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Abstract

In studying the motion law of rock-fall, the restitution coefficient is an important control parameter. It not only has something to do with stone's speed and quality but also is closely related with stone and overlay layer's physico-mechanical properties. However, currently there is no reasonable calculation theory about restitution coefficient which will still be a difficult problem in studying. According to Hertz contact theory and Cattaneo and Mindlin tangential contact theory, considering the elasto-plastic properties of materials, the calculation model of restitution coefficient both in normal and tangential impacting direction is studied. Then a specific formula is given, and some major factors about the coefficient are also illustrated. At last, by calculating an example and comparing with the existed conclusion, the theory is proved to be reasonable.
第 3卷第 3期 
年 3月  岩  土  力  学 
Ro  So M e 
  N 
M a  
文章编号 1-0-0 
 
明 L ,吴 永 ,李新坡  
中国科 学院 山地 灾害与地表过程 点实验室 ,成都 6;2 国科 院 水利部成 山地灾害与 环境究所,成 都 6 
摘 要在研究滚石的运动规律过程滚石的碰撞恢复系数是重要控制参数它不仅与滚石的冲击速度、质量有关,还 
与滚石坡面覆盖层物理力学性质密切相关目前尚无理的计算理论这是滚石灾究的一个难点问题。根据 H 
接触、C 向接触料弹塑的基础上 向碰恢复和切 向碰撞恢 
复系数的计算模式,给出了具体的计算公式阐明了碰撞恢复系数的主要控制因素。通过一的计算分析并与 
有结论的比较,表明该计算模式的合理性 
关 键 词:滚碰撞恢复系数 触力;冲击   
中图分类号:X   文献标识码 
   ic    
HE  一, W U Yo 一, LI   
KeyL MoHa  Acmy Ch Ch 
  Mo Ha  me Acmy    
Ab:I    w  ,t  i c   mpta  me    
me   wi                
.Howe         i c wh wi    i c 
      y  Cao & M i t  y,c  
   ma  m o   i c      m p  
    mu        ic       
mp  mp wi    y       
Ke wo  i c m emp 
  前  言 
石碰 撞恢系数 是正确估石运 动轨迹 的 
要参 数。与边坡 的坡面覆 盖层 的力学性 
、滚 石 自的大 小等因素 有关 。室试 验表 明 
随着的增 大,法 向恢复 系数略 有增大 ,但 
切 向恢复系 数的影 响却很 明显 ;边 坡坡面 覆 
土越 松散 ,越趋 向完非弹 性碰,相 应 
的法 向和切 向恢复系 数就越 小。相反 ,边坡面 出 
岩越硬 ,碰撞 就越趋 向弹性相应 的法 
向和切 向恢复系 数就大 。由于滚 石 问的复性 , 
采用理论方法研究滚石碰撞恢复系数的难度非常 
, 目主要 依赖于试验 ,在试验 的基础 上通反 
分析 ,得到 恢复系数 。 
碰撞恢复系数在颗粒流化学工程等众多领域 
研 究非常 广泛叫],并 提了众 多的算 公式 ,但 
在滚石灾害研究中确鲜有研究成果。为此,文在 
其他领域研究成果的础上考虑滚石灾害的具体 
特 点,根据 H接 触力 学l、C Mi 
向接触 理 论l¨,在考 虑材 料 弹塑特 性 的基础 
上 ,分 别研 究了滚石 法向碰恢 复系数和切 向碰撞 
复系 数的算模 式 ,给 出了滚 石碰撞恢复 系数 
算理 论 以具 体的计算公 式和步骤 ,为研 究滚石 
发 区滚石面运 动规提供 了一套理论 方法 ,对 
正确 认识滚石运 动机提供 理论基础 。 
收稿 日期 :2 
基金项 目:国家 自然科学基 金项 目 (、N :交通 部西部科 技项 目 ( 
第一作者 简介 :何思明,,1,博士 ,研究 员,主要山地灾害 形成机 理及防治技术研究 。 
  岩  土  力     
2 问题的提  
察 如 l所示 的滚对 坡的冲撞 问 
为研 究问题 的方 便,做   
形状段组 成的直线 坡面 ,坡 
材料为均匀、各 向同性弹塑性体 
  (5) 
  、  分别为滚输入、输 出的切向冲击 
量 。 
向    
布均匀 ,为均匀 、  3 滚石向碰系数 。   H   厶       、M  
滚石冲击速度面成一角 图 l 
 
冲击 速度沿法 向和切 向解 为 
=V n ;  =VC   
中 :    分 别为滚 石对构筑 物的切 向冲速 度 
和法 向冲击速度 ;  为滚石 的冲击速度 ;  为滚 石 
冲击速度向与坡面的夹 
图 1 滚石坡面运动计算模型 
  M o m o     
法 向碰撞恢 复系定义 为 
式 中:  为滚石法 向碰撞 恢复 系数  为滚石碰 
回弹法 向速 
滚石碰撞恢复定义 
:    
中:e 为滚石向碰撞恢系数  为滚石撞 
回弹切 向速 
可 以始 能量 与 回弹动 能进 行定 
义    
 以 H接 触理为基 础设 
料满足理想塑性特性的基础上,推导了球体法 
向碰撞 恢复系的计 算公式 : 
  
  
中 : 为滚石法 向碰撞恢 复系数 ;  滚石法 向 
速度 ;  为坡面 土体初始 屈服法 向冲速度 。 
笔者在假设坡面覆盖层满足莫尔库仑准则的 
,推了坡面 土体服法冲击 速度 
式㈣ : 
      ㈩ 
m 2       
其 中: 
    
     
 +(  ) 3
 
  
中 :  为等效半径 , 1=    分 
两 个 球 体 的半 径 ; E 为 等效 弹 性模 量 ,   
          :分别为两触 
        ‘ 
体 各 自弹性 模量松 比;  等效 质量定 
义为      ;c  分坡面岩体的凝聚 
和 内摩 擦角 。 
本 问题中 ,可 以看 成是一个平 面,质 
为无限大的球。为此,相应的等效半径、等效 
可 以简化 为 
 
\ 
6  
(  
  4  
● ●●●●●●●●、,,●●●●●●●●●J  
 
]●●●●●●●J  
2  
第 3  何思石冲 撞恢复系数研究   
m=    再根据式 ()可以求石斜冲击产 
4 滚向碰恢复系数 
年 )和 Min (9年)各 
独 立地 研 究了两球 体 在法 向压 P一 定条 件 
向荷载作用下接触面的切向应力分布 
究 结果表 明,接触 区间存 在所微 滑  
区(≤r黏结 (区( 
切 向荷 载作用下 ,球体相 对切 向位移计算 公 
为 
=   G            
中:        、 G、 G 
G  G1  G1 
为滚石防护构各 自的泊松比和剪切模 
为切 向荷载 ;为 法向接触 压力 ;为 切向变形 量; 
为摩擦系;其他号意同前 
对上式作变化 
     
滚石在切 向冲击速 度作用下 ,根据 牛顿第 二定 
律 ,并经推 导有下关 系成 
   ! 
式中:    )对切运动速度,在此即为滚 
切 向冲击度  ;其符 号意义 同前 
准静的情况下 向冲击力与 
接触变形间满系式 (于是有下关系 
 
 
_  } 
对  积分 ,据最切 向位   
条件 ,给 出最大向位移 量的计算公 式 
l(a) PS
m   + 
    P     
 
过式 )可 以算 出切向冲击下 的最大切 
形量  。 
切 向冲 
 ̄=/_  … 
当滚石切 向冲力达 到最大后 ,开始 卸载 ,回 
变形按下式计  
=   Ga   __      J   
 
   
  d,     
2 滚石切向荷载变形关系线 
Fi  T me  
  m p 
切 向荷卸载 到 0,即 a=切 向变形量 
并不为 0,对应 的变为切 向残余变形 。 
=   Ga _     川 
中:  为残余 形量  
根据公式 可 以采用滚石 切向输入 能量与切 
向可恢复能量来计算切 恢复系。在 
切 向碰 撞过程 中,输入 能量图 中 OB所围成 的图 
部分 的面而 可恢为 AC图围成 的面 
积,即: 
     ( 
部分 图形 的面积可通 过积计算 : 
… =  _ 
中 :   为滚 石切 向输入能量 。 
  岩  土  力  学  9年 
… =  一 
 ・ 
_  
中 :S 为滚石 切向可恢 复能量 。 
法 向压 力 P根据计算 : 
尸 :   
通过上述计算步骤,就可向碰撞恢复 
 
  
为验 证本的合,现提供 以下 
料予 以检验 。 
知 滚石的最冲击速度 为 4 冲击 角度与 
  。的夹角 :坡 面平直 ,为残 坡积覆盖土 
,相关计算参数见表 1 
表 1 滚石冲击计算参数 
   Ca m e   m p   
1 法恢复系数计算 
根据式 (,首先确定滚石对 向冲击碰 
塑冲击度   ,这个是 比较 小 
的,一旦滚冲击大于  ,就会在坡 面产 
塑 性 变 形 区 。 实 际 滚 石 的 法 冲 击 速 度   
 =1 远远超过冲击必定在 
上形成大围的塑性 根据 公式 
合坡面覆盖层初服冲击速度计式( 
计算出滚石法向碰撞恢复系数  = 
出了滚石法向碰撞恢复系数随冲击速度 
变化关 系曲线 。从 图中可 以看 出:法 向撞 系数 
冲击速度的增加而逐 
叵 
趟 
  3  4  5  6  7       
滚     
3 滚石向碰撞恢复系数.向冲击速度关系曲线 
Fi  Re i c ma mp  
  k. 
切 向碰撞恢复系 数计算 
切 向碰 撞恢系数根据 节 的内容计算 ,具 
计算步骤 
① 由式 )计算法 向冲击压力 P;②根式 
),采用 数值方法确 定滚石大切 向冲击位移 
  ③ 由式 计 算滚石切 向冲击下 ,产 
最大切 向力 Q ④ 由式 计 算残余切 向位移 
⑤ 由式算滚石切 向冲击输入 能量 S  ; 
由式 (算滚石切向可恢能量 伽 最 
后根据式 )计算石切 向碰撞恢 复系数 e。经 
过计,滚石向碰撞恢:e  
上述结果与文的覆盖土层碰 
恢复系数 的大比较一致 ,说 明文模 型用于计算 
滚石碰恢复系数是可行 
6 结 论 
)以 T碰撞恢复式 为基础 ,结 
合本文提的坡面岩土体材料初始屈服冲击速度 
出了滚向碰撞 恢复系数计算方,通过 
算例验证,本方法是合理有效的。 
)滚石法 向碰撞恢 复系数与滚石初 始冲击速 
度 、坡面初 始屈服法 向速度有 关,随滚石 初始冲击 
速度 增加 减少  
)以 C& M切 向加卸载一 
为基,通过滚石输入向冲能量可恢 
复切 向冲击能 量关系定义 滚石向碰撞 恢复系数 , 
了一种滚石向碰撞系数的计算方法 
确 了具体 的计算公式和计 算步过算 例检 算 
结果合。说明本文方法是可行 的。 
)本文关于滚石碰撞复系数的计算方法 
工程 
  如 
O   0   0   O   O   O   O  
第 3期  何思明等:滚石冲击碰撞恢复系数研究   
参 考 文 献 
  MNGWAN HEONG  AMS    
Th       
  .Cmi   
-4 
  WU  Y,LI  Y,HOON C.Re   
   mp   
mp -9 
】 TNTON ,NING WU  Y,  
me      
Lu  Be01 
4一 l 
  TNT NING A  mo   
  b  o  a  e 
wd   
  TH0NT0N    or  
     
 Ap  , T  A SM E ,  
3— 3 
  BME GEMAN ON    
  con    ma      t 
     .I 
     ,2,( 
6-- 5 
  WAG        mo 
 t   mp i v 
    .2.3:1— 
 
】 KHAR  H,0RAM  AN  D.An 
me        
Pow d Te01- 2 
  S  R  HIHL  W ,  EAR  P 
Nu  me     
上接第页 
  程强,周永江,黄绍槟.近水平红层开挖边坡变形破坏 
土力学,—1 
 HO   
Di        
ma   Ro    
 
】王汉辉,王均星,王开治.边坡稳定的有元塑性极 
分析 土力—7 
WA NG  Ha, W ANG J, W ANG  Ka 
 lmi  a  s  s u f 
me    
--  
 王均星,李泽,陈炜.考虑孔隙水压力土坡稳定性的 
有 限元 下析 【岩 土:1 
 
   r i 
 mp —5 
HN K .C M].Cmb 
Ca Un  
U— ,ZHA X, ma  
     
      
  -6 
HAUA  W0NG        
   mo    
   Ro M es & M i 
9—7 
H0NT      
     
 Appl M e,T  AS E,1  
 
明,李新坡吴永冲击载作用下土体屈服 
力学与工学报, 
 
HE ,L Xi.W U Yo.Re   
   u r imp  
  Ro M e  ne 
-2 
】VUQU  HA X.  
 fdim e m o f gr-w 
Co  wi  m a 
 mu  mp 
hy- 3 
庆,孙 运动 模 
自然灾害—8 
 Q, S Ho, ZHA , e a 
      
 Di,19~8 
W ANG ,LI Ze, CHEN W l we b 
ly o s s  u fe e 
       
— 1 
NGUYEN  ,BOGE0N L,CHIⅡM ATU    
Hyme       
m a t e o  t p Kam a M i 
me     Ro 
M e  M i 01— 94. 
REED B Hy fe     
       
M e  M i 7— 2 
NDRATNA   RAN G  S  wa 
   f. Ge  
Ge ,1  
n  
... 0.54-0.96 Lithic sandstones He et al. [42] 0.31 0.83 Weathered limestone ...
... On the other hand, the COR, as an important parameter to describe the energy dissipation in rockfall collisions, directly affects the prediction accuracy [3]. Therefore, after considering the geological conditions (lithology, weathering degree, and other parameters) of the predicted slope, the COR value of the field test was also considered (we referred to He et al. [42] with similar geological conditions, that is, R n = 0.35) through field tests [38] to verify the rationality of the trajectory prediction model at this COR value. The trajectory of the proposed approach was more consistent with the field test and more suitable than Yan's [20] numerical prediction. ...
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The rockfall process is characterized by bounces of a block on the ground. The coefficient of restitution (COR), which indicates the degree of rockfall energy dissipation, has a significant effect on the rockfall trajectory. The 3-dimensional Distinct Element Code (3DEC) is an effective tool to study the rockfall trajectory, and the damping can reflect the COR in numerical modeling. However, the relationship between damping and COR is not understood. A field test is numerically modelled to investigate the correspondence between damping and COR. A series of damping–COR correspondences are obtained and compared with the field test and its previous numerical simulation to verify the rationality of the correspondences. Then, the damping–COR correspondence is adopted in a typical rockslide in Yunnan province, China. The numerical results show that the proposed method is in good agreement with practical engineering. This study provides a new method for predicting rockfall trajectory.
... The slope body includes the stable slope and collapsing slope, and the collapsing slope is generated by rainfall, earthquakes, and so on (Chen et al. 2022(Chen et al. , 2019Gu et al. 2023). However, most of the existing collision theory models simplify the collision as a collision of particles (Chau et al. 2002;Dorren et al. 2004;He et al. 2009) or the disk/ball impact static slab (Chattopadhyay and Saxena 1991;Labiouse et al. 1996;Mougin et al. 2005;Plantard and Papini 2005;Sun 1977; Wang et al. 2016) and then investigate the influence of particle shape, incident angle, and velocity on the collision process. Nevertheless, theoretical collision models of rockfall-inclined plates are rare (Qing et al. 2003). ...
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... the motion characteristics of rockfalls in the study area. The model uses the lumped mass method, and the interaction between a rock and a slope is specifically described by two coefficients of restitution (R N and R T ) (Ji et al. 2019; Siming et al. 2009;Wenli andRong 2015Wong et al. 2000) and a coefficient of friction. Rockfall dynamics are not affected by the masses and shapes of the boulders, and only the inherent randomness of the source location and initial orientation is considered in the analysis (Lan et al. 2007). ...
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... Based on the Young's modulus, yield stress, and the principle of quasi-static contact mechanics, Mangwandi et al. (2007) studied the restitution coefficient of typical material particles after collisions at different speeds. According to the Cattaneo (1938a, b, c) and Mindlin (1949) tangential contact theory and the Hertz (1882) contact theory, He et al. (2009) studied the calculation model of the normal restitution coefficient and the tangential restitution coefficient and gave the calculation formula. Combining basic kinematics formulas, they also gave the incidence rebound law of rockfall slopes. ...
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In this work multi-modal systems subject to impact are considered. Using energy balance techniques for an arbitrary contact interval the effects of modal vibration can be included. The energy balance is used to obtain a relationship between the coefficient of restitution and the modal energy during the contact period. This allows the effects of impact induced vibration to be considered. The subsequent analytical relationships demonstrate that increasing contact duration and excitation of higher modes can reduce the effective value of the coefficient of restitution. It is also shown how this approach can be related to work on energetically consistent impacts.