RESEARCH PROGRESS OF CONTACT FORCE MODELS IN THE COLLISION MECHANICS OF MULTIBODY SYSTEM
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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.Abstract: Impact behavior is a ubiquitous phenomenon in multibody systems. The contact force model is a pivotal tool to predict the contact characteristics of multibody systems. At present, there are two kinds of contact models used for calculating impact behaviors: the static elastoplastic contact force model and the continuous contact force models with energy dissipation. There are many similarities and discrepancies among them in the impact dynamics of multibody systems. This review starts with the introduction of development history of these two kinds of contact models followed by their development progress and background illustrated in detail. Firstly, whether the initial impact velocity is contained in the denominator of damping term severs as a criterion to classify the continuous contact force model as two types of models that are the contact force model with hysteresis damping factor and the other one with viscous damping factor. The relationship between the power exponent and Hertz contact stiffness is analyzed. The problems in calculating the elastic-plastic contact collision behavior by using the existing continuous contact force models are discussed. Secondly, the static elastoplastic contact force models with the continuous transition between the pure elastic and full plastic are introduced, and its characteristic is illustrated when calculating the elastoplastic collision events. The coefficient of restitution acts as the bridge to connect the static elastoplastic contact model and dynamic dashpot model as a complete system. In order to sidestep the error from the Hertz contact stiffness in calculating the elastoplastic impact behavior, a new viscous damping factor is derived by means of the linear elastoplastic contact stiffness based on energy conservation. The intrinsic connection between the static elastoplastic model and the dashpot model is explored, which proves that the artificial damping describing energy dissipation is equivalent to the one generated by the discrepancy between the loading and unloading paths. In order to avoid the numerical singularity caused by the initial impact velocity in the denominator of damping when calculating the dynamic performance of granular matter, a continuous contact force model with viscous damping is obtained by solving a linear single degree of freedom underdamped vibration system. One-dimension chain is taken as the numerical example to validate that the new dashpot model is more accurate than the one used in the EDEM software. Finally, the current research status of impact dynamics of multibody systems is reviewed, and the development trend and future challenges of contact force models are briefly summarized.
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Key words:
- multibody system /
- impact /
- energy dissipation /
- coefficient of restitution /
- contact force model
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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 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 n Impact parameter m Hysteresis damping factor$ \chi $ Hunt-Crossley model [61] 3/2 3/2 $ \chi = \dfrac{{3\left( {1 - {c_r}} \right)}}{2}\dfrac{K}{{{{\dot \delta }^{\left( - \right)}}}} $ Herbert-McWhannell model [76] 3/2 3/2 $ \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] 3/2 3/2 $ \chi = \dfrac{{3\left( {1 - {c_r}} \right)}}{4}\dfrac{K}{{{{\dot \delta }^{\left( - \right)}}}} $ Lankarani-Nikravesh model[66] 3/2 3/2 $ \chi = \dfrac{{3\left( {1 - c_r^2} \right)}}{4}\dfrac{K}{{{{\dot \delta }^{\left( - \right)}}}} $ Gonthier et al. model[75] 3/2 3/2 $ \chi \approx \dfrac{{1 - c_r^2}}{{{c_r}}}\dfrac{K}{{{{\dot \delta }^{\left( - \right)}}}} $ Zhiying-Qishao model [67] 3/2 3/2 $\chi = \dfrac{ {3\left( {1 - c_r^2} \right){{\rm{e}}^{2\left( {1 - {c_r} } \right)} } } }{4}\dfrac{K}{ { { {\dot \delta }^{\left( - \right)} } } }$ Flores et al. model [68] 3/2 3/2 $ \chi = \dfrac{{8\left( {1 - {c_r}} \right)}}{{5{c_r}}}\dfrac{K}{{{{\dot \delta }^{\left( - \right)}}}} $ Gharib-Hurmuzlu model [77] 3/2 3/2 $ \chi = \dfrac{1}{{{c_r}}}\dfrac{K}{{{{\dot \delta }^{\left( - \right)}}}} $ Hu-Guo model [70] 3/2 3/2 $ \chi = \dfrac{{3\left( {1 - {c_r}} \right)}}{{2{c_r}}}\dfrac{K}{{{{\dot \delta }^{\left( - \right)}}}} $ Hu et al. model [78] 3/2 3/2 $\chi {\text{ = } } - \dfrac{ {6.66\;26\ln {c_r} } }{ {3.852\;38 + \ln {c_r} } }\dfrac{K}{ { { {\dot \delta }^{\left( - \right)} } } }$ Shen et al. model [71] 3/2 3/2 $ \chi {\text{ = }}\dfrac{{3\left( {1 - {c_r}} \right)}}{{2 c_r^{0.89}}}\dfrac{K}{{{{\dot \delta }^{\left( - \right)}}}} $ Carvalho-Martins model [15] 3/2 3/2 $ \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] 3/2 3/2 $ \chi {\text{ = }}\dfrac{{5\left( {1 - {c_r}} \right)}}{{4{c_r}}}\dfrac{K}{{{{\dot \delta }^{\left( - \right)}}}} $ Zhang et al. model [72] 3/2 3/2 $\chi {\text{ = } }\dfrac{ {3\left( {1 - {c_r} } \right)} }{ {2\left( {0.618{{\rm{e}}^{ - 3.25{c_r} } } + 0.899{{\rm{e}}^{0.090\;25{c_r} } } } \right){c_r} } }\dfrac{K}{ { { {\dot \delta }^{\left( - \right)} } } }$ Zhao et al. model [22] 3/2 3/2 $ \chi {\text{ = }}\dfrac{{4\left( {1 - {c_r}} \right)}}{{1.302{c_r}}}\dfrac{K}{{{{\dot \delta }^{\left( - \right)}}}} $ 
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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 $Power exponent n Impact parameter m Viscous damping factor$ \chi $ Kuwabara and Kono [99] 3/2 1/2 $\chi = \dfrac{K}{2}\dfrac{ { { {\left( {3{\eta _2} - {\eta _1} } \right)}^2} } }{ { {3{\eta _2} + 2{\eta _1} } } }\dfrac{ {\left( {1 - {\nu ^2} } \right)\left( {1 - 2\nu } \right)} }{ {E{\nu ^2} } }$($ {\eta _1},{\eta _2} $ are the viscous material constant values) Tsuji et al. [96] 3/2 1/4 $\chi = \dfrac{ {\sqrt 5 } }{2}D,D = 2\left| {\ln {c_r} } \right|\sqrt {\dfrac{ {KM} }{ { { {\text{π}} ^2} + { {\ln }^2}{c_r} } } }$ Jankowski [95] 3/2 1/4 $ \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] 3/2 0 $\chi = TD,D = 2\left| {\ln {c_r} } \right|\sqrt {\dfrac{ {KM} }{ { { {\text{π} } ^2} + { {\ln }^2}{c_r} } } } ,T = \dfrac{ {\delta + \left| \delta \right|} }{ {2\delta } }\exp \left\{ {\dfrac{q}{\varepsilon }\left[ {\left( {\delta - \varepsilon } \right) - \left| {\delta - \varepsilon } \right|} \right]} \right\}$
($ \varepsilon $, $ q $ are constants with unit m)Schwager and Poschel [74] 3/2 0.65 empirical Lee and Herrmann [80] 3/2 1 empirical Ristow [100] 3/2 1 empirical
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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 model Loading phase Unloading 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 forceStronge (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 deformationVu-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 deformationKogut 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 deformationJackson 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 phaseDu 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 deformationBrake (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 phaseBurgoyne 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 phaseMa 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
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表 4 恢复系数模型
Table 4. Coefficient of restitution (CoR) model
Authors CoR mathematical model Parameters 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 ratioThornton (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 velocityWu 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 velocityWeir 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 $densityJackson 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 stressMa 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
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表 5 接触参数
Table 5. Contact parameters
Young’s
modulus/GPaPoisson
ratioRadius/cm Yield
strength/MPaMass/kg Density/
kg·m−3200 0.29 2.05 1030 0.097 7800 65 0.33 2.00 30 0.261 2700
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表 6 误差分析对比
Table 6. Comparative analysis of the error percentage
No. 9 16 24 31 40 50 56 63 New 4.23 6.68 19.44 1.44 9.12 0.50 12.33 25.39 EDEM 5.36 7.39 22.44 10.03 10.19 16.38 26.54 36.79
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