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Attenuation of Solitary Waves and Localization of Breathers in 1D Granular Crystals Visualized via High Speed Photography

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Abstract

We investigate the propagation, attenuation, and localization of nonlinear elastic waves in a 1D granular crystal using high speed photography. We measure temporal displacement profiles of individual particles with a micrometer-scale resolution, and we reconstruct force profiles of propagating solitary waves and localized breathers by synchronizing and analyzing the acquired data. These investigations provide quantitative evidence for the transmission and attenuation trends of travelling solitary waves in a soft polymeric chain, which are significantly different from those in a hard metallic chain. We additionally study energy localization in a chain of hard particles embedded with a soft polymeric impurity. Specifically, we show that the proposed experimental technique is able to visualize the formation of localized breathers and quantify the energy highly concentrated in the vicinity of the impurity site—a phenomenon which can be exploited for harvesting vibrational energy in engineering applications. Finally, we compare, with good agreement, the experimental results with discrete element numerical simulations that account for dissipative effects due to viscoelasticity. The findings reported in this study imply that high speed photography can be an efficient and effective tool for non-contact measurements of nonlinear wave dynamics in granular lattices, despite their short characteristic times and minute displacements.

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References

  1. Sutton MA, Orteu JJ, Schreier HW (2009) Image correlation for shape, motion and deformation measurements. Springer, New York

    Google Scholar 

  2. Helm JD, Sutton MA, McNeill SR (2003) Deformations in wide, center-notched, thin panels: Parts I and II: three dimensional shape and deformation measurements by computer vision. Opt Eng 42(5):1293–1320

    Article  Google Scholar 

  3. Sutton M et al (2007) Scanning electron microscopy for quantitative small and large deformation measurements Part I: SEM imaging at magnifications from 200 to 10,000. Exp Mech 47(6):775–787

    Article  MathSciNet  Google Scholar 

  4. Sutton M et al (2007) Scanning electron microscopy for quantitative small and large deformation measurements Part II: experimental validation for magnifications from 200 to 10,000. Exp Mech 47(6):789–804

    Google Scholar 

  5. Schreier HW, Garcia D, Sutton MA (2004) Advances in light microscope stereo vision. Exp Mech 44(3):278–288

    Article  Google Scholar 

  6. Tiwari V, Sutton MA, McNeill SR (2007) Assessment of high speed imaging systems for 2D and 3D deformation measurements: methodology development and validation. Exp Mech 47(4):561–579

    Article  Google Scholar 

  7. Tiwari V et al (2009) Application of 3D image correlation for full-field transient plate deformation measurements during blast loading. Int J Impact Eng 36(6):862–874

    Article  Google Scholar 

  8. Zhao X et al (2013) Scaling of the deformation histories for clamped circular plates subjected to blast loading by buried charges. Int J Impact Eng 54:31–50

    Article  Google Scholar 

  9. Nishida M, Tanaka Y (2010) DEM simulations and experiments for projectile impacting two-dimensional particle packings including dissimilar material layers. Granul Matter 12(4):357–368

    Article  Google Scholar 

  10. Zhu Y, Shukla A, Sadd MH (1996) The effect of microstructural fabric on dynamic load transfer in two dimensional assemblies of elliptical particles. J Mech Phys Solids 44(8):1283–1303

    Article  Google Scholar 

  11. Daraio C et al (2005) Strongly nonlinear waves in a chain of Teflon beads. Phys Rev E 72(1):016603

    Article  Google Scholar 

  12. Leonard A, Fraternali F, Daraio C (2013) Directional wave propagation in a highly nonlinear square packing of spheres. Exp Mech 53(3):327–337

    Article  Google Scholar 

  13. Nesterenko VF (2001) Dynamics of heterogeneous materials. Springer-Verlag New York, Inc., New York

    Book  Google Scholar 

  14. Hladky-Hennion AC, de Billy M (2007) Experimental validation of band gaps and localization in a one-dimensional diatomic phononic crystal. J Acoust Soc Am 122(5):2594–2600

    Article  Google Scholar 

  15. Boechler N et al (2011) Tunable vibrational band gaps in one-dimensional diatomic granular crystals with three-particle unit cells. J Appl Phys 109(7):074906–074907

    Article  Google Scholar 

  16. Daraio C et al (2006) Tunability of solitary wave properties in one-dimensional strongly nonlinear phononic crystals. Phys Rev E Stat Nonlin Soft Matter Phys 73(2 Pt 2):026610

    Article  Google Scholar 

  17. Nesterenko VF et al (2005) Anomalous wave reflection at the interface of two strongly nonlinear granular media. Phys Rev Lett 95(15):158702

    Article  Google Scholar 

  18. Porter MA, Daraio C, Szelengowicz I, Herbold EB, Kevrekidis PG (2009) Highly nonlinear solitary waves in heterogeneous periodic granular media. Physica D 238:666–676

    Article  MATH  Google Scholar 

  19. Jayaprakash KR, Starosvetsky Y, Vakakis AF (2011) New family of solitary waves in granular dimer chains with no precompression. Phys Rev E 83(3):036606

    Article  MathSciNet  Google Scholar 

  20. Potekin R et al (2013) Experimental study of strongly nonlinear resonances and anti-resonances in granular dimer chains. Exp Mech 53(5):861–870

    Google Scholar 

  21. Flach S, Gorbach AV (2008) Discrete breathers — advances in theory and applications. Phys Rep 467(1–3):1–116

    Article  Google Scholar 

  22. Kevrekidis PG (2011) Non-linear waves in lattices: past, present, future. IMA J Appl Math 76:389–423

    Article  MATH  MathSciNet  Google Scholar 

  23. Theocharis G et al (2010) Intrinsic energy localization through discrete gap breathers in one-dimensional diatomic granular crystals. Phys Rev E 82(5):056604

    Article  MathSciNet  Google Scholar 

  24. Theocharis G et al (2009) Localized breathing modes in granular crystals with defects. Phys Rev E Stat Nonlin Soft Matter Phys 80(6 Pt 2):066601

    Article  Google Scholar 

  25. Starosvetsky Y, Jayaprakash KR, Vakakis AF (2011) Scattering of solitary waves and excitation of transient breathers in granular media by light intruders and no precompression. J Appl Mech 79(1):011001

    Article  Google Scholar 

  26. Chong C, Li F, Yang J., Williams MO, Kevrekidis IG, Kevrekidis PG, Daraio C (2013) Damped-driven granular crystals: an ideal playground for dark breathers and multibreathers. http://arxiv.org/abs/1307.4780

  27. Spadoni A, Daraio C (2010) Generation and control of sound bullets with a nonlinear acoustic lens. Proc Natl Acad Sci U S A 107:7230–7234

    Article  Google Scholar 

  28. Daraio C et al (2006) Pulse mitigation by a composite discrete medium. J Phys IV 134:473–479

    Google Scholar 

  29. Yang J, Dunatunga S, Daraio C (2012) Amplitude-dependent attenuation of compressive waves in curved granular crystals constrained by elastic guides. Acta Mech 223(3):549–562

    Article  MATH  Google Scholar 

  30. Fraternali F, Porter MA, Daraio C (2009) Optimal design of composite granular protectors. Mech Adv Mater Struct 17(1):1–19

    Article  Google Scholar 

  31. Hong J (2005) Universal power-law decay of the impulse energy in granular protectors. Phys Rev Lett 94(10):108001

    Article  Google Scholar 

  32. Feng L, Lingyu Y, Jinkyu Y (2013) Solitary wave-based strain measurements in one-dimensional granular crystals. J Phys D Appl Phys 46(15):155106

    Article  Google Scholar 

  33. Khatri D, Rizzo P, Daraio C (2008) Highly nonlinear waves’ sensor technology for highway infrastructures. In SPIE Smart Structures/NDE, 15th annual international symposium, San Diego, CA

  34. Yang J et al (2012) Site-specific quantification of bone quality using highly nonlinear solitary waves. J Biomech Eng 134(10):101001–101008

    Article  Google Scholar 

  35. Manciu M, Sen S, Hurd AJ (2001) Impulse propagation in dissipative and disordered chains with power-law repulsive potentials. Phys D Nonlinear Phenom 157(3):226–240

    Article  MATH  Google Scholar 

  36. Rosas A et al (2008) Short-pulse dynamics in strongly nonlinear dissipative granular chains. Phys Rev E 78(5):051303

    Article  Google Scholar 

  37. Hong J, Kim H, Hwang J-P (2000) Characterization of soliton damping in the granular chain under gravity. Phys Rev E 61(1):964–967

    Article  Google Scholar 

  38. Herbold EB, Nesterenko VF (2007) Shock wave structure in a strongly nonlinear lattice with viscous dissipation. Phys Rev E 75(2):021304

    Article  Google Scholar 

  39. Vergara L (2010) Model for dissipative highly nonlinear waves in dry granular systems. Phys Rev Lett 104(11):118001

    Article  MathSciNet  Google Scholar 

  40. Carretero-González R et al (2009) Dissipative solitary waves in granular crystals. Phys Rev Lett 102(2):024102

    Article  Google Scholar 

  41. Hascoët E, Herrmann HJ (2000) Shocks in non-loaded bead chains with impurities. Eur Phys J B Condens Matter Complex Syst 14(1):183–190

    Article  Google Scholar 

  42. Job S et al (2009) Wave localization in strongly nonlinear Hertzian chains with mass defect. Phys Rev E 80(2):025602

    Article  Google Scholar 

  43. Feng L et al (2013) Visualization of solitary waves via laser Doppler vibrometry for heavy impurity identification in a granular chain. Smart Mater Struct 22(3):035016

    Article  Google Scholar 

  44. Johnson KL (1985) Contact mechanics. Cambridge University Press, Cambridge

    Book  MATH  Google Scholar 

  45. Sen S et al (2008) Solitary waves in the granular chain. Phys Rep 462(2):21–66

    Article  MathSciNet  Google Scholar 

  46. Job S et al (2005) How Hertzian solitary waves interact with boundaries in a 1D granular medium. Phys Rev Lett 94:178002(17)

    Article  Google Scholar 

  47. Yang J et al (2011) Interaction of highly nonlinear solitary waves with linear elastic media. Phys Rev E 83, 046606

    Article  Google Scholar 

  48. Tsuji Y, Tanaka T, Ishida T (1992) Lagrangian numerical simulation of plug flow of cohesionless particles in a horizontal pipe. Powder Technol 71(3):239–250

    Article  Google Scholar 

  49. Shampine LF, Reichelt MW (1997) The MATLAB ODE Suite. SIAM J Sci Comput 18(1):1–22

    Article  MATH  MathSciNet  Google Scholar 

  50. Sutton MA et al (1988) Effects of subpixel image restoration on digital correlation error estimates. Opt Eng 27(10):271070

    Article  Google Scholar 

  51. Schreier HW, Braasch JR, Sutton MA (2000) Systematic errors in digital image correlation caused by intensity interpolation. Opt Eng 39(11):2915–2921

    Article  Google Scholar 

  52. Wang YQ et al (2009) Quantitative error assessment in pattern matching: effects of intensity pattern noise, interpolation, strain and image contrast on motion measurements. Strain 45(2):160–178

    Article  Google Scholar 

  53. Wang YQ et al (2011) On error assessment in stereo-based deformation measurements. Exp Mech 51(4):405–422

    Article  Google Scholar 

  54. Ke XD et al (2011) Error assessment in stereo-based deformation measurements. Exp Mech 51(4):423–441

    Article  Google Scholar 

  55. Chatterjee A (1999) Asymptotic solution for solitary waves in a chain of elastic spheres. Phys Rev E 59(5):5912–5919

    Article  Google Scholar 

  56. Remoissenet M (1999) Waves called solitons (concepts and experiments). 3rd revised and enlarged edition ed, Springer-Verlag, Berlin

  57. Carter WJ, Marsh SP (1995) Hugoniot equation of state of polymers. In Other Information: PBD: Jul 1995. p. Medium: ED; Size: 25 p

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Acknowledgments

The authors would like to thank C. Daraio for useful discussions. We also thank M. Meidani and S. Guo for assisting the construction of the experimental setup. The authors acknowledge support from the National Science Foundation (Grant No. 1234452).

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Authors and Affiliations

  1. William E. Boeing Department of Aeronautics & Astronautics, University of Washington, Seattle, WA, 98195, USA

    J. Yang & E. Kim

  2. Department of Mechanical Engineering, University of South Carolina, Columbia, SC, 29208, USA

    J. Yang, E. Kim, C. Agbasi & M. Sutton

  3. School of Mechanical Engineering, Purdue University, West Lafayette, IN, 47907, USA

    M. Gonzalez

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  1. J. Yang
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Correspondence to J. Yang.

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Yang, J., Gonzalez, M., Kim, E. et al. Attenuation of Solitary Waves and Localization of Breathers in 1D Granular Crystals Visualized via High Speed Photography. Exp Mech 54, 1043–1057 (2014). https://doi.org/10.1007/s11340-014-9866-2

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