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An Experimental and Theoretical Study of the Normal Coefficient of Restitution for Marble Spheres

  • Yang Ye
  • Yawu Zeng
  • Klaus Thoeni
  • Anna Giacomini
Original Paper
  • 38 Downloads

Abstract

The normal coefficient of restitution (NCOR) is a useful index to quantify the energy dissipation during impact. This study presents experimental tests of marble spheres impacting a plate. The effects of the sphere diameter, elastic properties of the plate, impact velocity, and repeated impacts on the NCOR were investigated. Three fracture phases were observed: no macrocrack, macrocrack, and fragmentation. A clear influence of the propagation of the macrocracks on the NCOR was observed. Macrocracks also cause increased NCOR variability. The cumulative damage caused by macrocracks can affect the propagation of macrocracks and fragmentation. In the no macrocrack phase, the NCOR decreases with increasing velocity and with decreasing diameter, and the velocity effect of the NCOR is also related to the size of the marble sphere. The proposed average rate of contact stress correlates well with the NCOR and can describe fully the velocity and size effects of the NCOR, which provides a simple way to consider the complex velocity and size effects in rockfall simulations. The dissipation caused by microcracks and viscosity can be considered simultaneously through the average rate of contact stress. The marble sphere NCOR decreases with increasing elasticity modulus of the plate, and this conclusion is verified by the elastic-perfectly plastic contact theory. Finally, viscoelastic contact theory is used to describe the NCOR, which proves that the decrease in the NCOR with decreasing diameter is reasonable and can be used to predict a decrease in the NCOR with an increase in velocity.

Keyword

Impact Viscoelastic contact theory Fracture phases Energy dissipation mechanism Fragmentation Size effect 

List of Symbols

a

Radius of the contact area

c

Propagation velocity of quasi-longitudinal waves

dn

Average diameter of a sphere

ds

Maximum diameter of a sphere

E1

Elasticity modulus of the sphere

E2

Elasticity modulus of the plate

E

Equivalent modulus of elasticity

F

Contact force

Fel

Elastic contact force

Fel,max

Maximum elastic contact force

g

Acceleration of gravity

h

Free fall height

hp

Half the thickness of the plate

H

Thickness of the plate

m1

Mass of the sphere

m2

Mass of plate

m

Equivalent mass

NCOR

Normal coefficient of restitution

P0

Largest contact stress in the contact area

P0,max

Maximum contact stress during collision

P0,ci

Crack initiation stress for contact damage

P0,cd

Crack damage stress for contact damage

R1

Radius of the sphere

R2

Radius of the plate

R

Equivalent radius

smax

Maximum contact displacement

t

Time

Δt

Measured time interval

tc

Time when the contact force is at its maximum

tR

Time when the contact force reaches zero

Δtn

Time interval between the n−1th and the nth collision

Δtn+1

Time interval between the nth and the n + 1th collision

tel

Total time of elastic collision

v

Velocity of a sphere

va

Velocity after collision

vb

Velocity before collision or impact velocity

vci

Crack initiation velocity of a sphere

vcd

Crack damage velocity of a sphere

Wkin,a

Kinetic energy after collision

Wkin

Kinetic energy before collision

Wdiss

Dissipative kinetic energy during collision

Wel

Elastic deformation energy

λ

Inelasticity parameter

κ

Average rate of contact stress

θ, ψ, ξ

Fitting parameters

ρ

Sphericity of a sphere

ρ1

Density of the sphere

ρ2

Density of the plate

μ1

Poisson’s ratio of the sphere

µ2

Poisson’s ratio of the plate

σci

Crack initiation stress

σcd

Crack damage stress

σcf

Yield stress of the local contact deformation

σf

Peak stress

σy

Yield stress according to elastic–plastic theory

Notes

Acknowledgements

This work was funded by the National Natural Science Foundation of China (CN) (Grant 41772308). Yang Ye was supported by the China Scholarship Council as a visiting student at the University of Newcastle (Grant number: 201706270107). All this support is gratefully acknowledged. Finally, the authors would also like to thank the anonymous reviewers for their comments and suggestions to improve the manuscript.

References

  1. Alizadeh E, Bertrand F, Chaouki J (2013) Development of a granular normal contact force model based on a non-Newtonian liquid filled dashpot. Powder Technol 237(3):202–212CrossRefGoogle Scholar
  2. Ansari MK, Ahmad M, Singh R, Singh TN (2015) correlation between schmidt hardness and coefficient of restitution of rocks. J Afr Earth Sci 104(C):1–5CrossRefGoogle Scholar
  3. Antonyuk S, Heinrich S, Tomas J, Deen NG, Buijtenen MSV, Kuipers JAM (2010) Energy absorption during compression and impact of dry elastic-plastic spherical granules. Granul Matter 12(1):15–47CrossRefGoogle Scholar
  4. Asteriou P, Tsiambaos G (2018) Effect of impact velocity, block mass and hardness on the coefficients of restitution for rockfall analysis. Int J Rock Mech Min 106:41–50CrossRefGoogle Scholar
  5. Asteriou P, Saroglou H, Tsiambaos G (2012) Geotechnical and kinematic parameters affecting the coefficients of restitution for rock fall analysis. Int J Rock Mech Min 54(3):103–113CrossRefGoogle Scholar
  6. Asteriou P, Saroglou H, Tsiambaos G (2013) Rockfall: scaling factors for the coefficient of restitution. In ISRM International Symposium-EUROCK, pp 109–113Google Scholar
  7. Auberger M, Rinehart JS (1960) Energy loss associated with impact of steel spheres on rocks. J Geophys Res 65(12):4157–4164CrossRefGoogle Scholar
  8. Brace WF, Paulding JBW, Scholz CH (1966) Dilatancy in the fracture of crystalline rocks. J Geophys Res 71(16):3939–3953CrossRefGoogle Scholar
  9. Brake MR (2012) An analytical elastic-perfectly plastic contact model. Int J Solids Struct 49(22):3129–3141CrossRefGoogle Scholar
  10. Brake MRW (2015) An analytical elastic plastic contact model with strain hardening and frictional effects for normal and oblique impacts. Int J Solids Struct 62:104–123CrossRefGoogle Scholar
  11. Cai M, Kaiser PK, Tasaka Y, Maejima T, Morioka H, Minami M (2004) Generalized crack initiation and crack damage stress thresholds of brittle rock masses near underground excavations. Int J Rock Mech Min 41(5):833–847CrossRefGoogle Scholar
  12. Chai B, Tang Z, Zhang A, Du J, Su H, Yi H (2015) An uncertainty method for probabilistic analysis of buildings impacted by rockfall in a limestone quarry in fengshan, southwestern china. Rock Mech Rock Eng 48(5):1981–1996CrossRefGoogle Scholar
  13. Chau KT, Wong RHC, Wu JJ (2002) Coefficient of restitution and rotational motions of rockfall impacts. Int J Rock Mech Min 39(1):69–77CrossRefGoogle Scholar
  14. Darlington WJ, Ranjith PG, Choi SK (2011) The effect of specimen size on strength and other properties in laboratory testing of rock and rock-like cementitious brittle materials. Rock Mech Rock Eng 44(5):513–529CrossRefGoogle Scholar
  15. Dmytro A, Elliott JA, Hancock BC (2011) Effect of particle size on energy dissipation in viscoelastic granular collisions. Phys Rev E 84(2):1713–1724Google Scholar
  16. Du Y, Wang S (2009) Energy dissipation in normal elastoplastic impact between two spheres. J Appl Mech 76(6):1–8CrossRefGoogle Scholar
  17. Eberhardt E, Stead D, Stimpson B (1999) Quantifying progressive pre-peak brittle fracture damage in rock during uniaxial compression. Int J Rock Mech Min 36(3):361–380CrossRefGoogle Scholar
  18. Farin M, Mangeney A, Toussaint R et al (2015) Characterization of rockfalls from seismic signal: Insights from laboratory experiments. J Geophys Res-Sol Ea 120(10):7102–7137CrossRefGoogle Scholar
  19. Giacomini A, Buzzi O, Renard B, Giani GP (2009) Experimental studies on fragmentation of rock falls on impact with rock surfaces. Int J Rock Mech Min 46(4):708–715CrossRefGoogle Scholar
  20. Giacomini A, Thoeni K, Lambert C, Sloan SW, Booth S (2012) Experimental study on rockfall drapery systems for open pit highwalls. Int J Rock Mech Min 56(12):171–181CrossRefGoogle Scholar
  21. Hardy C, Baronet CN, Tordion GV (1971) The elasto-plastic indentation of a half-space by a rigid sphere. Int J Numer Meth Eng 3(4):451–462CrossRefGoogle Scholar
  22. Heidenreich B (2004) Small-and half-scale experimental studies of rockfall impacts on sandy slopes. PhD thesis, Swiss Federal Institute of Technology of Lausanne, SwissGoogle Scholar
  23. Hungr O, Evans SG (1989) Engineering aspects of rockfall hazards in Canada. Geological survey of CanadaGoogle Scholar
  24. Imre B, Räbsamen S, Springman SM (2008) A coefficient of restitution of rock materials. Comput Geosci 34(4):339–350CrossRefGoogle Scholar
  25. Johnson KL (1987) Contact mechanics. Cambridge University, CambridgeGoogle Scholar
  26. Jones CL, Higgins JD, Andrew RD (2000) Colorado rockfall simulation program version 4.0. Colorado Geological Survey, ColoradoGoogle Scholar
  27. Kamijo A, Onda S, Masuya H, Tanaka Y (2000) Fundamental test on restitution coefficient and frictional coefficient of rock fall. In 5th Symposium on Impact Problems in Civil Engineering, pp 83–86Google Scholar
  28. Kanada T (1995) Evaluation of spherical form errors—computation of sphericity by means of minimum zone method and some examinations with using simulated data. Precis Eng 17(4):281–289CrossRefGoogle Scholar
  29. Khanal M, Schubert W, Tomas J (2008) Compression and impact loading experiments of high strength spherical composites. Int J Miner Process 86(1–4):104–113CrossRefGoogle Scholar
  30. Kuwabara G, Kono K (1987) Restitution coefficient in a collision between two spheres. Jpn J Appl Phys 26(8):1230–1233CrossRefGoogle Scholar
  31. Labiouse V, Heidenreich B (2009) Half-scale experimental study of rockfall impacts on sandy slopes. Nat Hazard Earth Sys 9(6):1981–1993CrossRefGoogle Scholar
  32. Labous L, Rosato AD, Dave RN (1997) Measurements of collisional properties of spheres using high-speed video analysis. Phys Rev E 56(5):5717–5725CrossRefGoogle Scholar
  33. Liu J, Xie H, Hou Z, Yang C, Chen L (2014) Damage evolution of rock salt under cyclic loading in unixial tests. Acta Geotech 9(1):153–160CrossRefGoogle Scholar
  34. Loland KE (1980) Continuous damage model for load—response estimation of concrete. Cement Cement Concrete Res 10(3):395–402CrossRefGoogle Scholar
  35. Luding S, Clément E, Blumen A, Rajchenbach J, Duran J (1994) Anomalous energy dissipation in molecular-dynamics simulations of grains: the ‘‘detachment’’effect. Phys Rev E 50(5):4113–4122CrossRefGoogle Scholar
  36. Macciotta R, Martin CD, Cruden DM (2015) Probabilistic estimation of rockfall height and kinetic energy based on a three-dimensional trajectory model and Monte Carlo simulation. Landslides 12(4):757–772CrossRefGoogle Scholar
  37. Müller P, Trüe M, Böttcher R et al (2015) Acoustic evaluation of the impact of moist spherical granules and glass beads. Powder Technol 278:138–149CrossRefGoogle Scholar
  38. Peng B (2000) Rockfall trajectory analysis: Parameter determination and application. Master thesis, University of CanterburyGoogle Scholar
  39. Pfeiffer TJ, BOWEN TD (1989) Computer simulation of rockfalls. Bull Assoc Eng Geol 26(1):135–146Google Scholar
  40. Rammer W, Brauner M, Dorren LKA, Berger F, Lexer MJ (2010) Evaluation of a 3-D rockfall module within a forest patch model. Nat Hazard Earth Sys 10(4):699–711CrossRefGoogle Scholar
  41. Reynolds GK, Fu JS, Cheong YS, Hounslow MJ, Salman AD (2005) Breakage in granulation: a review. Chem Eng Sci 60(14):3969–3992CrossRefGoogle Scholar
  42. Richards LR, Peng B, Bell DH (2001) Laboratory and field evaluation of the normal coefficient of restitution for rocks. In Proceedings of Eurock, pp 149–156Google Scholar
  43. Ritchie AM (1963) Evaluation of rockfall and its control. Highway Res Rec 7:13–28Google Scholar
  44. Seifried R, Schiehlen W, Eberhard P (2005) Numerical and experimental evaluation of the coefficient of restitution for repeated impacts. Int J Impact Eng 32(1–4):508–524CrossRefGoogle Scholar
  45. Shi F (2016) A review of the applications of the JK size-dependent breakage model: Part 1: Ore and coal breakage characterisation. Int J Miner Process 155:118–129CrossRefGoogle Scholar
  46. Thornton C (1997) Coefficient of restitution for collinear collisions of elastic-perfectly plastic spheres. J Appl Mech 64(2):383–386CrossRefGoogle Scholar
  47. Tomas J, Schreier M, Gröger T, Ehlers S (1999) Impact crushing of concrete for liberation and recycling. Powder Technol 105(1–3):39–51CrossRefGoogle Scholar
  48. Ushiro T, Shinohara S, Tanida K, Yagi N (2000) A study on the motion of rockfalls on slopes. In 5th Symposium on Impact Problems in Civil Engineering, pp 91–96Google Scholar
  49. Volkwein A, Schellenberg K, Labiouse V, Agliardi F, Berger F, Bourrier F, Jaboyedoff M (2011) Rockfall characterisation and structural protection-a review. Nat Hazard Earth Sys 11:2617–2651CrossRefGoogle Scholar
  50. Wasantha PLP, Ranjith PG, Zhao J, Shao SS, Permata G (2015) Strain rate effect on the mechanical behaviour of sandstones with different grain sizes. Rock Mech Rock Eng 48(5):1833–1848CrossRefGoogle Scholar
  51. Wu SZ, Chau KT, Yu TX (2004) Crushing and fragmentation of brittle spheres under double impact test. Powder Technol 143:41–55CrossRefGoogle Scholar
  52. Ye Y, Zeng Y (2017) A size-dependent viscoelastic normal contact model for particle collision. Int J Impact Eng 106:120–132CrossRefGoogle Scholar
  53. Zener C (1941) The intrinsic inelasticity of large plates. Phys Rev 59(8):669–673CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Austria, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of Civil EngineeringWuhan UniversityWuhanPeople’s Republic of China
  2. 2.Centre for Geotechnical Science and EngineeringThe University of NewcastleCallaghanAustralia

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