Abstract
We studied the dynamic response of a two-dimensional square packing of uncompressed stainless steel spheres excited by impulsive loadings. We developed a new experimental measurement technique, employing miniature tri-axial accelerometers, to determine the stress wave properties in the array resulting from both an in-plane and out-of-plane impact. Results from our numerical simulations, based on a discrete particle model, were in good agreement with the experimental results. We observed that the impulsive excitations were resolved into solitary waves traveling only through initially excited chains. The observed solitary waves were determined to have similar (Hertzian) properties to the extensively studied solitary waves supported by an uncompressed, uniform, one-dimensional chain of spheres. The highly directional response of this system could be used as a basis to design granular crystals with predetermined wave propagation paths capable of mitigating stress wave energy.
Similar content being viewed by others
References
Nesterenko V (1983) Propagation of nonlinear compression pulses in granular media. J Appl Mech Tech Phys 24:733
Nesterenko VF (2001) Dynamics of heterogeneous materials. Springer, New York
Lazaridi A, Nesterenko V (1985) Observation of a new type of solitary waves in a one-dimensional granular medium. J Appl Mech Tech Phys 26:405
Coste C, Falcon E, Fauve S (1997) Solitary waves in a chain of beads under Hertz contact. Phys Rev E 56:6104
Coste C, Gilles B (1999) On the validity of Hertz contact law for granular material acoustics. Eur Phys J B 7:155
Daraio C, Nesterenko VF, Herbold EB, Jin S (2005) Strongly nonlinear waves in a chain of Teflon beads. Phys Rev E 72:016603
Chatterjee A (1999) Asymptotic solution for solitary waves in a chain of elastic spheres. Phys Rev E 59:5912
Nakagawa M, Agui J, Wu D, Extramiana D (2003) Impulse dispersion in a tapered granular chain. Granul Matter 4:167
Sen S, Manciu F, Manciu M (2001) Thermalizing an impulse. Physica A 299:551
Nesterenko V, Lazaridi A, Sibiryakov E (1995) The decay of soliton at the contact of two “acoustic vacuums”. J Appl Mech Tech Phys 36:166
Vergara L (2005) Scattering of solitary waves from interfaces in granular media. Phys Rev Lett 95:108002
Nesterenko V, Daraio C, Herbold E, Jin S (2005) Anomalous wave reflection at the interface of two strongly nonlinear granular media. Phys Rev Lett 95:158702
Job S, Melo F, Sokolow A, Sen S (2007) Solitary wave trains in granular chains: experiments, theory and simulations. Granul Matter 10:13
Hascoët E, Herrmann H, Loreto V (1999) Shock propagation in a granular chain. Phys Rev E 59:3202
Job S, Santibanez F, Tapia F, Melo F (2009) Wave localization in strongly nonlinear Hertzian chains with mass defect. Phys Rev E 80:25602
Job S, Melo F, Sokolow A, Sen S (2005) How Hertzian solitary waves interact with boundaries in a 1D granular medium. Phys Rev Lett 94:178002
Daraio C, Nesterenko V, Herbold E, Jin S (2006) Energy trapping and shock disintegration in a composite granular medium. Phys Rev Lett 96:058002
Hinch E, Saint-Jean S (1999) The fragmentation of a line of balls by an impact. Proc R Soc Lond A 455:3201
Manciu M, Sen S, Hurd A (2000) Crossing of identical solitary waves in a chain of elastic beads. Phys Rev E 63:016614
Daraio C, Nesterenko V, Herbold E, Jin S (2006) Tunability of solitary wave properties in one-dimensional strongly nonlinear phononic crystals. Phys Rev E 73:026610
Sinkovits R, Sen S (2003) Nonlinear dynamics in granular columns. Phys Rev Lett 74:2686
Gilles B, Coste C (2003) Low-frequency behavior of beads constrained on a lattice. Phys Rev Lett 90:174302
Velický B, Caroli C (2002) Pressure dependence of the sound velocity in a two-dimensional lattice of Hertz–Mindlin balls: mean-field description. Phys Rev E 65(2):021307
Goddard J (1990) Nonlinear elasticity and pressure-dependent wave speeds in granular media. Proc R Soc Lond, A Math Phys Sci 430:105
Zhu Y, Shukla A, Sadd M (1996) The effect of microstructural fabric on dynamic load transfer in two dimensional assemblies of elliptical particles. J Mech Phys Solids 44:1283
Sadd M, Gao J, Shukla A (1997) Numerical analysis of wave propagation through assemblies of elliptical particles. Comput Geotech 20:323
Nishida M, Tanaka K, Ishida T (2009) DEM simulation of wave propagation in two-dimensional ordered array of particles. Shock Waves 815
Nishida M, Tanaka Y (2010) DEM simulations and experiments for projectile impacting two-dimensional particle packings including dissimilar material layers. Granul Matter 12:357
Merkel A, Tournat V, Gusev V (2010) Elastic waves in noncohesive frictionless granular crystals. Ultrasonics 50:133
Sen S, Sinkovits R (1996) Sound propagation in impure granular columns. Phys Rev E 54:6857
Sadd M, Tai Q, Shukla A (1993) Contact law effects on wave propagation in particulate materials using distinct element modeling. Int J Non-Linear Mech 28:251
Daraio C, Ngo D, Nesterenko V, Fraternali F (2010) H ighly nonlinear pulse splitting and recombination in a two-dimensional granular network. Phys Rev E 82:036603
Shukla A, Zhu C, Sadd M (1988) Angular dependence of dynamic load transfer due to explosive loading in granular aggregate chains. J Strain Anal Eng Des 23:121
Shukla A, Sadd M, Singh R, Tai Q, Vishwanathan S (1993) Role of particle shape and contact profile on the dynamic response of particulate materials. Opt Lasers Eng 19:99
Bardenhagen S, Brackbill J (1998) Dynamic stress bridging in granular material. J Appl Phys 83:5732
Geng J, Reydellet G, Clément E, Behringer R (2003) Green’s function measurements of force transmission in 2D granular materials. Physica D: Nonlinear Phenom 182:274
Johnson K (1987) Contact mechanics. The Press Syndicate of the University of Cambridge, The Pitt Building
Hong J, Xu A (2001) Effects of gravity and nonlinearity on the waves in the granular chain. Phys Rev E 63:061310
Carretero-González R, Khatri D, Porter M, Kevrekidis P, Daraio C (2009) Dissipative solitary waves in granular crystals. Phys Rev Lett 102:024102
Acknowledgements
We thank Dr. A. Spadoni for his suggestions in the design of the confining box used in experiments. We also thank R. Giordano and A. Esposito for their preliminary work on the subject. This work was supported by an award from the Department of Energy Office of Science (DOE SCGF) and the Army Research Office MURI (Dr. David Stepp).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Leonard, A., Fraternali, F. & Daraio, C. Directional Wave Propagation in a Highly Nonlinear Square Packing of Spheres. Exp Mech 53, 327–337 (2013). https://doi.org/10.1007/s11340-011-9544-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11340-011-9544-6