann-Poincaré-type operator and analysis of cloaking due to anomalous localized resonance. Arch. Ration. Mech. Anal. 208, 667–692 (2013)

    Article  MathSciNet  Google Scholar 

  4. Ammari, H., Deng, Y., Millien, P.: Surface plasmon resonance of nanoparticles and applications in imaging. Arch. Ration. Mech. Anal. 220, 109–153 (2016)

    Article  MathSciNet  Google Scholar 

  5. Ammari, H., Millien, P., Ruiz, M., Zhang, H.: Mathematical analysis of plasmonic nanoparticles: the scalar case. Arch. Ration. Mech. Anal. 224, 597–658 (2017)

    Article  MathSciNet  Google Scholar 

  6. Ammari, H., Ruiz, M., Yu, S., Zhang, H.: Mathematical analysis of plasmonic resonances for nanoparticles: the full Maxwell equations. J. Differential Equations 261, 3615–3669 (2016)

    Article  ADS  MathSciNet  Google Scholar 

  7. Ando, K., Kang, H.: Analysis of plasmon resonance on smooth domains using spectral properties of the Neumann-Poincaré operator. J. Math. Anal. Appl. 435, 162–178 (2016)

    Article  MathSciNet  Google Scholar 

  8. Ando, K., Ji, Y., Kang, H., Kim, K., Yu, S.: Spectral properties of the Neumann-Poincaré operator and cloaking by anomalous localized resonance for the elasto-static system. European J. Appl. Math. 29, 189–225 (2018)

    Article  MathSciNet  Google Scholar 

  9. Ando, K., Kang, H., Kim, K., Yu, S.: Cloaking by anomalous localized resonance for linear elasticity on a coated structure. arXiv:1612.08384

  10. Ando, K., Kang, H., Liu, H.: Plasmon resonance with finite frequencies: a validation of the quasi-static approximation for diametrically small inclusions. SIAM J. Appl. Math. 76, 731–749 (2016)

    Article  MathSciNet  Google Scholar 

  11. Bruno, O., Lintner, S.: Superlens-cloaking of small dielectric bodies in the quasistatic regime. Journal of Applied Physics 102(12), 124502 (2007)

    Article  ADS  Google Scholar 

  12. Blåsten, E., Li, H., Liu, H., Wang, Y.: Localization and geometrization in plasmon resonances and geometric structures of Neumann-Poincaré eigenfunctions. ESAIM: Math. Model. Numer. Anal. https://doi.org/10.1051/m2an/2019091

  13. Bouchitté, G., Schweizer, B.: Cloaking of small objects by anomalous localized resonance. Quart. J. Mech. Appl. Math. 63, 438–463 (2010)

    Article  MathSciNet  Google Scholar 

  14. Bruno, O.P., Lintner, S.: Superlens-cloaking of small dielectric bodies in the quasistatic regime. J. Appl. Phys. 102, 124502 (2007)

    Article  ADS  Google Scholar 

  15. Colton, D., Kress, R.: Inverse Acoustic and Electromagnetic Scattering Theory, 2nd edn. Springer, Berlin (1998)

    Book  Google Scholar 

  16. Deng, Y., Li, H., Liu, H.: On spectral properties of Neumann-Poincare operator and plasmonic cloaking in 3D elastostatics. J. Spectr. Theory 9(3), 767–789 (2019)

    Article  MathSciNet  Google Scholar 

  17. Deng, Y., Li, H., Liu, H.: Analysis of surface polariton resonance for nanoparticles in elastic system. SIAM J. Math. Anal. (2020), in press, arXiv:1804.05480

  18. Kettunen, H., Lassas, M., Ola, P.: On absence and existence of the anomalous localized resonance without the quasi-static approximation. SIAM J. Appl. Math. 78, 609–628 (2018)

    Article  MathSciNet  Google Scholar 

  19. Kochmann, D.M., Milton, G.W.: Rigorous bounds on the effective moduli of composites and inhomogeneous bodies with negative-stiffness phases. J. Mech. Phys. Solids 71, 46–63 (2014)

    Article  ADS  MathSciNet  Google Scholar 

  20. Kohn, R.V., Lu, J., Schweizer, B., Weinstein, M.I.: A variational perspective on cloaking by anomalous localized resonance. Comm. Math. Phys. 328, 1–27 (2014)

    Article  ADS  MathSciNet  Google Scholar 

  21. Kupradze, V.D.: Three-dimensional Problems of the Mathematical Theory of Elasticity and Thermoelasticity. North-Holland, Amsterdam (1979)

    Google Scholar 

  22. Lakes, R.S., Lee, T., Bersie, A., Wang, Y.: Extreme damping in composite materials with negative-stiffness inclusions. Nature 410, 565–567 (2001)

    Article  ADS  Google Scholar 

  23. Li, H., Liu, H.: On anomalous localized resonance for the elastostatic system. SIAM J. Math. Anal. 48, 3322–3344 (2016)

    Article  MathSciNet  Google Scholar 

  24. Li, H., Liu, H.: On three-dimensional plasmon resonance in elastostatics. Annali di Matematica Pura ed Applicata 196, 1113–1135 (2017)

    Article  MathSciNet  Google Scholar 

  25. Li, H., Liu, H.: On anomalous localized resonance and plasmonic cloaking beyond the quasi-static limit. Proc. R. Soc. A 474, 20180165 (2018)

    ADS  MATH  Google Scholar 

  26. Li, H., Li, J., Liu, H.: On novel elastic structures inducing polariton resonances with finite frequencies and cloaking due to anomalous localized resonance. Journal de Mathématiques Pures et Appliquées 120, 195–219 (2018)

    Article  MathSciNet  Google Scholar 

  27. Li, H., Li, J., Liu, H.: On quasi-static cloaking due to anomalous localized resonance in \(\mathbb{R}^{3}\). SIAM J. Appl. Math. 75(3), 1245–1260 (2015)

    Article  MathSciNet  Google Scholar 

  28. Li, H., Li, S., Liu, H., Wang, X.: Analysis of electromagnetic scattering from plasmonic inclusions beyond the quasi-static approximation and applications. ESAIM: Math. Model. Numer. Anal. 53(4), 1351–1371 (2019)

    Article  MathSciNet  Google Scholar 

  29. Li, H., Liu, H., Zou, J.: Minnaert resonances for bubbles in soft elastic materials. arXiv:1911.03718

  30. McPhedran, R.C., Nicorovici, N.-A.P., Botten, L.C., Milton, G.W.: Cloaking by plasmonic resonance among systems of particles: cooperation or combat? C.R. Phys. 10, 391–399 (2009)

    Article  ADS  Google Scholar 

  31. Milton, G.W., Nicorovici, N.-A.P.: On the cloaking effects associated with anomalous localized resonance. Proc. R. Soc. A 462, 3027–3059 (2006)

    Article  ADS  MathSciNet  Google Scholar 

  32. Milton, G.W., Nicorovici, N.-A.P., McPhedran, R.C., Cherednichenko, K., Jacob, Z.: Solutions in folded geometries, and associated cloaking due to anomalous resonance. New. J. Phys. 10, 115021 (2008)

    Article  ADS  Google Scholar 

  33. Milton, G.W., Nicorovici, N.-A.P., McPhedran, R.C., Podolskiy, V.A.: Proof of superlensing in the quasistatic regime, and limitations of superlenses in this regime due to anomalous localized resonance. Proc. R. Soc. A 461, 3999–4034 (2005)

    Article  ADS  MathSciNet  Google Scholar 

  34. Nédélec, J.C.: Acoustic and Electromagnetic Equations: Integral Representations for Harmonic Problems. Springer, New York (2001)

    Book  Google Scholar 

  35. Nguyen, H.: Cloaking an arbitrary object via anomalous localized resonance: the cloak is independent of the object. SIAM J. Math. Anal. 49, 3208–3232 (2017)

    Article  MathSciNet  Google Scholar 

  36. Nguyen, H.: Cloaking via anomalous localized resonance for doubly complementary media in the finite frequency regime. arXiv:1511.08053

  37. Nicorovici, N.-A.P., McPhedran, R.C., Enoch, S., Tayeb, G.: Finite wavelength cloaking by plasmonic resonance. New. J. Phys. 10, 115020 (2008)

    Article  ADS  Google Scholar 

  38. Nicorovici, N.-A.P., McPhedran, R.C., Milton, G.W.: Optical and dielectric properties of partially resonant composites. Phys. Rev. B 49, 8479–8482 (1994)

    Article  ADS  Google Scholar 

  39. Nicorovici, N.-A.P., Milton, G.W., McPhedran, R.C., Botten, L.C.: Quasistatic cloaking of two-dimensional polarizable discrete systems by anomalous resonance. Optics Express 15, 6314–6323 (2007)

    Article  ADS  Google Scholar 

  40. Smith, D.R., Pendry, J.B., Wiltshire, M.C.K.: Metamaterials and negative refractive index. Science 305, 788–792 (2004)

    Article  ADS  Google Scholar 

  41. Veselago, V.G.: The electrodynamics of substances with simultaneously negative values of \(\epsilon\) and \(\mu\). Sov. Phys. Usp. 10, 509–514 (1968)

    Article  ADS  Google Scholar 

Download references

Acknowledgements

The authors would like to express their gratitudes to the anonymous referees and the Associate Editor who handled our manuscript, as well as the Editor-in-Chief Professor Fosdick for many insightful and constructive comments and suggestions, which have significantly improved the results and presentation of the paper. The work of Y. Deng was supported by NSF grant of China No. 11971487, PSCF of Hunan No. 18YBQ077 and RFEB of Hunan No. 18B337. The work of Hongyu Liu was support by a startup fund of City University of Hong Kong, and Hong Kong RGC General Research Funds, 12302017, 12301218 and 12302919.

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Central South University, Changsha, Hunan, China

    Youjun Deng

  2. Department of Mathematics, The Chinese University of Hong Kong, Shatin, N. T., Hong Kong, China

    Hongjie Li

  3. Department of Mathematics, City University of Hong Kong, Kowloon Tong, Hong Kong, China

    Hongyu Liu

Authors
  1. Youjun Deng
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hongjie Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Hongyu Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Youjun Deng.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Deng, Y., Li, H. & Liu, H. Spectral Properties of Neumann-Poincaré Operator and Anomalous Localized Resonance in Elasticity Beyond Quasi-Static Limit. J Elast 140, 213–242 (2020). https://doi.org/10.1007/s10659-020-09767-8

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10659-020-09767-8

Keywords

Mathematics Subject Classification

Search

Navigation