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An improved calculation model for predicting the coefficient of restitution and maximum impact force of rockfalls

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Abstract

Rockfalls, which are a common natural hazard, collide and continuously bounce along slope surfaces during their movement; the coefficient of restitution (COR) and impact force are two important characteristic parameters related to the impact process of a rockfall. In this paper, considering the elastoplastic characteristics of the ground material, a theoretical model is proposed to obtain the normal COR and normal impact force at different stages of the impact process. Subsequently, the tangential COR and tangential force are calculated based on the improved friction theory. Finally, an engineering case is studied to illustrate the applicability of the proposed model. The calculation results show that the COR is positively correlated with the incident angle under the same incident velocity and negatively correlated with the incident velocity under the same incident angle; the impact force is linearly and positively correlated with the incident velocity. Under a given slope condition, the theory proposed in this paper can explain the locking phenomenon between the boulder and the slope during impact.

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Abbreviations

p 0 :

Maximum normal contact stress (Pa)

a :

Maximum contact radius in the elastic stage (m)

R*:

Equivalent radius (m)

R i :

Equivalent radius of the rockfall block (m)

δ :

Maximum indentation depth in the elastic phase (m)

E i :

Elastic modulus of rockfall (Pa)

E j :

Elastic modulus of cushion (Pa)

E* :

Equivalent elastic modulus (Pa)

v i :

Poisson’s ratio of rockfall block

v j :

Poisson’s ratio of cushion

δ y :

Indentation depth of the ground material in initial yield stage (m)

δ m :

Maximum indentation depth of ground material at any moment (m)

δ r :

Plastic residual deformation (m)

σ y :

Initial contact yield stress (Pa)

V 1 :

Initial normal velocity of the rockfall centroid (m/s)

V 2 :

Final normal velocity (m/s)

V :

Instantaneous velocity (m/s)

V y :

Initial yield velocity (m/s)

F y :

Initial yield force (N)

m :

Quality of rockfall (kg)

F em :

Maximum force in the elastic stage (N)

F epm :

Maximum force in the plastic stage (N)

F rep :

Maximum force in the rebound stage (N)

a m :

Maximum contact radius in yield stage (m)

R r :

Equivalent contact radius in the rebound stage (m)

F τ :

Tangential impact force (N)

F η :

Normal impact force (N)

κ :

Stiffness ratio

μ :

Friction coefficient of ground

G 1 :

Shear modulus of rockfall (Pa)

G 2 :

Shear modulus of ground (Pa)

\( \psi \) i :

Dimensionless factor related to the incident angle

θ i :

Incident angle (°)

λ :

Lamé coefficient (Pa)

H :

Vertical falling height (m)

g :

Gravitational acceleration (m/s2)

ρ :

Density of rockfall (kg/m3)

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Acknowledgments

We are very grateful to the editors and the anonymous reviewers for comments that considerably improved the initial manuscript.

Funding

This work was supported by the National Natural Science Foundation of China (41672283, 41731285) and the Major Scientific and Technological Support Research Subject for the Prevention and Control of Ecological Geological Disasters in “8.8” Jiuzhaigou Earthquake Stricken Area of Department of Natural Resources of Sichuan Province (KJ-2018-20).

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Authors and Affiliations

  1. Faculty of Geosciences and Environment Engineering, Southwest Jiaotong University, Chengdu, Sichuan, China

    Xuefeng Mei, Xiewen Hu, Gang Luo & Jianli Wu

  2. Engineering Laboratory combined with national and local of spatial information technology of high speed railway operation safety, Southwest Jiaotong University, Chengdu, Sichuan, China

    Xiewen Hu

  3. State Key Laboratory of Geomechanics and Geotechnical Engineering, Institute of Rock and Soil Mechanics, Chinese Academy of Sciences, Hubei, Wuhan, China

    Zhenhua Zhou

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  1. Xuefeng Mei
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Correspondence to Xuefeng Mei.

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Responsible Editor: Zeynal Abiddin Erguler

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Mei, X., Hu, X., Zhou, Z. et al. An improved calculation model for predicting the coefficient of restitution and maximum impact force of rockfalls. Arab J Geosci 13, 1081 (2020). https://doi.org/10.1007/s12517-020-06020-3

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