Research Papers: Contact Mechanics

Experimental Investigations on the Coefficient of Restitution for Sphere–Thin Plate Elastoplastic Impact

[+] Author and Article Information
Deepak Patil

Mechanical Engineering Department,
Carnegie Mellon University,
Pittsburgh, PA 15213

C. Fred Higgs, III

Mechanical Engineering Department,
Carnegie Mellon University,
Pittsburgh, PA 15213
e-mail: higgs@rice.edu

1Corresponding author.

Contributed by the Tribology Division of ASME for publication in the JOURNAL OF TRIBOLOGY. Manuscript received March 11, 2017; final manuscript received June 29, 2017; published online August 22, 2017. Assoc. Editor: Sinan Muftu.

J. Tribol 140(1), 011406 (Aug 22, 2017) (13 pages) Paper No: TRIB-17-1081; doi: 10.1115/1.4037212 History: Received March 11, 2017; Revised June 29, 2017

In multiparticle simulations of industrial granular systems such as hoppers, tumblers, and mixers, the particle energy dissipation is governed by an important input parameter called the coefficient of restitution (COR). Oftentimes, the wall thickness in these systems is on the order of a particles diameter or less. However, the COR value implemented in event-driven simulations is either constant or a monotonically decreasing function of the impact velocity. The present work experimentally investigates the effect of wall thickness on the COR through sphere–thin plate elastoplastic impacts and elucidates the underlying impact phenomena. Experiments were performed on 0.635 cm and 0.476 cm diameter (d) spheres of various materials impacting aluminum 6061 plates of different thicknesses (t) with the low impact velocities up to 3.1 m/s. Besides COR, indentation measurements and numerical simulations are performed to gain a detailed understanding of the contact process and energy dissipation mechanism. As the “t/d” ratio decreases, a considerable amount of energy is dissipated into flexural vibrations leading to a significantly lower COR value. Based on the results, it can be concluded that using a constant COR input value in particle simulations may not always be an appropriate choice, especially, in the case of thin plates. However, these new COR results validate that when the wall thickness is more than twice the sphere diameter (i.e., t/d > 2), a constant COR value obtained for an impact with semi-infinite plate can be reasonably used.

Copyright © 2018 by ASME
Your Session has timed out. Please sign back in to continue.


Bertrand, F. , Leclaire, L.-A. , and Levecque, G. , 2005, “ DEM-Based Models for the Mixing of Granular Materials,” Chem. Eng. Sci., 60(8), pp. 2517–2531. [CrossRef]
Radjaï, F. , and Dubois, F. , 2011, Discrete-Element Modeling of Granular Materials, Wiley/ISTE, Hoboken, NJ.
Ketterhagen, W. R. , Curtis, J. S. , and Wassgren, C. R. , 2005, “ Stress Results From Two-Dimensional Granular Shear Flow Simulations Using Various Collision Models,” Phys. Rev. E, 71(6), p. 061307. [CrossRef]
Dahl, S. , Clelland, R. , and Hrenya, C. , 2003, “ Three-Dimensional, Rapid Shear Flow of Particles With Continuous Size Distributions,” Powder Technol., 138(1), pp. 7–12. [CrossRef]
Clelland, R. , and Hrenya, C. , 2002, “ Simulations of a Binary-Sized Mixture of Inelastic Grains in Rapid Shear Flow,” Phys. Rev. E, 65(3), p. 031301. [CrossRef]
Anand, A. , Curtis, J. S. , Wassgren, C. R. , Hancock, B. C. , and Ketterhagen, W. R. , 2008, “ Predicting Discharge Dynamics From a Rectangular Hopper Using the Discrete Element Method (DEM),” Chem. Eng. Sci., 63(24), pp. 5821–5830. [CrossRef]
Ketterhagen, W. R. , Curtis, J. S. , Wassgren, C. R. , Kong, A. , Narayan, P. J. , and Hancock, B. C. , 2007, “ Granular Segregation in Discharging Cylindrical Hoppers: A Discrete Element and Experimental Study,” Chem. Eng. Sci., 62(22), pp. 6423–6439. [CrossRef]
McCarthy, J. , Jasti, V. , Marinack, M. , and Higgs, C. , 2010, “ Quantitative Validation of the Discrete Element Method Using an Annular Shear Cell,” Powder Technol., 203(1), pp. 70–77. [CrossRef]
Jasti, V . K. , and Higgs, C. F. , 2010, “ A Fast First Order Model of a Rough Annular Shear Cell Using Cellular Automata,” Granular Matter, 12(1), pp. 97–106. [CrossRef]
Marinack, M. C. , and Higgs, C. F. , 2011, “ The Inclusion of Friction in Lattice-Based Cellular Automata Modeling of Granular Flows,” ASME J. Tribol., 133(3), p. 031302. [CrossRef]
Marinack, M. C. , and Higgs, C. F. , 2015, “ Three-Dimensional Physics-Based Cellular Automata Model for Granular Shear Flow,” Powder Technol., 277, pp. 287–302. [CrossRef]
Marinack, M. C. , Mpagazehe, J. N. , and Higgs, C. F. , 2012, “ An Eulerian, Lattice-Based Cellular Automata Approach for Modeling Multiphase Flows,” Powder Technol., 221, pp. 47–56. [CrossRef]
Dougherty, P. S. , Marinack, M. C. , Patil, D. , Evans, R. D. , and Higgs, C. F. , 2016, “ The Influence of W-DLC and CrxN Thin Film Coatings on Impact Damage Between Bearing Materials,” Tribol. Trans., 59(6), pp. 1104–1113. [CrossRef]
Patil, D. , Marinack, M. C., Jr. , DellaCorte, C. , and Higgs, C. F., III , 2017, “ Experimental Investigations of the Superelastic Impact Performance of Nitinol 60,” Tribol. Trans., 60(4), pp. 615–620.
Goldsmith, W. , and Frasier, J. , 1961, Impact: The Theory and Physical Behavior of Colliding Solids, Vol. 28, Dover Publications, Mineola, NY.
Tabor, D. , 1948, “ A Simple Theory of Static and Dynamic Hardness,” Proc. R. Soc. London, Ser. A, 192(1029), pp. 247–274. [CrossRef]
Johnson, K. L. , 1987, Contact Mechanics, Cambridge University Press, Cambridge, UK.
Kharaz, A. , and Gorham, D. , 2000, “ A Study of the Restitution Coefficient in Elastic–Plastic Impact,” Philos. Mag. Lett., 80(8), pp. 549–559. [CrossRef]
Koller, M. , and Kolsky, H. , 1987, “ Waves Produced by the Elastic Impact of Spheres on Thick Plates,” Int. J. Solids Struct., 23(10), pp. 1387–1400. [CrossRef]
Vincent, J. , 1900, “ Experiments on Impact,” Proc. Cambridge Philos. Soc., 8, pp. 332–357.
Reed, J. , 1985, “ Energy Losses Due to Elastic Wave Propagation During an Elastic Impact,” J. Phys. D: Appl. Phys., 18(12), pp. 2329–2337. [CrossRef]
Raman, C. , 1920, “ On Some Applications of Hertz's Theory of Impact,” Phys. Rev., 15(4), pp. 277–284. [CrossRef]
Tillett, J. , 1954, “ A Study of the Impact of Spheres on Plates,” Proc. Phys. Soc. Sect. B, 67(9), pp. 677–688. [CrossRef]
Zener, C. , 1941, “ The Intrinsic Inelasticity of Large Plates,” Phys. Rev., 59(8), pp. 669–673. [CrossRef]
Sondergaard, R. , Chaney, K. , and Brennen, C. , 1990, “ Measurements of Solid Spheres Bouncing Off Flat Plates,” ASME J. Appl. Mech., 112(3), pp. 694–699. [CrossRef]
Marinack, M. C., Jr. , Musgrave, R. E. , and Higgs, C. F., III , 2013, “ Experimental Investigations on the Coefficient of Restitution of Single Particles,” Tribol. Trans., 56(4), pp. 572–580. [CrossRef]
Patil, D. , and Higgs, C. F., III , 2017, “ Critical Plate Thickness for Energy Dissipation During Sphere–Plate Elastoplastic Impact Involving Flexural Vibrations,” ASME J. Tribol., 139(4), p. 041104. [CrossRef]
Zygo, 2010, “ MetroPro Reference Guide, OMP-0347L,” Zygo Corp., Middlefield, CT. http://zeus.phys.uconn.edu/halld/diamonds/Zygo/MetroPro_docs/MetroPro%20Reference%20Guide%200347_K.pdf
Montaine, M. , Heckel, M. , Kruelle, C. , Schwager, T. , and Pöschel, T. , 2011, “ Coefficient of Restitution as a Fluctuating Quantity,” Phys. Rev. E, 84(4), p. 041306. [CrossRef]
Marinack, M. C. , Jasti, V . K. , Choi, Y. E. , and Higgs, C. F. , 2011, “ Couette Grain Flow Experiments: The Effects of the Coefficient of Restitution, Global Solid Fraction, and Materials,” Powder Technol., 211(1), pp. 144–155. [CrossRef]
Kolsky, H. , 1963, Stress Waves in Solids, Vol. 1098, Dover Publications, Mineola, NY.
Müller, P. , Heckel, M. , Sack, A. , and Pöschel, T. , 2013, “ Complex Velocity Dependence of the Coefficient of Restitution of a Bouncing Ball,” Phys. Rev. Lett., 110(25), p. 254301. [CrossRef] [PubMed]
Weir, G. , and Tallon, S. , 2005, “ The Coefficient of Restitution for Normal Incident, Low Velocity Particle Impacts,” Chem. Eng. Sci., 60(13), pp. 3637–3647. [CrossRef]
ANSYS, 2012, “ ANSYS Academic Research, Release 14.5, Help System, ANSYS Theory Reference Guide,” ANSYS, Inc., Canonsburg, PA.
Hertz, H. , 1882, “ Über die berührung fester elastischer Körper (On the Contact of Solid Elastic Bodies),” J. Reine Angew. Math., 92, pp. 156–171.
Jackson, R. L. , and Green, I. , 2005, “ A Finite Element Study of Elasto-Plastic Hemispherical Contact Against a Rigid Flat,” ASME J. Tribol., 127(2), pp. 343–354. [CrossRef]
Ghaednia, H. , Marghitu, D. B. , and Jackson, R. L. , 2014, “ Predicting the Permanent Deformation After the Impact of a Rod With a Flat Surface,” ASME J. Tribol., 137(1), p. 011403. [CrossRef]
Kraus, D. , 2014, “ Consolidated Data Analysis and Presentation Using an Open-Source Add-In for the Microsoft Excel® Spreadsheet Software,” Med. Writing, 23(1), pp. 25–28. [CrossRef]


Grahic Jump Location
Fig. 2

Plate installation on the hollow metal support: (a) supporting metal frame, (b) plate mounted on the frame, and (c) clamped plate

Grahic Jump Location
Fig. 1

Drop test apparatus (COR rig): (a) laboratory photograph and (b) schematic side view (A) hollow metal support, (B) plate, (C) sphere, (D) sphere holder, (E) Plexiglass casing, (F) suction device, (G) hose, and (H) air pump

Grahic Jump Location
Fig. 3

COR as a function of distance from plate support for tungsten carbide sphere (d = 0.635 cm) impacting aluminum 6061 plates at Vi = 3.1 m/s

Grahic Jump Location
Fig. 12

Indentation depth on various plates after impact with S2 tool steel sphere (d = 0.635 cm)

Grahic Jump Location
Fig. 4

Coefficient of restitution results for brass spheres: (a) d = 0.476 cm and (b) d = 0.635 cm

Grahic Jump Location
Fig. 5

COR as a function t/d ratio for various materials, results plotted for impact velocity Vi = 1.1 m/s

Grahic Jump Location
Fig. 6

COR as a function t/d ratio for various materials, results plotted for impact velocity Vi = 2.3 m/s

Grahic Jump Location
Fig. 7

COR as a function t/d ratio for various materials, results plotted for impact velocity Vi = 3.1 m/s

Grahic Jump Location
Fig. 18

Relative energy loss between plastic deformation and flexural vibrations during the impact process, brass sphere (d = 0.635 cm) impacting at Vi = 2.3 m/s

Grahic Jump Location
Fig. 10

Indentation depth on various plates after impact with brass sphere (d = 0.635 cm)

Grahic Jump Location
Fig. 11

Indentation depth on various plates after impact with tungsten carbide sphere (d = 0.635 cm)

Grahic Jump Location
Fig. 13

Finite element model mesh layout

Grahic Jump Location
Fig. 14

Comparison of experimental and simulation COR results for various materials

Grahic Jump Location
Fig. 15

Contact forces developed during the impact for various plates, brass sphere (d = 0.635 cm) impacting at Vi = 2.3 m/s

Grahic Jump Location
Fig. 16

Sphere–plate energy transfer during the impact process, brass sphere (d = 0.635 cm) impacting at Vi = 2.3 m/s on plate t = 0.476 cm

Grahic Jump Location
Fig. 17

Stress (von Mises equivalent) and plastic strain contours at the instance of maximum contact force for brass sphere (d = 0.635 cm) impacting at Vi = 2.3 m/s on thickest and thinnest plate: (a) stress contours, t = 0.160 cm, (b) stress contours, t = 1.270 cm, (c) plastic strain contours, t = 0.160 cm, and (d) plastic strain contours, t = 1.270 cm

Grahic Jump Location
Fig. 8

Comparison between analytical solutions and the current work's experimental data of S2 steel sphere for impact velocity Vi = 2.3 m/s

Grahic Jump Location
Fig. 9

Plate indentation measurements: (a) contour plot, (b) cross section profile, and (c) crater image

Grahic Jump Location
Fig. 19

Experimental versus simulation results for indentation, brass sphere (d = 0.635 cm) impacting at Vi = 2.3 m/s



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