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On the compact wave dynamics of tensegrity beams in multiple dimensions

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Abstract

This work presents a numerical investigation on the nonlinear wave dynamics of tensegrity beams in 1D, 2D, and 3D arrangements. The simulation of impact loading on a chain of tensegrity prisms and lumped masses allows us to apply on a smaller scale recent results on the propagation of compression solitary waves in 1D tensegrity metamaterials. Novel results on the wave dynamics of 2D and 3D beams reveal—for the first time—the presence of compact compression waves in two- and three-dimensional tensegrity lattices with slender aspect ratio and stiffening-type elastic response. The dynamics of such systems is characterized by the thermalization of the lattice nearby the impacted regions of the boundary. The portion of the absorbed energy moving along the longitudinal direction is transported by compression waves with compact support. Such waves emerge with nearly constant speed, and slight modifications of their spatial shape and amplitude, after collisions with compression waves traveling in opposite direction. The analyzed behaviors suggest the use of multidimensional tensegrity lattices for the design and additive manufacturing of novel sound focusing devices.

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References

  1. Liu, Z., Zhang, X., Mao, Y., Zhu, Y.Y., Yang, Z., Chan, C.T., Sheng, P.: Locally resonant sonic materials. Science 289, 1734–1736 (2000)

    Google Scholar 

  2. Lu, M.H., Feng, L., Chen, Y.F.: Phononic crystals and acoustic metamaterials. Mater. Today 12, 34–42 (2009)

    Google Scholar 

  3. Maldovan, M.: Sound and heat revolution in phononics. Nature 503, 209–217 (2013)

    Google Scholar 

  4. Brunet, T., Leng, J., Mondain-Monva, O.: Soft acoustic metamaterials. Science 342, 323–324 (2013)

    Google Scholar 

  5. Friesecke, G., Pego, R.: Solitary waves on FPU lattices: I. Qualitative properties, renormalization and continuum limit. Nonlinearity 12, 1601–1627 (1999)

    MathSciNet  MATH  Google Scholar 

  6. Friesecke, G., Matthies, K.: Atomic-scale localization of high-energy solitary waves on lattices. Physica D 171, 211–220 (2002)

    MathSciNet  MATH  Google Scholar 

  7. Theocharis, G., Boechler, N., Daraio, C.: Nonlinear phononic structures and metamaterials. In: Deymier, P.A. (ed.) Acoustic Metamaterials and Phononic Crystals. Springer Series in Solid State Sciences, vol. 173. Springer, Berlin (2013)

    Google Scholar 

  8. Meza, L.R., Das, S., Greer, J.R.: Strong, lightweight, and recoverable three-dimensional ceramic nanolattices. Science 345(6202), 1322–1326 (2014)

    Google Scholar 

  9. Zheng, X., et al.: Ultralight, ultrastiff mechanical metamaterials. Science 344, 6190 (2014)

    Google Scholar 

  10. Christensen, J., Kadic, M., Kraft, O., Wegener, M.: Vibrant times for mechanical metamaterials. MRS Commun. 5(3), 453–462 (2015)

    Google Scholar 

  11. Cummer, S.A., Christensen, J., Alu, A.: Controlling sound with acoustic metamaterials. Nat. Rev. Mater. 1, 16001 (2016)

    Google Scholar 

  12. Phani, A.S., Hussein, M.T. (eds.): Dynamics of Lattice Materials. Wiley, Chichester (2017)

    Google Scholar 

  13. Narisetti, R.K., Leamy, M.J., Ruzzene, M.: A perturbation approach for predicting wave propagation in one-dimensional nonlinear periodic structures. J. Vib. Acoust. Trans. ASME 132(3), 0310011–03100111 (2010)

    Google Scholar 

  14. Hussein, M.I., Leamy, M.J., Ruzzene, M.: Dynamics of phononic materials and structures: historical origins, recent progress, and future outlook. Appl. Mech. Rev. 66(4), 040802 (2014)

    Google Scholar 

  15. Bertoldi, K., Vitelli, V., Christensen, J., Van Hecke, M.: Flexible mechanical metamaterials. Nat. Rev. Mater. 2, 17066 (2017)

    Google Scholar 

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

    Google Scholar 

  17. Herbold, E.B., Nesterenko, V.F.: Propagation of rarefaction pulses in discrete materials with strain-softening behavior. Phys. Rev. Lett. 110, 144101 (2012)

    Google Scholar 

  18. Rosenau, P.: Dynamics of dense lattices. Phys. Rev. B 36(11), 5868–5876 (1987)

    MathSciNet  Google Scholar 

  19. Rosenau, P.: WHAT IS... a compacton? Not. Am. Math. Soc. 52(7), 738–739 (2005)

    MathSciNet  MATH  Google Scholar 

  20. Zhang, T., Li, J.: Exact solitons, periodic peakons and compactons in an optical soliton model. Nonlinear Dyn. 91(2), 1371–1381 (2018)

    MATH  Google Scholar 

  21. Dang, Y.L., Li, H.J., Lin, J.: Soliton solutions in nonlocal nonlinear coupler. Nonlinear Dyn. 88(1), 489–501 (2017)

    Google Scholar 

  22. Ma, L., Li, H., Ma, J.: Single-peak solitary wave solutions for the generalized Korteweg–de vries equation. Nonlinear Dyn. 79(1), 349–357 (2015)

    MathSciNet  MATH  Google Scholar 

  23. Guo, D., Tian, S.T., Zhang, T.T., Li, J.: Modulation instability analysis and soliton solutions of an integrable coupled nonlinear schrödinger system. Nonlinear Dyn. 94(4), 2749–2761 (2018)

    Google Scholar 

  24. Sun, H.Q., Chen, A.H.: Interactional solutions of a lump and a solitary wave for two higher-dimensional equations. Nonlinear Dyn. 94(3), 1753–1762 (2018)

    Google Scholar 

  25. Fraternali, F., Senatore, L., Daraio, C.: Solitary waves on tensegrity lattices. J. Mech. Phys. Solids 60, 1137–1144 (2012)

    Google Scholar 

  26. Fraternali, F., Carpentieri, G., Amendola, A., Skelton, R.E., Nesterenko, V.F.: Multiscale tunability of solitary wave dynamics in tensegrity metamaterials. Appl. Phys. Lett. 105, 201903 (2014)

    Google Scholar 

  27. Davini, C., Micheletti, A., Podio-Guidugli, P.: On the impulsive dynamics of T3 tensegrity chains. Meccanica 51(11), 2763–2776 (2016)

    MathSciNet  Google Scholar 

  28. Amendola, A., Krushynska, A., Daraio, C., Pugno, N.M., Fraternali, F.: Tuning frequency band gaps of tensegrity metamaterials with local and global prestress. Int. J. Solids Struct. 155, 47–56 (2018)

    Google Scholar 

  29. Skelton, R.E., de Oliveira, M.C.: Tensegrity Systems. Springer, Berlin (2010)

    MATH  Google Scholar 

  30. Micheletti, A.: Bistable regimes in an elastic tensegrity system. Proc. R. Soc. 469(2154), 201300520 (2012)

    Google Scholar 

  31. Fraternali, F., De Chiara, E., Skelton, R.E.: On the use of tensegrity structures for kinetic solar facades of smart buildings. Smart. Mater. Struct. 24, 105032 (2015)

    Google Scholar 

  32. Amendola, A., Hernández-Nava, E., Goodall, R., Todd, I., Skeltonf, R.E., Fraternali, F.: On the additive manufacturing, post-tensioning and testing of bi-material tensegrity structures. Compos. Struct. 131, 66–71 (2015)

    Google Scholar 

  33. Fraternali, F., Carpentieri, G., Amendola, A.: On the mechanical modeling of the extreme softening/stiffening response of axially loaded tensegrity prisms. J. Mech. Phys. Solids 74, 136–157 (2014)

    MATH  Google Scholar 

  34. Amendola, A., Carpentieri, G., De Oliveira, M., Skelton, R.E., Fraternali, F.: Experimental investigation of the softening-stiffening response of tensegrity prisms under compressive loading. Compos. Struct. 117, 234–243 (2014)

    Google Scholar 

  35. Rimoli, J.J., Pal, R.K.: Mechanical response of 3-dimensional tensegrity lattices. Compos. Part B Eng. 115, 30–42 (2017)

    Google Scholar 

  36. Rimoli, J.J.: A reduced-order model for the dynamic and post-buckling behavior of tensegrity structures. Mech. Mater. 116, 146–157 (2018)

    Google Scholar 

  37. Pal, R.K., Ruzzene, M., Rimoli, J.J.: Tunable wave propagation by varying prestrain in tensegrity-based periodic media. Extreme Mech. Lett. 22, 149–156 (2018)

    Google Scholar 

  38. Salahshoor, H., Pal, R.K., Rimol, J.J.: Material symmetry phase transitions in three-dimensional tensegrity metamaterials. J. Mech. Phys. Solids 119, 382–399 (2018)

    MathSciNet  Google Scholar 

  39. Micheletti, A.: Simple analytical models of tensegrity structures. In: Frémond, M., Maceri, F. (eds.) Novel Approaches in Civil Engineering. Lecture Notes in Applied and Computational Mechanics, vol. 14, pp. 351–358. Springer, Berlin (2004)

    Google Scholar 

  40. Ekici, M., Mirzazadeh, M., Eslami, M.: Solitons and other solutions to Boussinesq equation with power law nonlinearity and dual dispersion. Nonlinear Dyn. 84(2), 669–676 (2016)

    MathSciNet  MATH  Google Scholar 

  41. Li, L., Xie, Y., Zhu, S.: New exact solutions for a generalized KdV equation. Nonlinear Dyn. 92(2), 215–219 (2018)

    MATH  Google Scholar 

  42. Sultan, C., Skelton, R.E.: Deployment of tensegrity structures. Int. J. Solid Struct. 40, 4637–4657 (2003)

    MATH  Google Scholar 

  43. Pellegrino, S., Calladine, C.R.: Matrix analysis of statically and kinematically indeterminate frameworks. Int. J. Solid Struct. 22, 409–428 (1986)

    Google Scholar 

  44. Materials Spotlight: The Properties of Nylon 12. https://www.cableorganizer.com/learning-center/articles/materials-nylon12.html. Date accessed: 14 Jan 2019

  45. Fraternali, F., Porter, M.A., Daraio, C.: Optimal design of composite granular protector. Mech. Adv. Mater. Struct. 17, 1–19 (2010)

    Google Scholar 

  46. Daraio, C., Fraternali, F.: Method and Apparatus for Wave Generation and Detection Using Tensegrity Structures, US Pat. No. 8,616,328, granted on December 31, 2013 (2013)

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

    Google Scholar 

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Acknowledgements

AM and GR gratefully acknowledge the financial support from the Italian Ministry of Education, University, and Research (MIUR) under the ‘FFABR’ Grant L.232/2016. FF gratefully acknowledges financial support from the Italian Ministry of Education, University, and Research (MIUR) under the ‘Departments of Excellence’ Grant L.232/2016.

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Authors and Affiliations

  1. DICII, University of Rome Tor Vergata, Rome, Italy

    Andrea Micheletti

  2. DISA, University of Bergamo, Bergamo, Italy

    Giuseppe Ruscica

  3. DICIV, University of Salerno, Fisciano, Italy

    Fernando Fraternali

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  1. Andrea Micheletti
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  2. Giuseppe Ruscica
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  3. Fernando Fraternali
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Correspondence to Fernando Fraternali.

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Appendix A. Supplementary material

Appendix A. Supplementary material

Animations of the wave dynamics of the systems analyzed in this paper can be found in the online version.

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Micheletti, A., Ruscica, G. & Fraternali, F. On the compact wave dynamics of tensegrity beams in multiple dimensions. Nonlinear Dyn 98, 2737–2753 (2019). https://doi.org/10.1007/s11071-019-04986-8

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