Long Wavelength Coherent Pulse of Sound Propagating in Granular Media

Authors: Rohit Kumar Shrivastava, Amalia Thomas, Nathalie Vriend, Stefan Luding

Abstract:

A mechanical wave or vibration propagating through granular media exhibits a specific signature in time. A coherent pulse or wavefront arrives first with multiply scattered waves (coda) arriving later. The coherent pulse is micro-structure independent i.e. it depends only on the bulk properties of the disordered granular sample, the sound wave velocity of the granular sample and hence bulk and shear moduli. The coherent wavefront attenuates (decreases in amplitude) and broadens with distance from its source. The pulse attenuation and broadening effects are affected by disorder (polydispersity; contrast in size of the granules) and have often been attributed to dispersion and scattering. To study the effect of disorder and initial amplitude (non-linearity) of the pulse imparted to the system on the coherent wavefront, numerical simulations have been carried out on one-dimensional sets of particles (granular chains). The interaction force between the particles is given by a Hertzian contact model. The sizes of particles have been selected randomly from a Gaussian distribution, where the standard deviation of this distribution is the relevant parameter that quantifies the effect of disorder on the coherent wavefront. Since, the coherent wavefront is system configuration independent, ensemble averaging has been used for improving the signal quality of the coherent pulse and removing the multiply scattered waves. The results concerning the width of the coherent wavefront have been formulated in terms of scaling laws. An experimental set-up of photoelastic particles constituting a granular chain is proposed to validate the numerical results.

Keywords: Discrete elements, Hertzian Contact, polydispersity, weakly nonlinear, wave propagation.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1132665

References:

[1] P. Shearer, Introduction to Seismology. Cambridge University Press, 2009. (Online). Available: https://books.google.nl/books?id=VV0mV4lF0RUC.
[2] A. M. Dainty and M. Toksz, “Seismic codas on the earth and the moon: a comparison,” Physics of the Earth and Planetary Interiors, vol. 26, no. 4, pp. 250 – 260, 1981. (Online). Available: http://www.sciencedirect.com/science/article/pii/0031920181900297.
[3] J. O’Donovan, E. Ibraim, C. O’Sullivan, S. Hamlin, D. Muir Wood, and G. Marketos, “Micromechanics of seismic wave propagation in granular materials,” Granular Matter, vol. 18, no. 3, p. 56, Jun 2016. (Online). Available: https://doi.org/10.1007/s10035-015-0599-4.
[4] P. Sheng, Intoduction to Wave Scattering, Localization and Mesoscopic Phenomena, ser. Springer Series in Materials Science. Springer, 2006. (Online). Available: https://doi.org/10.1007/3-540-29156-3.
[5] V. Langlois and X. Jia, “Sound pulse broadening in stressed granular media,” Phys. Rev. E, vol. 91, p. 022205, Feb 2015. (Online). Available: https://link.aps.org/doi/10.1103/PhysRevE.91.022205
[6] I. G¨uven, “Hydraulical and acoustical properties of porous sintered glass bead systems: experiments, theory, & simulations,” Ph.D. dissertation, Enschede, 2016. (Online). Available: http://doc.utwente.nl/100546/.
[7] X. Jia, “Codalike multiple scattering of elastic waves in dense granular media,” Phys. Rev. Lett., vol. 93, p. 154303, Oct 2004. (Online). Available: http://link.aps.org/doi/10.1103/PhysRevLett.93.154303.
[8] X. Jia, C. Caroli, and B. Velicky, “Ultrasound propagation in externally stressed granular media,” Phys. Rev. Lett., vol. 82, pp. 1863–1866, Mar 1999. (Online). Available: http://link.aps.org/doi/10.1103/PhysRevLett.82.1863.
[9] H. A. Makse, N. Gland, D. L. Johnson, and L. Schwartz, “Granular packings: Nonlinear elasticity, sound propagation, and collective relaxation dynamics,” Phys. Rev. E, vol. 70, p. 061302, Dec 2004. (Online). Available: http://link.aps.org/doi/10.1103/PhysRevE.70.061302.
[10] R. K. Shrivastava and S. Luding, “Effect of disorder on bulk sound wave speed: a multiscale spectral analysis,” Nonlinear Processes in Geophysics, vol. 24, no. 3, pp. 435–454, 2017. (Online). Available: https://www.nonlin-processes-geophys.net/24/435/2017/.
[11] V. Nesterenko, “Propagation of nonlinear compression pulses in granular media,” J. Appl. Mech. Tech. Phys., vol. 24, pp. 733–743, March 1983.
[12] E. T. Owens and K. E. Daniels, “Sound propagation and force chains in granular materials,” EPL (Europhysics Letters), vol. 94, no. 5, p. 54005, 2011. (Online). Available: http://stacks.iop.org/0295- 5075/94/i=5/a=54005.
[13] E. Somfai, J.-N. Roux, J. H. Snoeijer, M. van Hecke, and W. van Saarloos, “Elastic wave propagation in confined granular systems,” Phys. Rev. E, vol. 72, p. 021301, Aug 2005. (Online). Available: http://link.aps.org/doi/10.1103/PhysRevE.72.021301.
[14] T. S. Majmudar and R. P. Behringer, “Contact force measurements and stress-induced anisotropy in granular materials,” Nature, vol. 435, no. 1079, pp. 1079–1082, 06 2005. (Online). Available: http://dx.doi.org/10.1038/nature03805.
[15] B. P. Lawney and S. Luding, “Massdisorder effects on the frequency filtering in onedimensional discrete particle systems,” AIP Conference Proceedings, vol. 1542, no. 1, pp. 535–538, 2013. (Online). Available: http://scitation.aip.org/content/aip/proceeding/aipcp/10.1063/1.4811986.
[16] Shrivastava, Rohit Kumar and Luding, Stefan, “Wave propagation of spectral energy content in a granular chain,” EPJ Web Conf., vol. 140, p. 02023, 2017. (Online). Available: https://doi.org/10.1051/epjconf/201714002023.
[17] L. D. Landau and E. M. Lifshitz, Theory of elasticity. Pergamon Press, 1970.
[18] P. W. Anderson, “Absence of diffusion in certain random lattices,” Phys. Rev., vol. 109, pp. 1492–1505, Mar 1958. (Online). Available: http://link.aps.org/doi/10.1103/PhysRev.109.1492.
[19] J. O’Donovan, C. O’Sullivan, G. Marketos, and D. Muir Wood, “Analysis of bender element test interpretation using the discrete element method,” Granular Matter, vol. 17, no. 2, pp. 197–216, 2015. (Online). Available: http://dx.doi.org/10.1007/s10035-015-0552-6.
[20] E. W. Weisstein, “Tree. From MathWorld—A Wolfram Web Resource,” last visited on 6/9/2017. (Online). Available: http://mathworld.wolfram.com/GaussianFunction.html.
[21] O. Mouraille, W. A. Mulder, and S. Luding, “Sound wave acceleration in granular materials,” Journal of Statistical Mechanics: Theory and Experiment, vol. 2006, no. 07, p. P07023, 2006. (Online). Available: http://stacks.iop.org/1742-5468/2006/i=07/a=P07023.