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多体系统碰撞动力学中接触力模型的研究进展

王庚祥 马道林 刘洋 刘才山

王庚祥, 马道林, 刘洋, 刘才山. 多体系统碰撞动力学中接触力模型的研究进展. 力学学报, 2022, 54(12): 3239-3266 doi: 10.6052/0459-1879-22-266
引用本文: 王庚祥, 马道林, 刘洋, 刘才山. 多体系统碰撞动力学中接触力模型的研究进展. 力学学报, 2022, 54(12): 3239-3266 doi: 10.6052/0459-1879-22-266
Wang Gengxiang, Ma Daolin, Liu Yang, Liu Caishan. Research progress of contact force models in the collision mechanics of multibody system. Chinese Journal of Theoretical and Applied Mechanics, 2022, 54(12): 3239-3266 doi: 10.6052/0459-1879-22-266
Citation: Wang Gengxiang, Ma Daolin, Liu Yang, Liu Caishan. Research progress of contact force models in the collision mechanics of multibody system. Chinese Journal of Theoretical and Applied Mechanics, 2022, 54(12): 3239-3266 doi: 10.6052/0459-1879-22-266

多体系统碰撞动力学中接触力模型的研究进展

doi: 10.6052/0459-1879-22-266
基金项目: 国家自然科学基金资助项目(11932001, 12172004, 12111530108)
详细信息
    作者简介:

    刘才山, 教授, 主要研究方向: 多体系统动力学、碰撞力学. E-mail: liucs@pku.edu.cn

  • 中图分类号: TH113.1

RESEARCH PROGRESS OF CONTACT FORCE MODELS IN THE COLLISION MECHANICS OF MULTIBODY SYSTEM

  • 摘要: 接触碰撞行为作为大自然与多体系统中的常见现象, 其接触力模型对于多体系统的碰撞行为机理研究与性能预测至关重要. 静态弹塑性接触模型与考虑能量耗散的连续接触力模型是研究接触碰撞行为的两类不同方法, 在多体系统碰撞动力学中存在诸多共性与差异. 本文分别从上述两类接触模型的发展历程入手, 详细介绍了两类模型的区别与联系. 首先, 根据阻尼项分母中是否含有初始碰撞速度将连续接触力模型分为黏性接触力模型与迟滞接触力模型, 讨论了能量指数与Hertz接触刚度之间的关系, 阐述了现有连续接触力模型在计算弹塑性材料接触碰撞行为时存在的问题. 其次, 着重介绍了分段连续的准静态弹塑性接触力模型(可连续从完全弹性转换到完全塑性接触阶段), 分析了利用此类弹塑性接触力模型计算碰撞行为的技术特点. 同时, 以恢复系数为桥梁和借助线性化的弹塑性接触刚度, 避免了Hertz刚度对弹塑性接触刚度的计算误差, 根据碰撞前后多体系统的能量与动能守恒推导了弹塑性接触模型等效的迟滞阻尼因子. 探索了连续接触力模型与准静态弹塑性接触力模型之间的内在联系, 数值计算结果定量说明了人为阻尼项代表的能量耗散与弹塑性接触力模型中加卸载路径代表的能量耗散具有等效性. 另外, 为了避免阻尼项分母中初始碰撞速度在计算颗粒物质动态性能时导致的数值奇异问题, 通过求解等效的线性单自由度欠阻尼非受迫振动方程获得了阻尼项分母中不含初始碰撞速度的连续接触力模型, 并以一维球链为例, 证明了该模型相比EDEM软件使用的连续接触力模型具有更高的精度. 最后, 本文分析了当前多体系统碰撞动力学的研究现状, 并简要展望了多体系统碰撞动力学中接触力模型的发展趋势与面临的挑战.

     

  • 图  1  接触力模型的分类

    Figure  1.  Classification of contact force models

    图  2  碰撞体接触过程

    Figure  2.  Contact process between two contact bodies

    图  3  常见连续接触力模型的力与变形量之间的关系[51]

    Figure  3.  Force-deformation relationship of the common contact force models[51]

    图  4  弹簧−阻尼模型

    Figure  4.  The spring-damper model

    图  5  两类阻尼的比较

    Figure  5.  Comparative analysis between two different damping systems

    图  6  拥有不同阻尼类型的接触力模型

    Figure  6.  Contact force models with different damping systems

    图  7  表面粗糙度形貌

    Figure  7.  The surface roughness topography

    图  8  准静态弹塑性接触力模型[146]

    Figure  8.  Quasi-static elastoplatsic contact force model[146]

    图  9  一般弹塑性接触力模型中力与变形量之间的关系

    Figure  9.  Force-deformation relationship for a general elastoplastic contact force model

    图  10  接触力−变形量(关于弹塑性阶段)[43]

    Figure  10.  Relationship between the force and deformation (in elastoplastic phase)[43]

    图  11  一维球链的碰撞实验装置示意图[117]

    Figure  11.  Experimental setup of the one-dimension granular chain[117]

    图  12  一维球链中孤立波的传播特征

    Figure  12.  Propagation of solitary wave in the one-dimension granular chain

    图  13  连续接触力模型之间的对比分析

    Figure  13.  Comparative analysis between two continuous contact force models

    表  1  迟滞接触力模型(阻尼项分母中含有初始相对碰撞速度)

    Table  1.   Contact force models with hysteresis damping factors (the denominators of the damping force do not include the initial impact velocity)

    Continuous contact force model $ F = K{\delta ^n} + \chi {\delta ^m}\dot \delta $Power exponent nImpact parameter mHysteresis damping factor$ \chi $
    Hunt-Crossley model [61]3/23/2$ \chi = \dfrac{{3\left( {1 - {c_r}} \right)}}{2}\dfrac{K}{{{{\dot \delta }^{\left( - \right)}}}} $
    Herbert-McWhannell model [76]3/23/2$ \chi = \dfrac{{6\left( {1 - {c_r}} \right)}}{{\left[ {{{\left( {2{c_r} - 1} \right)}^2} + 3} \right]}}\dfrac{K}{{{{\dot \delta }^{\left( - \right)}}}} $
    Lee-Wang model [79]3/23/2$ \chi = \dfrac{{3\left( {1 - {c_r}} \right)}}{4}\dfrac{K}{{{{\dot \delta }^{\left( - \right)}}}} $
    Lankarani-Nikravesh model[66]3/23/2$ \chi = \dfrac{{3\left( {1 - c_r^2} \right)}}{4}\dfrac{K}{{{{\dot \delta }^{\left( - \right)}}}} $
    Gonthier et al. model[75]3/23/2$ \chi \approx \dfrac{{1 - c_r^2}}{{{c_r}}}\dfrac{K}{{{{\dot \delta }^{\left( - \right)}}}} $
    Zhiying-Qishao model [67]3/23/2$\chi = \dfrac{ {3\left( {1 - c_r^2} \right){{\rm{e}}^{2\left( {1 - {c_r} } \right)} } } }{4}\dfrac{K}{ { { {\dot \delta }^{\left( - \right)} } } }$
    Flores et al. model [68]3/23/2$ \chi = \dfrac{{8\left( {1 - {c_r}} \right)}}{{5{c_r}}}\dfrac{K}{{{{\dot \delta }^{\left( - \right)}}}} $
    Gharib-Hurmuzlu model [77]3/23/2$ \chi = \dfrac{1}{{{c_r}}}\dfrac{K}{{{{\dot \delta }^{\left( - \right)}}}} $
    Hu-Guo model [70]3/23/2$ \chi = \dfrac{{3\left( {1 - {c_r}} \right)}}{{2{c_r}}}\dfrac{K}{{{{\dot \delta }^{\left( - \right)}}}} $
    Hu et al. model [78]3/23/2$\chi {\text{ = } } - \dfrac{ {6.66\;26\ln {c_r} } }{ {3.852\;38 + \ln {c_r} } }\dfrac{K}{ { { {\dot \delta }^{\left( - \right)} } } }$
    Shen et al. model [71]3/23/2$ \chi {\text{ = }}\dfrac{{3\left( {1 - {c_r}} \right)}}{{2 c_r^{0.89}}}\dfrac{K}{{{{\dot \delta }^{\left( - \right)}}}} $
    Carvalho-Martins model [15]3/23/2$ \chi {\text{ = }}\dfrac{{3\left( {1 - {c_r}} \right)\left( {11 - {c_r}} \right)}}{{2\left( {1{\text{ + }}9{c_r}} \right)}}\dfrac{K}{{{{\dot \delta }^{\left( - \right)}}}} $
    Safaeifar-Farshidianfar model [21]3/23/2$ \chi {\text{ = }}\dfrac{{5\left( {1 - {c_r}} \right)}}{{4{c_r}}}\dfrac{K}{{{{\dot \delta }^{\left( - \right)}}}} $
    Zhang et al. model [72]3/23/2$\chi {\text{ = } }\dfrac{ {3\left( {1 - {c_r} } \right)} }{ {2\left( {0.618{{\rm{e}}^{ - 3.25{c_r} } } + 0.899{{\rm{e}}^{0.090\;25{c_r} } } } \right){c_r} } }\dfrac{K}{ { { {\dot \delta }^{\left( - \right)} } } }$
    Zhao et al. model [22]3/23/2$ \chi {\text{ = }}\dfrac{{4\left( {1 - {c_r}} \right)}}{{1.302{c_r}}}\dfrac{K}{{{{\dot \delta }^{\left( - \right)}}}} $
    下载: 导出CSV

    表  2  黏性接触力模型(阻尼项分母中不含初始相对碰撞速度)

    Table  2.   Contact force models with viscous damping factors (the denominators of the damping force do not include the initial impact velocity)

    Continuous contact force model
    $ F = K{\delta ^n} + \chi {\delta ^m}\dot \delta $
    Power exponent nImpact parameter mViscous damping factor$ \chi $
    Kuwabara and Kono [99]3/21/2$\chi = \dfrac{K}{2}\dfrac{ { { {\left( {3{\eta _2} - {\eta _1} } \right)}^2} } }{ { {3{\eta _2} + 2{\eta _1} } } }\dfrac{ {\left( {1 - {\nu ^2} } \right)\left( {1 - 2\nu } \right)} }{ {E{\nu ^2} } }$($ {\eta _1},{\eta _2} $ are the viscous material constant values)
    Tsuji et al. [96]3/21/4$\chi = \dfrac{ {\sqrt 5 } }{2}D,D = 2\left| {\ln {c_r} } \right|\sqrt {\dfrac{ {KM} }{ { { {\text{π}} ^2} + { {\ln }^2}{c_r} } } }$
    Jankowski [95]3/21/4$ \chi = 9\sqrt 5 \dfrac{{1 - c_r^2}}{{{c_r}\left[ {{c_r}\left( {9{\text{π}} - 16} \right) + 16} \right]}}\sqrt {KM} $
    Lee and Wang[79]3/20$\chi = TD,D = 2\left| {\ln {c_r} } \right|\sqrt {\dfrac{ {KM} }{ { { {\text{π} } ^2} + { {\ln }^2}{c_r} } } } ,T = \dfrac{ {\delta + \left| \delta \right|} }{ {2\delta } }\exp \left\{ {\dfrac{q}{\varepsilon }\left[ {\left( {\delta - \varepsilon } \right) - \left| {\delta - \varepsilon } \right|} \right]} \right\}$
    ($ \varepsilon $, $ q $ are constants with unit m)
    Schwager and Poschel [74]3/20.65empirical
    Lee and Herrmann [80]3/21empirical
    Ristow [100]3/21empirical
    下载: 导出CSV

    表  3  连续准静态弹塑性接触力模型

    Table  3.   Continuous quasi-static elastoplastic contact force models

    Quasi-static elastoplastic contact modelLoading phaseUnloading phase
    Thornton (1997)[135]$ F = \left\{ \begin{gathered} \dfrac{4}{3}{E^ * }\sqrt R {\delta ^{\frac{3}{2}}}\begin{array}{*{20}{c}} {}&{\begin{array}{*{20}{c}} {}&{} \end{array}}&{\left( {\delta < {\delta _y}} \right)} \end{array} \\ {F_y} + {\text{π}} {\sigma _y}R\left( {\delta - {\delta _y}} \right)\begin{array}{*{20}{c}} {}&{\left( {\delta > {\delta _y}} \right)} \end{array} \\ \end{gathered} \right. $
    $ {\delta _y} $critical elastic deformation;$ {\sigma _y} $ yield stress;$ {F_y} $loading force
    $ F = \dfrac{4}{3}{E^ * }\sqrt {{R_p}} {\left( {\delta - {\delta _p}} \right)^{\frac{3}{2}}},{R_p} = \dfrac{{4{E^ * }}}{{3{F_{\max }}}}{\left( {\dfrac{{2{F_{\max }} + {F_y}}}{{2{\text{π}} {\sigma _y}}}} \right)^{\frac{3}{2}}} $
    $ {\delta _p} $permanent deformation;$ {F_{\max }} $maximum
    normal contact force
    Stronge (2000)[63]$F = \left\{ \begin{gathered} \dfrac{4}{3}{E^ * }\sqrt R {\delta ^{\frac{3}{2} } }\begin{array}{*{20}{c} } {}&{\begin{array}{*{20}{c} } {}&{} \end{array} }&{\left( {\delta < {\delta _y} } \right)} \end{array} \\ {F_y}\left( {\dfrac{ {2\delta } }{ { {\delta _y} } } - 1} \right)\begin{array}{*{20}{c} } {\left[ {1 + \dfrac{1}{ {3{\vartheta _y} } }\ln \left( {\dfrac{ {2\delta } }{ { {\delta _y} } } - 1} \right)} \right]}&{\left( { {\delta _y} \leqslant \delta < {\delta _p} } \right)} \end{array} \\ \dfrac{ {2.8{F_y} } }{ { {\vartheta _y} } }\left( {\dfrac{ {2\delta } }{ { {\delta _y} } } - 1} \right)\begin{array}{*{20}{c} } {}&{}&{\left( {\delta \geqslant {\delta _p} } \right)} \end{array} \\ \end{gathered} \right.$
    $ {F_y} = {\text{π}} {\vartheta _y}{\sigma _y}{R^2}{\left( {\dfrac{{3{\text{π}} }}{4}} \right)^2}{\left( {\dfrac{{{\vartheta _y}{\sigma _y}}}{{{E^ * }}}} \right)^2} $,$ {\vartheta _y} $coefficient;$ {\delta _y} $critical elastic;
    $ {\sigma _y} $yield stress;$ {\delta _p} $critical plastic deformation;
    $ \begin{gathered} F = \dfrac{4}{3}{E^ * }\sqrt {\bar R} {\left( {\delta - {\delta _f}} \right)^{\frac{3}{2}}} \\ {\delta _f} = {\delta _c} - {\delta _r},\bar R = {\left( {\dfrac{{2{\delta _c}}}{{{\delta _y}}} - 1} \right)^{\frac{1}{2}}}R \\ \end{gathered} $
    $ {\delta _c} $maximum contact deformation;
    $ {\delta _r} $permanent deformation
    Vu-Quoc and Zhang (2002)[147-148]$F = \left[ {\dfrac{ {2{E^ * } } }{ {3 R\left( {1 - {\nu ^2} } \right)} } } \right]{a^3},a = \sqrt {2 R\delta } \begin{array}{*{20}{c} } {}&{F < {F_y} } \end{array}$

    $ \left\{ \begin{gathered} {a^{ep}} = {a^e} + {a^p},{a^p} = \left\{ \begin{gathered} 0\begin{array}{*{20}{c}} {}&{}&{}&{\left( {F < {F_y}} \right)} \end{array} \\ {C_a}\left( {F - {F_y}} \right)\begin{array}{*{20}{c}} {}&{\left( {F > {F_y}} \right)} \end{array} \\ \end{gathered} \right. \\ \delta = \dfrac{{{{\left( {{a^{ep}}} \right)}^2}}}{{2{R^{ep}}}} \\ {R^{ep}} = {C_R}R \\ {C_R}{\text{ = }}\left\{ \begin{gathered} 1.0\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {}&{} \end{array}}&{}&{\left( {F < {F_y}} \right)} \end{array} \\ 1.0{\text{ + }}{K_c}\left( {F - {F_y}} \right)\begin{array}{*{20}{c}} {}&{\left( {F > {F_y}} \right)} \end{array} \\ \end{gathered} \right. \\ \end{gathered} \right. $


    Fy Normal yield load; Kc and Ca are the empirical parameters;
    ae contact radius corresponding to the elastic region; $ \nu $Poisson ratio
    $\begin{gathered} F = \left[ {\dfrac{ {2{E^ * } } }{ {3 R\left( {1 - {\nu ^2} } \right)} } } \right]a_e^3 \\ {a_e} = {2{ {\left( { {C_R} } \right)}_{\max } }R\left( {\delta - {\delta _r} } \right)} \\ \end{gathered}$
    $ {\delta _r} $ permanent deformation
    Kogut and Etsion (2002)[139]$ F = \left\{ \begin{gathered} \dfrac{4}{3}{E^ * }\sqrt R {\delta ^{\frac{3}{2}}}\begin{array}{*{20}{c}} {}&{\begin{array}{*{20}{c}} {}&{} \end{array}}&{\left( {\dfrac{\delta }{{{\delta _y}}} < 1} \right)} \end{array} \\ {F_c}1.03{\left( {\dfrac{\delta }{{{\delta _y}}}} \right)^{1.425}}\begin{array}{*{20}{c}} {}&{\left( {1 \leqslant \dfrac{\delta }{{{\delta _y}}} < 6} \right)} \end{array} \\ {F_c}1.40{\left( {\dfrac{\delta }{{{\delta _y}}}} \right)^{1.263}}\begin{array}{*{20}{c}} {}&{\left( {6 \leqslant \dfrac{\delta }{{{\delta _y}}} < 110} \right)} \end{array} \\ \end{gathered} \right. $
    $ {\delta _y} $critical elastic deformation; Fc loading force corresponding to the
    beginning phase of the plastic deformation
    $ F = {F_{\max }}{\left( {\dfrac{{\omega - {\omega _{res}}}}{{{\omega _{\max }} - {\omega _{res}}}}} \right)^{np}},np = 1.5{\left( {{\omega _{\max }}} \right)^{ - 0.0331}} $
    $ \omega = {\delta \mathord{\left/ {\vphantom {\delta {{\delta _y}}}} \right. } {{\delta _y}}};{\omega _{\max }} = {{{\delta _{\max }}} \mathord{\left/ {\vphantom {{{\delta _{\max }}} {{\delta _y}}}} \right. } {{\delta _y}}} $
    Fmax maximum contact force; $ {\delta _{\max }} $maximum contact
    deformation;$ {\omega _{res}} $residual deformation
    Jackson and Green (2005)[138]$F = \left\{ \begin{gathered} \dfrac{4}{3}{E^ * }\sqrt R {\delta ^{\frac{3}{2} } }\begin{array}{*{20}{c} } {}&{\begin{array}{*{20}{c} } {}&{} \end{array} }&{\left( {\dfrac{\delta }{ { {\delta _y} } } \leqslant 1.9} \right)} \end{array} \\ {F_c}\left\{ {\exp \left[ { - \dfrac{1}{4}{\omega ^{\frac{5}{ {12} } } } } \right]{\omega ^{\frac{3}{2} } } + \dfrac{ {4 H} }{ {C{S_y} } }\left[ {1 - \exp \left( { - \dfrac{1}{ {25} }{\omega ^{\frac{5}{9} } } } \right)} \right]\omega } \right\}\begin{array}{*{20}{c} } {}&{\left( {\dfrac{\delta }{ { {\delta _y} } } \geqslant 1.9} \right)} \end{array} \\ \end{gathered} \right.$
    $ C = 1.295\exp \left( {0.736\nu } \right) $;$ {\delta _y} $critical elastic deformation; H hardness;
    Sy limit of yielding;$ \nu $Poisson ratio; Fc critical elastic load
    $\begin{gathered} F = \dfrac{4}{3}{E^ * }\sqrt { {R} } {\left( {\delta - {\delta _r} } \right)^{\frac{3}{2} } } \\ {R_b} = R\cos \theta ,\theta = \dfrac{ { {a_c} } }{R} \\ \end{gathered}$
    $ {\delta _r} $permanent deformation; ac contact radius in
    the unloading phase
    Du and Wang (2009)[143]$ F = \left\{ \begin{gathered} \dfrac{4}{3}{E^ * }\sqrt R {\delta ^{\frac{3}{2}}}\begin{array}{*{20}{c}} {}&{\begin{array}{*{20}{c}} {}&{} \end{array}}&{\left( {\delta \leqslant {\delta _e}} \right)} \end{array} \\ {\text{π}} R{p_p}\delta - \dfrac{{p_p^3{{\text{π}} ^3}{R^2}}}{{12{{\left( {{E^ * }} \right)}^2}}}\begin{array}{*{20}{c}} {}&{\left( {\delta \leqslant {\delta _e}} \right)} \end{array} \\ \end{gathered} \right. $
    $ {p_p} = \left( {1 + \dfrac{{\text{π}} }{2}} \right){\sigma _y} $;$ {\sigma _y} $yield strength; $ {\delta _e} $critical elastic deformation
    $F = \dfrac{4}{3}{E^ * }\sqrt { {R} } {\left( {\delta - {\delta _{res} } } \right)^{\frac{3}{2} } }$
    $ {\delta _{res}} $permanent deformation
    Brake (2012)[146]$F = \left\{ \begin{gathered} \dfrac{4}{3}{E^ * }\sqrt R {\delta ^{\frac{3}{2} } }\begin{array}{*{20}{c} } {}&{\begin{array}{*{20}{c} } {}&{} \end{array} }&{\left( {\delta < {\delta _y} } \right)} \end{array} \\ \left[{2{F_y} - 2{F_p} + \left( { {\delta _p} - {\delta _y} } \right)\left( { { {F'}_y} + { {F'}_p} } \right)} \right]{\left( {\dfrac{ {\delta - {\delta _y} } }{ { {\delta _p} - {\delta _y} } } } \right)^3}+ \\ \qquad \left[ { - 3{F_y} + 3{F_p} + \left( { {\delta _p} - {\delta _y} } \right)\left( { - 2{ {F'}_y} - { {F'}_p} } \right)} \right]{\left( {\dfrac{ {\delta - {\delta _y} } }{ { {\delta _p} - {\delta _y} } } } \right)^2}+ \\ \qquad \left( { {\delta _p} - {\delta _y} } \right)F'\left[ {\dfrac{ {\delta - {\delta _y} } }{ { {\delta _p} - {\delta _y} } } } \right] + {F_y}\begin{array}{*{20}{c} } {}&{\left( { {\delta _y} \leqslant \delta <{\delta _p} } \right)} \end{array} \\ \dfrac{ {3{F_y}{\text{π} } {p_0} } }{ {4 E} }\sqrt {\dfrac{R}{ { {\delta _y} } } } \left[ {4\left( {\dfrac{\delta }{ { {\delta _y} } } } \right) + \dfrac{c}{ {R{\delta _y} } } } \right] \\ \end{gathered} \right.$
    $ \begin{gathered} {p_0} \approx H,c = a_p^2 - 2 R{\delta _p},{F_y} = \dfrac{4}{3}{E^ * }\sqrt R \delta _y^{\frac{3}{2}},{F_p} = {p_0}{\text{π}} {a_p} \\ {a_p} = \dfrac{{3 R{\text{π}} {p_0}}}{{4{E^ * }}},{{F'}_p} = 2 R{\text{π}} {p_0},{{F'}_y} = 2{E^ * }\sqrt {R{\delta _y}} \\ \end{gathered} $

    H hardness;$ {\delta _y} $ critical elastic deformation;$ {\delta _p} $ critical
    plastic deformation
    $ \begin{gathered} F = \dfrac{4}{3}{E^ * }\sqrt {{{\bar R}_b}} {\left( {\delta - \bar \delta } \right)^{\frac{3}{2}}} \\ \bar \delta = {\delta _m} - {\left( {\dfrac{{3{F_m}}}{{4{E^ * }\sqrt {{{\bar R}_b}} }}} \right)^{\frac{3}{2}}},{{\bar R}_b} = R + \dfrac{1}{2}{\delta _m} \\ \end{gathered} $
    $ {\delta _m} $ maximum contact deformation in the
    loading phase; Fm maximum contact force in
    the loading phase
    Burgoyne and Daraio (2014)[144]$ F = \left\{ \begin{gathered} \dfrac{4}{3}{E^ * }\sqrt R {\delta ^{\frac{3}{2}}}\begin{array}{*{20}{c}} {}&{}&{\left( {0 < \delta < {\delta _y}} \right)} \end{array} \\ \delta \left( {\alpha + \beta \ln \delta } \right)\begin{array}{*{20}{c}} {}&{}&{\left( {{\delta _y} < \delta < {\delta _p}} \right)} \end{array} \\ {p_0}{\text{π}} \left( {2 R\delta + {c_2}} \right)\begin{array}{*{20}{c}} {}&{}&{\left( {\delta > {\delta _p}} \right)} \end{array} \\ \end{gathered} \right. $
    $\alpha {\text{ = } }{ {\left( { {\delta _p}{F_y}\ln {\delta _p} - {\delta _y}{F_p}\ln {\delta _y} } \right)} \mathord{\left/ {\vphantom { {\left[ { {\delta _p}{F_y}\ln {\delta _p} - {\delta _y}{F_p}\ln {\delta _y} } \right]} {\left[ { {\delta _y}{\delta _p}(\ln {\delta _p} - \ln {\delta _y})} \right]} } } \right. } {\left[ { {\delta _y}{\delta _p}(\ln {\delta _p} - \ln {\delta _y})} \right]} }$
    $\beta {\text{ = } }{ {\left( { {\delta _y}{F_p} - {\delta _p}{F_y} } \right)} \mathord{\left/ {\vphantom { {\left( { {\delta _y}{F_p} - {\delta _p}{F_y} } \right)} {\left[ { {\delta _y}{\delta _p}(\ln {\delta _p} - \ln {\delta _y})} \right]} } } \right. } {\left[ { {\delta _y}{\delta _p}(\ln {\delta _p} - \ln {\delta _y})} \right]} }$
    $ {F_y} = \dfrac{1}{6}{\left( {{R \mathord{\left/ {\vphantom {R {{E^ * }}}} \right. } {{E^ * }}}} \right)^2}{\left( {1.6{\text{π}} {\sigma _y}} \right)^3},{F_p} = {p_0}{\text{π}} \left( {2 R{\delta _p} + {c_2}} \right),{p_0}{\text{ = }}{c_1}{\sigma _y} $
    c1 and c2 are empirical parameters; $ {\delta _y} $ critical elastic deformation;
    $ {\delta _p} $ critical plastic deformation;$ {\sigma _y} $yield strength
    $ F = \dfrac{4}{3}{E^ * }\sqrt {{R_p}} {\left( {\delta - {\delta _r}} \right)^{\frac{3}{2}}} $
    $ {R_p} = \dfrac{{4{E^ * }}}{{3{F_{\max }}}}\left( {\dfrac{{2{F_{\max }} + {F_y}}}{{2{\text{π}} {p_y}}}} \right),{p_y} = 1.6{\sigma _y} $
    $ {\delta _r} = {\delta _{\max }} - {\left( {\dfrac{{3{F_{\max }}}}{{4{E^ * }\sqrt {{R_p}} }}} \right)^{\frac{2}{3}}} $
    $ {\delta _m} $ maximum contact deformation in the
    loading phase; Fm maximum contact force in
    the loading phase
    Ma and Liu (2015)[113]$ F\left( \delta \right) = \left\{ \begin{gathered} \dfrac{4}{3}{E^ * }{R^{\frac{1}{2}}}{\delta ^{\frac{3}{2}}}\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {}&{} \end{array}}&{}&{}&{\delta < {\delta _y}} \end{array} \\ \delta \left( {{c_1} + {c_2}\ln \dfrac{\delta }{{{\delta _c}}}} \right) + {c_3}\begin{array}{*{20}{c}} {}&{{\delta _y} \leqslant \delta < {\delta _p}} \end{array} \\ {F_p} + {k_1}\left( {\delta - {\delta _p}} \right)\begin{array}{*{20}{c}} {}&{}&{\delta \geqslant {\delta _p}} \end{array} \\ \end{gathered} \right. $
    $\left\{\begin{array}{l}{k}_{1}=2{\text{π} } R\psi {\sigma }_{y},\\ {F}_{p}={\delta }_{p}\left[{c}_{1} + {c}_{2}\mathrm{ln}\left({\xi }^{2}/2\right)\right] + {c}_{3}\\ {c}_{1}=\dfrac{ {p}_{y}\left[1 + \mathrm{ln}\left({\xi }^{2}/2\right)\right]-2\psi {\sigma }_{y} }{\mathrm{ln}\left({\xi }^{2}/2\right)}{\text{π} } R,\\{c}_{2}=\dfrac{2\psi {\sigma }_{y}-{p}_{y} }{\mathrm{ln}\left({\xi }^{2}/2\right)}{\text{π} } R\\ {c}_{3}={F}_{y}-{c}_{1}{\delta }_{y}, \\{F}_{y}={ {\text{π} } }^{3}{R}^{2}{p}_{y}^{3}/6{E}^{\ast }{}^{2}\end{array} \right.$


    $ {p_y} = 1.61{\sigma _y} $;$ \xi $and$ \psi $are the dimensionless coeffcients;
    $ {\delta _y} $ critical elastic deformation;$ {\delta _p} $ critical plastic deformation;
    $ {\sigma _y} $yield strength
    $ F\left( \delta \right) = \left\{ \begin{gathered} \dfrac{4}{3}{E^ * }{R^{\frac{1}{2}}}{\delta ^{\frac{3}{2}}}\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {}&{}&{} \end{array}}&{}&{\delta < {\delta _y}} \end{array} \\ \dfrac{4}{3}{E^ * }{\left( {{R^e}} \right)^{\frac{1}{2}}}{\left( {\delta - {\delta _r}} \right)^{\frac{3}{2}}}\begin{array}{*{20}{c}} {}&{{\delta _y} \leqslant \delta < {\delta _p}} \end{array} \\ \dfrac{4}{3}{E^ * }{\left( {R_p^e} \right)^{\frac{1}{2}}}{\left( {\delta - {\delta _r}} \right)^{\frac{3}{2}}}\begin{array}{*{20}{c}} {}&{\delta \geqslant {\delta _p}} \end{array} \\ \end{gathered} \right. $

    $ R_{ep}^e = \dfrac{{{F^e}R}}{{{F_{\max }}}},{F^e} = \dfrac{4}{3}{E^ * }{R^{\frac{1}{2}}}\delta _y^{\frac{3}{2}} $

    $ R_p^e = \dfrac{{F_p^eR}}{{{F_{ep}}}},F_p^e = \dfrac{4}{3}{E^ * }{R^{\frac{1}{2}}}\delta _p^{\frac{3}{2}} $
    $ {\delta _r} $permanent deformation
    下载: 导出CSV

    表  4  恢复系数模型

    Table  4.   Coefficient of restitution (CoR) model

    AuthorsCoR mathematical modelParameters
    Chang and Ling (1992)[159]${c_r} = \sqrt { { {\left\{ {\dfrac{8}{ {15} }{E^ * }\sqrt R { {\left[ { {\omega _c}{\omega _m}\left( {2 - \dfrac{ { {\omega _c} } }{ { {\omega _m} } } } \right)} \right]}^{\frac{3}{2} } } } \right\} } \mathord{\left/ {\vphantom { {\left[ {\dfrac{8}{ {15} }{E^ * }\sqrt R { {\left[ { {\omega _c}{\omega _m}\left( {2 - \dfrac{ { {\omega _c} } }{ { {\omega _m} } } } \right)} \right]}^{\frac{3}{2} } } } \right] } {\left[ {\dfrac{8}{ {15} }{E^ * }\sqrt R \omega _c^{\dfrac{5}{2} } + {\text{π} } KYR{\omega _m}\left( { {\omega _m} - {\omega _c} } \right)} \right] } } } \right. } {\left[ {\dfrac{8}{ {15} }{E^ * }\sqrt R \omega _c^{\frac{5}{2} } + {\text{π} } KYR{\omega _m}\left( { {\omega _m} - {\omega _c} } \right)} \right] } } }$$ {\omega _c} $critical elastic deformation; $ {\omega _m} $maximum deformation in the
    compression phase; Y yield strength;$ K = 1.282 + 1.158 v $;
    v Poisson ratio
    Thornton (1997)[135]$ {c_r} = {\left( {\dfrac{{6\sqrt 3 }}{5}} \right)^{\frac{1}{2}}}\sqrt {1 - \dfrac{1}{6}{{\left( {\dfrac{{{V_y}}}{{{V_l}}}} \right)}^2}} {\left[ {\dfrac{{\left( {\dfrac{{{V_y}}}{{{V_l}}}} \right)}}{{\left( {\dfrac{{{V_y}}}{{{V_l}}}} \right) + 2\sqrt {\dfrac{6}{5} - \dfrac{1}{5}{{\left( {\dfrac{{{V_y}}}{{{V_l}}}} \right)}^2}} }}} \right]^{\frac{1}{4}}} $Vy yield impact velocity;
    Vl impact velocity
    Wu et al. (2005)[165]$ {c_r} = \left\{ \begin{gathered} 2.08{\left( {\dfrac{{{V_1}}}{{{V_c}}}} \right)^{0.156}}\begin{array}{*{20}{c}} {}&{\left( {{V_1} < {V_f}} \right)} \end{array} \\ 0.62{\left( {\dfrac{{{V_1}{S_y}}}{{{V_c}{E^ * }}}} \right)^{ - \frac{1}{2}}}\begin{array}{*{20}{c}} {}&{\left( {{V_1} > {V_f}} \right)} \end{array} \\ \end{gathered} \right. $Vc initial impact velocity leading to the plastic deformation;
    Vf critical impact velocity; Sy yield stress; V1 impact velocity
    Weir and Tallon (2005)[161]$ {c_r} = 3.1{\left( {\dfrac{{{S_y}}}{{{E^ * }}}} \right)^{\frac{5}{8}}}{\left( {\dfrac{{{R_1}}}{R}} \right)^{\frac{3}{8}}}{\left( {\dfrac{{{c_0}}}{{{v_0}}}} \right)^{\frac{1}{4}}},{c_0} = \sqrt {\dfrac{E}{\rho }} $v0 initial impact velocity; Sy yield stress; R1 contact
    radius after contact; $ \rho $density
    Jackson et al. (2010)[152]$ {c_r} = \left\{ \begin{gathered} 1\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {}&{}&{} \end{array}}&{}&{} \end{array}}&{}&{\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {}&{}&{} \end{array}}&{} \end{array}}&{\left( {0 < {V_1} < 1} \right)} \end{array} \\ 1 - 0.1\ln \left( {{V_1}} \right){\left( {\dfrac{{{V_1} - 1}}{{59}}} \right)^{0.156}}\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {}&{}&{} \end{array}}&{\left( {1 < {V_1} \leqslant 60} \right)} \end{array} \\ 1 - 0.1\ln \left( {60} \right) - 0.1\ln \left( {\dfrac{{{V_1}}}{{60}}} \right){\left( {{V_1} - 60} \right)^{2.36{\varepsilon _y}}}\begin{array}{*{20}{c}} {}&{\left( {60 \leqslant {V_1} \leqslant 1000} \right)} \end{array} \\ \end{gathered} \right. $V1 impact velocity;
    $ {\varepsilon _y} = {{{S_y}} \mathord{\left/ {\vphantom {{{S_y}} {{E^ * }}}} \right. } {{E^ * }}} $; Sy yield stress
    Ma and Liu (2015)[113]$ {c_r} = 0.81{E^ * }^{ - \frac{1}{3}}{\left( {R_p^e} \right)^{ - \frac{1}{6}}}k_1^{\frac{5}{{12}}}{m^{ - \frac{1}{{12}}}}v_0^{ - \frac{1}{6}} $Rep contact radius after plastic deformation;
    k1 contact parameter; m mass; v0 initial impact velocity
    下载: 导出CSV

    表  5  接触参数

    Table  5.   Contact parameters

    Young’s
    modulus/GPa
    Poisson
    ratio
    Radius/cmYield
    strength/MPa
    Mass/kgDensity/
    kg·m−3
    2000.292.0510300.0977800
    650.332.00300.2612700
    下载: 导出CSV

    表  6  误差分析对比

    Table  6.   Comparative analysis of the error percentage

    No.916243140505663
    New4.236.6819.441.449.120.5012.3325.39
    EDEM5.367.3922.4410.0310.1916.3826.5436.79
    下载: 导出CSV
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出版历程
  • 收稿日期:  2022-06-12
  • 录用日期:  2022-08-24
  • 网络出版日期:  2022-08-25
  • 刊出日期:  2022-12-15

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