Experimental Mechanics

, Volume 54, Issue 6, pp 1043–1057 | Cite as

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

  • J. YangEmail author
  • M. Gonzalez
  • E. Kim
  • C. Agbasi
  • M. Sutton


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.


Solitary waves Localized breathers High speed photography Digital image processing 



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).


  1. 1.
    Sutton MA, Orteu JJ, Schreier HW (2009) Image correlation for shape, motion and deformation measurements. Springer, New YorkGoogle Scholar
  2. 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):1293–1320CrossRefGoogle Scholar
  3. 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):775–787CrossRefMathSciNetGoogle Scholar
  4. 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):789–804Google Scholar
  5. 5.
    Schreier HW, Garcia D, Sutton MA (2004) Advances in light microscope stereo vision. Exp Mech 44(3):278–288CrossRefGoogle Scholar
  6. 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):561–579CrossRefGoogle Scholar
  7. 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):862–874CrossRefGoogle Scholar
  8. 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 54:31–50CrossRefGoogle Scholar
  9. 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):357–368CrossRefGoogle Scholar
  10. 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):1283–1303CrossRefGoogle Scholar
  11. 11.
    Daraio C et al (2005) Strongly nonlinear waves in a chain of Teflon beads. Phys Rev E 72(1):016603CrossRefGoogle Scholar
  12. 12.
    Leonard A, Fraternali F, Daraio C (2013) Directional wave propagation in a highly nonlinear square packing of spheres. Exp Mech 53(3):327–337CrossRefGoogle Scholar
  13. 13.
    Nesterenko VF (2001) Dynamics of heterogeneous materials. Springer-Verlag New York, Inc., New YorkCrossRefGoogle Scholar
  14. 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):2594–2600CrossRefGoogle Scholar
  15. 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):074906–074907CrossRefGoogle Scholar
  16. 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):026610CrossRefGoogle Scholar
  17. 17.
    Nesterenko VF et al (2005) Anomalous wave reflection at the interface of two strongly nonlinear granular media. Phys Rev Lett 95(15):158702CrossRefGoogle Scholar
  18. 18.
    Porter MA, Daraio C, Szelengowicz I, Herbold EB, Kevrekidis PG (2009) Highly nonlinear solitary waves in heterogeneous periodic granular media. Physica D 238:666–676CrossRefzbMATHGoogle Scholar
  19. 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):036606CrossRefMathSciNetGoogle Scholar
  20. 20.
    Potekin R et al (2013) Experimental study of strongly nonlinear resonances and anti-resonances in granular dimer chains. Exp Mech 53(5):861–870Google Scholar
  21. 21.
    Flach S, Gorbach AV (2008) Discrete breathers — advances in theory and applications. Phys Rep 467(1–3):1–116CrossRefGoogle Scholar
  22. 22.
    Kevrekidis PG (2011) Non-linear waves in lattices: past, present, future. IMA J Appl Math 76:389–423CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Theocharis G et al (2010) Intrinsic energy localization through discrete gap breathers in one-dimensional diatomic granular crystals. Phys Rev E 82(5):056604CrossRefMathSciNetGoogle Scholar
  24. 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):066601CrossRefGoogle Scholar
  25. 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):011001CrossRefGoogle Scholar
  26. 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.
  27. 27.
    Spadoni A, Daraio C (2010) Generation and control of sound bullets with a nonlinear acoustic lens. Proc Natl Acad Sci U S A 107:7230–7234CrossRefGoogle Scholar
  28. 28.
    Daraio C et al (2006) Pulse mitigation by a composite discrete medium. J Phys IV 134:473–479Google Scholar
  29. 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):549–562CrossRefzbMATHGoogle Scholar
  30. 30.
    Fraternali F, Porter MA, Daraio C (2009) Optimal design of composite granular protectors. Mech Adv Mater Struct 17(1):1–19CrossRefGoogle Scholar
  31. 31.
    Hong J (2005) Universal power-law decay of the impulse energy in granular protectors. Phys Rev Lett 94(10):108001CrossRefGoogle Scholar
  32. 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):155106CrossRefGoogle Scholar
  33. 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, CAGoogle Scholar
  34. 34.
    Yang J et al (2012) Site-specific quantification of bone quality using highly nonlinear solitary waves. J Biomech Eng 134(10):101001–101008CrossRefGoogle Scholar
  35. 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):226–240CrossRefzbMATHGoogle Scholar
  36. 36.
    Rosas A et al (2008) Short-pulse dynamics in strongly nonlinear dissipative granular chains. Phys Rev E 78(5):051303CrossRefGoogle Scholar
  37. 37.
    Hong J, Kim H, Hwang J-P (2000) Characterization of soliton damping in the granular chain under gravity. Phys Rev E 61(1):964–967CrossRefGoogle Scholar
  38. 38.
    Herbold EB, Nesterenko VF (2007) Shock wave structure in a strongly nonlinear lattice with viscous dissipation. Phys Rev E 75(2):021304CrossRefGoogle Scholar
  39. 39.
    Vergara L (2010) Model for dissipative highly nonlinear waves in dry granular systems. Phys Rev Lett 104(11):118001CrossRefMathSciNetGoogle Scholar
  40. 40.
    Carretero-González R et al (2009) Dissipative solitary waves in granular crystals. Phys Rev Lett 102(2):024102CrossRefGoogle Scholar
  41. 41.
    Hascoët E, Herrmann HJ (2000) Shocks in non-loaded bead chains with impurities. Eur Phys J B Condens Matter Complex Syst 14(1):183–190CrossRefGoogle Scholar
  42. 42.
    Job S et al (2009) Wave localization in strongly nonlinear Hertzian chains with mass defect. Phys Rev E 80(2):025602CrossRefGoogle Scholar
  43. 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):035016CrossRefGoogle Scholar
  44. 44.
    Johnson KL (1985) Contact mechanics. Cambridge University Press, CambridgeCrossRefzbMATHGoogle Scholar
  45. 45.
    Sen S et al (2008) Solitary waves in the granular chain. Phys Rep 462(2):21–66CrossRefMathSciNetGoogle Scholar
  46. 46.
    Job S et al (2005) How Hertzian solitary waves interact with boundaries in a 1D granular medium. Phys Rev Lett 94:178002(17)CrossRefGoogle Scholar
  47. 47.
    Yang J et al (2011) Interaction of highly nonlinear solitary waves with linear elastic media. Phys Rev E 83, 046606CrossRefGoogle Scholar
  48. 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):239–250CrossRefGoogle Scholar
  49. 49.
    Shampine LF, Reichelt MW (1997) The MATLAB ODE Suite. SIAM J Sci Comput 18(1):1–22CrossRefzbMATHMathSciNetGoogle Scholar
  50. 50.
    Sutton MA et al (1988) Effects of subpixel image restoration on digital correlation error estimates. Opt Eng 27(10):271070CrossRefGoogle Scholar
  51. 51.
    Schreier HW, Braasch JR, Sutton MA (2000) Systematic errors in digital image correlation caused by intensity interpolation. Opt Eng 39(11):2915–2921CrossRefGoogle Scholar
  52. 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):160–178CrossRefGoogle Scholar
  53. 53.
    Wang YQ et al (2011) On error assessment in stereo-based deformation measurements. Exp Mech 51(4):405–422CrossRefGoogle Scholar
  54. 54.
    Ke XD et al (2011) Error assessment in stereo-based deformation measurements. Exp Mech 51(4):423–441CrossRefGoogle Scholar
  55. 55.
    Chatterjee A (1999) Asymptotic solution for solitary waves in a chain of elastic spheres. Phys Rev E 59(5):5912–5919CrossRefGoogle Scholar
  56. 56.
    Remoissenet M (1999) Waves called solitons (concepts and experiments). 3rd revised and enlarged edition ed, Springer-Verlag, BerlinGoogle Scholar
  57. 57.
    Carter WJ, Marsh SP (1995) Hugoniot equation of state of polymers. In Other Information: PBD: Jul 1995. p. Medium: ED; Size: 25 pGoogle Scholar

Copyright information

© Society for Experimental Mechanics 2014

Authors and Affiliations

  • J. Yang
    • 1
    • 2
    Email author
  • M. Gonzalez
    • 3
  • E. Kim
    • 1
    • 2
  • C. Agbasi
    • 2
  • M. Sutton
    • 2
  1. 1.William E. Boeing Department of Aeronautics & AstronauticsUniversity of WashingtonSeattleUSA
  2. 2.Department of Mechanical EngineeringUniversity of South CarolinaColumbiaUSA
  3. 3.School of Mechanical EngineeringPurdue UniversityWest LafayetteUSA

Personalised recommendations