0
Research Papers

Elastic–Plastic Wave Propagation in Uniform and Periodic Granular Chains

[+] Author and Article Information
Hayden A. Burgoyne

Mem. ASME
GALCIT,
California Institute of Technology,
1200 E. California Boulevard,
Pasadena, CA 91125
e-mail: hburgoyne@caltech.edu

Chiara Daraio

Mem. ASME
ETH Zurich,
Tannenstrasse 3,
Zurich 8092, Switzerland
e-mail: daraio@ethz.ch

1Corresponding author.

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received February 5, 2015; final manuscript received April 24, 2015; published online June 9, 2015. Editor: Yonggang Huang.

J. Appl. Mech 82(8), 081002 (Aug 01, 2015) (10 pages) Paper No: JAM-15-1070; doi: 10.1115/1.4030458 History: Received February 05, 2015; Revised April 24, 2015; Online June 09, 2015

We investigate the properties of high-amplitude stress waves propagating through chains of elastic–plastic particles using experiments and simulations. We model the system after impact using discrete element method (DEM) with strain-rate dependent contact interactions. Experiments are performed on a Hopkinson bar coupled with a laser vibrometer. The bar excites chains of 50 identical particles and dimer chains of two alternating materials. After investigating how the speed of the initial stress wave varies with particle properties and loading amplitude, we provide an upper bound for the leading pulse velocity that can be used to design materials with tailored wave propagation.

FIGURES IN THIS ARTICLE
<>
Copyright © 2015 by ASME
Your Session has timed out. Please sign back in to continue.

References

Nesterenko, V., 2001, Dynamics of Heterogeneous Materials, Springer, New York.
Job, S., Melo, F., Sokolow, A., and Sen, S., 2007, “Solitary Wave Trains in Granular Chains: Experiments, Theory, and Simulations,” Granular Matter, 10(1), pp. 13–20. [CrossRef]
Daraio, C., Nesterenko, V. F., Herbold, E. B., and Jin, S., 2006, “Tunability of Solitary Wave Properties in One-Dimensional Strongly Nonlinear Phononic Crystals,” Phys. Rev. E, 73(2), p. 026610. [CrossRef]
Doney, R. L., Agui, J. H., and Sen, S., 2009, “Energy Partitioning and Impulse Dispersion in the Decorated, Tapered, Strongly Nonlinear Granular Alignment: A System With Many Potential Applications,” J. Appl. Phys., 106(6), p. 064905. [CrossRef]
Nesterenko, V. F., Daraio, C., Herbold, E. B., and Jin, S., 2005, “Anomalous Wave Reflection at the Interface of Two Strongly Nonlinear Granular Media,” Phys. Rev. Lett., 95(15), p. 158702. [CrossRef] [PubMed]
Daraio, C., Nesterenko, V. F., Herbold, E. B., and Jin, S., 2006, “Energy Trapping and Shock Disintegration in a Composite Granular Medium,” Phys. Rev. Lett., 96(5), p. 058002. [CrossRef] [PubMed]
Fraternali, F., Porter, M. A., and Daraio, C., 2008, “Optimal Design of Composite Granular Protectors,” Mech. Adv. Mater. Struct., 17(1), pp. 1–19. [CrossRef]
Hong, J., 2005, “Universal Power-Law Decay of Impulse Energy Granular Protectors,” Phys. Rev. Lett., 94(10), p. 108001. [CrossRef] [PubMed]
Ngo, D., Fraternali, F., and Daraio, C., 2012, “Highly Nonlinear Solitary Wave Propagation in Y-Shaped Granular Crystals With Variable Branch Angles,” Phys. Rev. E, 85(3), p. 036602. [CrossRef]
Leonard, A., Ponson, L., and Daraio, C., 2014, “Wave Mitigation in Ordered Networks of Granular Chains,” J. Mech. Phys. Solids, 73, pp. 103–117. [CrossRef]
Jayaprakash, K. R., Starosvetsky, Y., and Vakakis, A. F., 2011, “New Family of Solitary Waves in Granular Dimer Chains With No Precompression,” Phys. Rev. E, 83(3), p. 036606. [CrossRef]
Molinari, A., and Daraio, C., 2009, “Stationary Shocks in Periodic Highly Nonlinear Granular Chains,” Phys. Rev. E, 80(5), p. 056602. [CrossRef]
Porter, M. A., Daraio, C., Herbold, E. B., Szelengowicz, I., and Kevrekidis, P. G., 2008, “Highly Nonlinear Solitary Waves in Periodic Dimer Granular Chains,” Phys. Rev. E, 77(1), p. 015601. [CrossRef]
Bragança, E. A., Rosas, A., and Lindenberg, K., 2013, “Binary Collision Approximation for Multi-Decorated Granular Chains,” Physica A, 392(24), pp. 6198–6205. [CrossRef]
Sen, S., Hong, J., Bang, J., Avalos, E., and Doney, R., 2008, “Solitary Waves in the Granular Chain,” Phys. Reports, 462(2), pp. 21–66. [CrossRef]
Boechler, N., Yang, J., Theocharis, G., Kevrekidis, P. G., and Daraio, C., 2011, “Tunable Vibrational Band Gaps in One-Dimensional Diatomic Granular Crystals With Three-Particle Unit Cells,” J. App. Phys., 109(7), p. 074906. [CrossRef]
Hoogeboom, C., Man, Y., Boechler, N., Theocharis, G., Kevrekidis, P. G., Kevrekidis, I. G., and Daraio, C., 2013, “Hysteresis Loops and Multi-Stability: From Periodic Orbits to Chaotic Dynamics (and Back) in Diatomic Granular Crystals,” Europhys. Lett., 101(4), p. 44003. [CrossRef]
Herbold, E. B., Kim, J., Nesterenko, V. F., Wang, S., and Daraio, C., 2009, “Pulse Propagation in a Linear and Nonlinear Diatomic Periodic Chain: Effects of Acoustic Frequency Band-Gap,” Acta Mech., 205(1–4), pp. 85–103. [CrossRef]
Breindel, A., Sun, D., and Sen, S., 2011, “Impulse Absorption Using Small, Hard Panels of Embedded Cylinders With Granular Alignments,” App. Phys. Lett., 99(6), p. 063510. [CrossRef]
Tournat, V., Gusev, V. E., and Castagnede, B., 2004, “Self-Demodulation of Elastic Waves in a One-Dimensional Granular Chain,” Phys. Rev. E, 70(5), p. 056603. [CrossRef]
Ganesh, R., and Gonella, S., 2014, “Invariants of Nonlinearity in the Phononic Characteristics of Granular Chains,” Phys. Rev. E, 90(2), p. 023205. [CrossRef]
Cabaret, J., Tournat, V., and Bequin, P., 2012, “Amplitude-Dependent Phononic Processes in a Diatomic Granular Chain in the Weakly Nonlinear Regime,” Phys. Rev. E, 86(4), p. 041305. [CrossRef]
Coste, C., and Gilles, B., 1998, “On the Validity of Hertz Contact Law for Granular Material Acoustics,” Eur. Phys. J. B, 7(1), pp. 155–168. [CrossRef]
Pal, R. K., Awasthi, A. P., and Geubelle, P. H., 2013, “Wave Propagation in Elasto-Plastic Granular Systems,” Granular Matter, 15(6), pp. 747–758. [CrossRef]
Pal, R. K., Awasthi, A. P., and Geubelle, P. H., 2014, “Characterization of Wave Propagation in Elastic and Elastoplastic Granular Chains,” Phys. Rev. E, 89(1), p. 012204. [CrossRef]
Wang, E., Manjunath, M., Awasthi, A. P., Pal, R. K., Geubelle, P. H., and Lambros, J., 2014, “High-Amplitude Elastic Solitary Wave Propagation in 1-D Granular Chains With Preconditioned Beads: Experiments and Theoretical Analysis,” J. Mech. Phys. Solids, 72, pp. 161–173. [CrossRef]
Shoaib, M., and Kari, L., 2011, “Discrete Element Simulations of Elastoplastic Shock Wave Propagation in Spherical Particles,” Adv. Acoust. Vib., 2011, pp. 1–9. [CrossRef]
Thornton, C., 1995, “Coefficient of Restitution for Collinear Collisions of Elastic-Perfectly Plastic Spheres,” ASME J. Appl. Mech., 62(2), pp. 383–386. [CrossRef]
Vu-Quoc, L., Zhang, X., and Lesburg, L., 1999, “A Normal Force–Displacement Model for Contacting Spheres Accounting for Plastic Deformation: Force-Driven Formulation,” ASME J. Appl. Mech., 67(2), pp. 363–371. [CrossRef]
Wang, E., On, T., and Lambros, J., 2013, “An Experimental Study of the Dynamic Elasto-Plastic Contact Behavior of Dimer Metallic Granules,” Exp. Mech., 53(5), pp. 883–892. [CrossRef]
Burgoyne, H. A., and Daraio, C., 2014, “Strain-Rate-Dependent Model for the Dynamic Compression of Elastoplastic Spheres,” Phys. Rev. E, 89(3), p. 032203. [CrossRef]
On, T., LaVigne, P. A., and Lambros, J., 2014, “Development of Plastic Nonlinear Waves in One-Dimensional Ductile Granular Chains Under Impact Loading,” Mech. Mater., 68, pp. 29–37. [CrossRef]
Pal, R. K., Morton, J., Wang, E., Lambros, J., and Geubelle, P. H., 2015, “Impact Response of Elasto-Plastic Granular Chains Containing an Intruder Particle,” ASME J. Appl. Mech., 82(1), p. 011002. [CrossRef]
On, T., Wang, E., and Lambros, J., “Plastic Waves in One-Dimensional Heterogeneous Granular Chains Under Impact Loading: Single Intruders and Dimer Chains,” Int. J. Solids Struct.,61, pp. 81–90. [CrossRef]
Wang, E., Geubelle, P., and Lambros, J., 2013, “An Experimental Study of the Dynamic Elasto-Plastic Contact Behavior of Metallic Granules,” ASME J. Appl. Mech., 80(2), p. 021009. [CrossRef]
Gray, G. T., 2000, “Classic Split-Hopkinson Pressure Bar Testing,” ASM Handbook: Mechanical Testing and Evaluation, Vol. 8, ASM International, Novelty, OH.
Daraio, C., Nesterenko, V. F., Herbold, E. B., and Jin, S., 2005, “Strongly Nonlinear Waves in a Chain of Teflon Beads,” Phys. Rev. E, 72(1), p. 016603. [CrossRef]
Ashcroft, N. W., and Mermin, N. D., 1976, Solid State Physics, Saunders College Publishing, Orlando, FL.
Hascoet, E., Herrmann, H. J., and Loreto, V., 1999, “Shock Propagation in a Granular Chain,” Phys. Rev. E, 59(3), pp. 3202–3206. [CrossRef]
Tasi, J., 1980, “Evolution of Shocks in a One Dimensional Lattice,” J. Appl. Phys., 51(11), pp. 5804–5815. [CrossRef]
Zhuang, S., Ravichandran, G., and Grady, D. E., 2003, “An Experimental Investigation of Shock Wave Propagation in Periodically Layered Composites,” J. Mech. Phys. Solids, 52(2), pp. 245–265. [CrossRef]

Figures

Grahic Jump Location
Fig. 1

(a) Schematics of the three force–displacement relations used in the numerical simulations: elastic–plastic (blue, right-most curve), Hertzian, elastic (green, left-most curve), and linear (red, centered curve). Dotted vertical lines show transitions between regions in the elastic–plastic model with δy representing the displacement at which plasticity initiates and δp representing the displacement at which the linear regime begins. (b) Schematic diagram of the experimental setup showing the Hopkinson pressure bar with the laser vibrometer. The samples consisted of chains of 50 spheres, 6.35 mm in diameter, enclosed in a 3D printed tube with a window for the laser to measure the particle velocity of the 40th sphere.

Grahic Jump Location
Fig. 2

Numerical simulations comparing the dynamic response of chains of 50 particles, subjected to a 20 m/s constant velocity, governed by different contact dynamics. The different color curves represent all particle velocities after the arrival of the initial wave front. The velocities curves were translated based on the arrival time of the wave on each particle. (a) Response of a Hertzian chain; (b) response of a harmonic chain of linear springs; and (c) response of a chain of elastic–plastic particles. Arrows indicate the movement of the velocity wave front at progressively later positions in the chain. The arrow points toward the steady wave front that is formed in (a) and (c) in the Hertzian and elastic–plastic cases, respectively, while in linear case the wave front continues to spread in the direction of the arrow due to dispersion.

Grahic Jump Location
Fig. 3

Numerical results of parametric studies of the propagating wave speed as a function of (a) the density and (b) stiffness of the particles material. The plots compare the results obtained for chains with elastic–plastic contacts (blue, bottom-most curves), Hertzian contacts (green, top-most curves), and linear contacts (red, centered curves), subjected to a 2Fp impulse. (a) Markers represent the average wave speed observed in each DEM simulation with solid lines showing the wave speed dependence on the square root of the inverse of density. (b) Markers represent the average wave speed observed in each DEM simulation with solid lines showing the wave speed dependence on either the square root or the cube root of the stiffness. (c) Numerical results showing the dependence of the normalized wave velocity on the normalized applied force, plotted on a logarithmic scale. Markers represent the average wave speed observed in each DEM simulation, with “×” representing elastic-plastic and “+” representing Hertzian. The green, left-most curve shows the dependence of the wave speed in the Hertzian material on F1/6. The red, horizontal line is the maximum velocity component of a harmonic chain given by Eq. (9). The black, vertical, dashed line shows Fp, the force at which the linear regime begins in the elastic–plastic material and the local wave speed begins to asymptotically approach the bound, as the amplitude increases.

Grahic Jump Location
Fig. 4

Experimentally measured forces within the incident and transmission bars of the Hopkinson bar setup as measured by the strain gauges for 25 and 50 particle chains of uniform stainless steel 440 c spheres. For the two experiments with 25 particles chains, the cyan and yellow curves (overlapping larger pulses) show the forces in the incident bar while the blue and red curves (overlapping small pulses also shown in inset) show the forces in the transmission bar. For the experiment with a 50 particle chain, the magenta curve (overlapping large pulse) shows the forces in the incident bar while the green curve (small pulse also shown in inset) shows the forces in the transmission bar. The inset shows the transmitted forces for the three experiments zoomed in to show repeatability between experiments as well as the influence of experimental noise.

Grahic Jump Location
Fig. 5

Experimental results obtained with the Hopkinson bar setup compared with corresponding numerical simulations for (a) uniform chains of aluminum particles, (b) uniform chains of stainless steel 302 particles, and (c) dimer chains. The blue and red (top-most and bottom-most) curves represent experimentally measured velocities for the first (blue, top) and the 40th (red, bottom) spheres. The yellow, magenta, cyan, and green (progressively lower) curves represent numerical simulation for the 10th, 20th, 30th, and 40th particles in the chains. The dashed dark-gray line in (b) shows the numerical results obtained when strain-rate dependence is ignored and the yield stress is assumed to be the same as the quasi-static yield stress of stainless steel 302. (d) Plot of the simulated contact force between the first and second particles after the experimentally measured velocity profile is applied, and the simulated force profile of the final bead in 50 sphere chains of aluminum particles (blue, bottom-most), stainless steel 302 particles (green, top-most), and an alternating dimer (red, centered).

Grahic Jump Location
Fig. 6

X-T diagrams showing the wave propagation in time through chains of 50 particles, assembled with particles of different materials. All chains were excited by a pulse of amplitude 2FP and 100 × 10−6 s duration. The color scale represents the contact forces between particles, normalized by the applied force. (a) Chain of aluminum particles, (b) chain of stainless steel 302 particles, and (c) dimer chain consisting of alternating stainless steel 302 and aluminum.

Grahic Jump Location
Fig. 7

(a) Arrival speed of the wave front as it travel through chains of 50-particles of different materials (Table 1). Numerical results (lines) are compared with experiments (markers) for the particle’s velocity measured by the laser vibrometer in the 40th particle. Solid lines represent the arrival speeds in uniform chains. Dashed lines show the same results for dimer chains. (b) Numerical results showing the local wave speed as it travel through the 50-particles chains, normalized by the wave speed bound given in Eq. (10).

Grahic Jump Location
Fig. 8

Surface plot relating the maximum wave front speed as a function of the linear stiffness of the contact and the average density of the constituent particles, calculated using Eq. (10). Markers and stems show locations of uniform chains of particles with materials listed in Table 1 (blue, circle markers), alternating dimers of those materials (red, triangle markers), and engineered particles such as heavy core aluminum and hollow tungsten carbide (black, square markers).

Tables

Errata

Discussions

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In