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On the coupling dynamics between thermally stressed beams and granular chains


The in-situ measurement of thermal stress in slender beams, or long continuous welded rails, may prevent structural anomalies. With this aim, we investigated the coupling dynamics between a beam and the highly nonlinear solitary waves propagating along a straight granular chain in contact with the beam. We hypothesized that these waves can be used to measure the stress of thermally loaded structures, or to infer the neutral temperature, i.e., the temperature at which this stress is null. We studied numerically and experimentally the mechanical interaction of one and two straight chains of spherical particles in contact with a prismatic beam that is subjected to heat. The results show that certain features of the waves are affected by the beam’s stress. In the future, these findings may be used to develop a novel nondestructive evaluation technique for the prediction of neutral temperature and thermal buckling.

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  1. 1.

    Bartoli, I., Phillips, R., Coccia, S., Srivastava, A., Lanza di Scalea, F., Fateh, M., Carr, G.: Stress dependence of ultrasonic guided waves in rails. Transp. Res. Rec. 2159, 91–97 (2010)

    Article  Google Scholar 

  2. 2.

    Carretero-González, R., Khatri, D., Porter, M.A., Kevrekidis, P.G., Daraio, C.: Dissipative solitary waves in granular crystals. Phys. Rev. Lett. 102(2), 024102 (2009)

    Article  Google Scholar 

  3. 3.

    Coste, C., Falcon, E., Fauve, S.: Solitary waves in a chain of beads under Hertz contact. Phys. Rev. E. 56(5), 6104–6117 (1997)

    Article  Google Scholar 

  4. 4.

    Daraio, C., Nesterenko, V.F., Herbold, E.B., Jin, S.: Strongly nonlinear waves in a chain of Teflon beads. Phys. Rev. E. 72(1), 016603 (2005)

    Article  Google Scholar 

  5. 5.

    Daraio, C., Nesterenko, V.F., Herbold, E.B., Jin, S.: Tunability of solitary wave properties in one-dimensional strongly nonlinear phononic crystals. Phys. Rev. E. 73(2), 026610 (2006)

    Article  Google Scholar 

  6. 6.

    Job, S., Melo, F., Sokolow, A., Sen, S.: How Hertzian solitary waves interact with boundaries in a 1D granular medium. Phys. Rev. Lett. 94(17), 178002 (2005)

    Article  Google Scholar 

  7. 7.

    Job, S., Melo, F., Sokolow, A., Sen, S.: Solitary wave trains in granular chains: experiments, theory and simulations. Granul. Matter 10(1), 13–20 (2007)

    Article  MATH  Google Scholar 

  8. 8.

    Lazaridi, A.N., Nesterenko, V.F.: Observation of a new type of solitary waves in a one-dimensional granular medium. J. Appl. Mech. Tech. Phys. 26(3), 405–408 (1985)

    Article  Google Scholar 

  9. 9.

    Nesterenko, V.F.: Propagation of nonlinear compression pulses in granular media. J. Appl. Mech. Tech. Phys. 24(5), 733–743 (1983)

    Article  Google Scholar 

  10. 10.

    Nesterenko, V.F.: Dynamics of Heterogeneous Materials. Springer, New York (2001)

    Book  Google Scholar 

  11. 11.

    Nesterenko, V.F., Lazaridi, A.N., Sibiryakov, E.B.: The decay of soliton at the contact of two “acoustic vacuums”. J. Appl. Mech. Tech. Phys. 36(2), 166–168 (1995)

    Article  Google Scholar 

  12. 12.

    Yang, J., Silvestro, C., Khatri, D., De Nardo, L., Daraio, C.: Interaction of highly nonlinear solitary waves with linear elastic media. Phys. Rev. E. 83(4), 046606 (2011)

    Article  Google Scholar 

  13. 13.

    Boechler, N., Theocharis, G., Daraio, C.: Bifurcation-based acoustic switching and rectification. Nat. Mater. 10(9), 665–668 (2011)

    Article  Google Scholar 

  14. 14.

    Donahue, C.M., Anzel, P.W., Bonanomi, L., Keller, T.A., Daraio, C.: Experimental realization of a nonlinear acoustic lens with a tunable focus. Appl. Phys. Lett. 104(1), 014103 (2014)

    Article  Google Scholar 

  15. 15.

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

    Article  Google Scholar 

  16. 16.

    Li, F., Zhao, L., Tian, Z., Yu, L., Yang, J.: Visualization of solitary waves via laser Doppler vibrometry for heavy impurity identification in a granular chain. Smart Mater. Struct. 22(3), 035016 (2013a)

    Article  Google Scholar 

  17. 17.

    Li, F., Yu, L., Yang, J.: Solitary wave-based strain measurements in one-dimensional granular crystals. J. Phys. D Appl. Phys. 46(15), 155106 (2013b)

    Article  Google Scholar 

  18. 18.

    Shelke, A., Uddin, A., Yang, J.: Impact identification in sandwich structures using solitary wave-supporting granular crystal sensors. AIAA J. 52(10), 2283–2290 (2014)

    Article  Google Scholar 

  19. 19.

    Spadoni, A., Daraio, C.: Generation and control of sound bullets with a nonlinear acoustic lens. Proc. Natl. Acad. Sci. U.S. 107(16), 7230–7234 (2010)

    Article  Google Scholar 

  20. 20.

    Ni, X., Rizzo, P., Yang, J., Katri, D., Daraio, C.: Monitoring the hydration of cement using highly nonlinear solitary waves. NDT E Int. 52, 76–85 (2012)

    Article  Google Scholar 

  21. 21.

    Ni, X., Rizzo, P.: Highly nonlinear solitary waves for the inspection of adhesive joints. Exp. Mech. 52(9), 1493–1501 (2012a)

    Article  Google Scholar 

  22. 22.

    Ni, X., Rizzo, P.: Use of highly nonlinear solitary waves in nondestructive testing. Mater. Eval. 70(5), 561–569 (2012b)

    Google Scholar 

  23. 23.

    Yang, J., Khatri, D., Anzel, P., Daraio, C.: Interaction of highly nonlinear solitary waves with thin plates. Int. J. Solids Struct. 49(13), 1463–1471 (2012a)

    Article  Google Scholar 

  24. 24.

    Yang, J., Silvestro, C., Sangiorgio, S.N., Borkowski, S.L., Ebramzadeh, E., De Nardo, L., Daraio, C.: Nondestructive evaluation of orthopaedic implant stability in THA using highly nonlinear solitary waves. Smart Mater. Struct. 21(1), 012002 (2012b)

    Article  Google Scholar 

  25. 25.

    Cai, L., Rizzo, P., Al-Nazer, L.: On the coupling mechanism between nonlinear solitary waves and slender beams. Int. J. Solids Struct. 50(25), 4173–4183 (2013)

    Article  Google Scholar 

  26. 26.

    Tedesco, J.W., McDougal, W.G., Ross, C.A.: Structural dynamics: theory and applications. Addison-Wesley, Montlo Park, CA (1999)

    Google Scholar 

  27. 27.

    Lanza di Scalea, F., Rizzo, P., Seible, F.: Stress measurement and defect detection in steel strands by guided stress waves. J. Mater. Civ. Eng. 15(3), 219–227 (2003)

    Article  Google Scholar 

  28. 28.

    Rizzo, P., Lanza di Scalea, F.: Effect of frequency on the acoustoelastic response of steel bars. Exp. Tech. 27(6), 40–43 (2003)

    Article  Google Scholar 

  29. 29.

    Rizzo, P., Lanza di Scalea, F.: Monitoring in cable stays via guided wave magnetostrictive ultrasonics. Mater. Eval. 62, 1057–1065 (2004)

    Google Scholar 

  30. 30.

    Szelengowicz, I., Kevrekidis, P.G., Daraio, C.: Wave propagation in square granular crystals with spherical interstitial intruders. Phys. Rev. E 86(6), 061306 (2012)

    Article  Google Scholar 

  31. 31.

    Job, S., Santibanez, F., Tapia, F., Melo, F.: Wave localization in strongly nonlinear Hertzian chains with mass defect. Phys. Rev. E 80(2), 025602 (2009)

    Article  Google Scholar 

  32. 32.

    Hascoët, E., Herrmann, H.J.: Shocks in non-loaded bead chains with impurities. Eur. Phys. J. B 14(1), 183–190 (2000)

    Article  Google Scholar 

  33. 33.

    Theocharis, G., Kavousanakis, M., Kevrekidis, P.G., Daraio, C., Porter, M.A., Kevrekidis, I.G.: Localized breathing modes in granular crystals with defects. Phys. Rev. E 80(6), 066601 (2009)

    Article  Google Scholar 

  34. 34.

    Tichler, A.M., Gómez, L.R., Upadhyaya, N., Campman, X., Nesterenko, V.F., Vitelli, V.: Transmission and reflection of strongly nonlinear solitary waves at granular interfaces. Phys. Rev. Lett. 111(4), 048001 (2013)

    Article  Google Scholar 

  35. 35.

    Yang, J., Gonzalez, M., Kim, E., Agbasi, C., Sutton, M.: Attenuation of solitary waves and localization of breathers in 1D granular crystals Visualized via High Speed Photography. Exp. Mech. 54(6), 1043–1057 (2014)

    Article  Google Scholar 

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This project was supported by the U.S. Federal Railroad Administration under contract DTFR53-12-C-00014. Partial support came from the U.S. National Science Foundation grant CMMI 1200259. We thank Mr. Charles “Scooter” Hager for helping in the design and construction of the house-built steel frame.

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Correspondence to Piervincenzo Rizzo.

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Bagheri, A., Ribolla, E.L.M., Rizzo, P. et al. On the coupling dynamics between thermally stressed beams and granular chains. Arch Appl Mech 86, 541–556 (2016).

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  • Highly nonlinear solitary waves
  • Vibration
  • Nondestructive testing
  • Thermal stress
  • Granular chains
  • Neutral temperature
  • Discrete particle model