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

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## 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

*d*_{n}Average diameter of a sphere

*d*_{s}Maximum diameter of a sphere

*E*_{1}Elasticity modulus of the sphere

*E*_{2}Elasticity modulus of the plate

*E*^{∗}Equivalent modulus of elasticity

*F*Contact force

*F*_{el}Elastic contact force

*F*_{el,max}Maximum elastic contact force

*g*Acceleration of gravity

*h*Free fall height

*h*_{p}Half the thickness of the plate

*H*Thickness of the plate

*m*_{1}Mass of the sphere

*m*_{2}Mass of plate

*m*^{∗}Equivalent mass

- NCOR
Normal coefficient of restitution

*P*_{0}Largest contact stress in the contact area

*P*_{0,max}Maximum contact stress during collision

*P*_{0,ci}Crack initiation stress for contact damage

*P*_{0,cd}Crack damage stress for contact damage

*R*_{1}Radius of the sphere

*R*_{2}Radius of the plate

*R*^{∗}Equivalent radius

*s*_{max}Maximum contact displacement

*t*Time

- Δ
*t* Measured time interval

*t*_{c}Time when the contact force is at its maximum

*t*_{R}Time when the contact force reaches zero

- Δ
*t*_{n} Time interval between the

*n−1*th and the*n*th collision- Δ
*t*_{n+1} Time interval between the

*n*th and the*n*+*1*th collision*t*_{el}Total time of elastic collision

*v*Velocity of a sphere

*v*_{a}Velocity after collision

*v*_{b}Velocity before collision or impact velocity

*v*_{ci}Crack initiation velocity of a sphere

*v*_{cd}Crack damage velocity of a sphere

*W*_{kin,a}Kinetic energy after collision

*W*_{kin}Kinetic energy before collision

*W*_{diss}Dissipative kinetic energy during collision

*W*_{el}Elastic deformation energy

- λ
Inelasticity parameter

- κ
Average rate of contact stress

- θ, ψ, ξ
Fitting parameters

*ρ*Sphericity of a sphere

*ρ*_{1}Density of the sphere

*ρ*_{2}Density of the plate

*μ*_{1}Poisson’s ratio of the sphere

*µ*_{2}Poisson’s ratio of the plate

*σ*_{ci}Crack initiation stress

*σ*_{cd}Crack damage stress

*σ*_{cf}Yield stress of the local contact deformation

*σ*_{f}Peak stress

*σ*_{y}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.

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