Rock Mechanics and Rock Engineering

, Volume 52, Issue 6, pp 1705–1722 | Cite as

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

  • Yang Ye
  • Yawu ZengEmail author
  • Klaus Thoeni
  • Anna Giacomini
Original Paper


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.


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

List of Symbols


Radius of the contact area


Propagation velocity of quasi-longitudinal waves


Average diameter of a sphere


Maximum diameter of a sphere


Elasticity modulus of the sphere


Elasticity modulus of the plate


Equivalent modulus of elasticity


Contact force


Elastic contact force


Maximum elastic contact force


Acceleration of gravity


Free fall height


Half the thickness of the plate


Thickness of the plate


Mass of the sphere


Mass of plate


Equivalent mass


Normal coefficient of restitution


Largest contact stress in the contact area


Maximum contact stress during collision


Crack initiation stress for contact damage


Crack damage stress for contact damage


Radius of the sphere


Radius of the plate


Equivalent radius


Maximum contact displacement




Measured time interval


Time when the contact force is at its maximum


Time when the contact force reaches zero


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


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


Total time of elastic collision


Velocity of a sphere


Velocity after collision


Velocity before collision or impact velocity


Crack initiation velocity of a sphere


Crack damage velocity of a sphere


Kinetic energy after collision


Kinetic energy before collision


Dissipative kinetic energy during collision


Elastic deformation energy


Inelasticity parameter


Average rate of contact stress

θ, ψ, ξ

Fitting parameters


Sphericity of a sphere


Density of the sphere


Density of the plate


Poisson’s ratio of the sphere


Poisson’s ratio of the plate


Crack initiation stress


Crack damage stress


Yield stress of the local contact deformation


Peak stress


Yield stress according to elastic–plastic theory



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.


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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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