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Article(id=1246028558989116395, tenantId=1146029695717560320, journalId=1241755870837649424, issueId=1246028557319783390, articleNumber=null, orderNo=null, doi=10.19636/j.cnki.cjsm42-1250/o3.2023.059, pmid=null, cstr=null, oa=null, hot=null, price=null, onlineType=0, articleFormat=0, articleType=null, articleTypeStr=null, receivedDate=1703088000000, receivedDateStr=2023-12-21, revisedDate=null, revisedDateStr=null, acceptedDate=null, acceptedDateStr=null, onlineDate=1775005958554, onlineDateStr=2026-04-01, pubDate=1719244800000, pubDateStr=2024-06-25, doiRegisterDate=null, doiRegisterDateStr=null, onlineIssueDate=1775005958554, onlineIssueDateStr=2026-04-01, onlineJustAcceptDate=null, onlineJustAcceptDateStr=null, onlineFirstDate=null, onlineFirstDateStr=null, sourceXml=null, magXml=null, createTime=1775005958553, creator=13701087609, updateTime=1775005958553, updator=13701087609, issue=Issue{id=1246028557319783390, tenantId=1146029695717560320, journalId=1241755870837649424, year='2024', volume='45', issue='3', pageStart='289', pageEnd='426', issueExtLink='null', onlineDate='null', pubDate='null', beforeIssueId=null, nextIssueId=null, price=null, status=1, issueComplete=1, articleOrder=1, issueType=-1, specialIssue=null, createTime=1775005958156, creator=13701087609, updateTime=1775006058227, updator=13701087609, preIssue=null, nextIssue=null, ext={EN=IssueExt(id=1246028977123471371, tenantId=1146029695717560320, journalId=1241755870837649424, issueId=1246028557319783390, language=EN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=), CN=IssueExt(id=1246028977123471372, tenantId=1146029695717560320, journalId=1241755870837649424, issueId=1246028557319783390, language=CN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=)}, issueFiles=null}, startPage=416, endPage=426, ext={EN=ArticleExt(id=1246028559278523378, articleId=1246028558989116395, tenantId=1146029695717560320, journalId=1241755870837649424, language=EN, title=Numerical Analysis of the Angle of Repose in Particle Systems Using the Discrete Element Method, columnId=1244229834482757770, journalTitle=Chinese Journal of Solid Mechanics, columnName=Research Paper, runingTitle=null, highlight=null, articleAbstract=

The angle of repose in particle systems is a fundamental scientific problem in particle science. A deep understanding of its influencing factors and patterns of variation is of great significance for optimization in fields such as civil and chemical engineering. However, existing research based on experiments is limited by available types of particles and measurement methods, making it difficult to comprehensively reveal the impact regulations of various physical parameters on the angle of repose. This paper conducts a high-precision numerical study on the angle of repose in particle systems using the discrete element method (DEM), uncovering the most important particle property parameters that affect the angle of repose. DEM is a numerical method that directly simulates the motion of complex particle systems. Specifically, this study adopts a rolling friction coefficient to characterize the influence of non-sphericity on the simulation and validates the established model with existing experimental data. In addition, binary images of the heap projection in the vertical plane are utilized to calculate the angle of repose reasonably. Simulation results show that both sliding and rolling friction coefficients are positively correlated with the angle of repose. Increasing the sliding friction coefficient can double the angle of repose, while increasing the angle of repose caused by the rolling friction coefficient will reach an upper threshold. Once reaching this value, the angle of repose cannot be further increased. The particle's Young's modulus and restitution coefficient have a relatively small impact on the angle of repose. At the same time, this study employs the DEM to investigate the heat transfer characteristics of granular heaps. It is observed that during heating, the region with the highest heat flux within granular heaps migrates from the bottom to the top. These results reveal the most important factors affecting the angle of repose and can guide engineering optimization.

, correspAuthors=Shun Yan, authorNote=null, correspAuthorsNote=null, copyrightStatement=null, copyrightOwner=null, extLink=null, articleAbsUrl=null, sourceXml=null, magXml=null, pdfUrl=null, pdf=null, pdfFileSize=null, pdfExtLink=null, richHtmlUrl=null, mobilePdfUrl=null, reviewReport=null, pdfFirstPage=null, abstractGraph=null, abstractGraphContent=null, abstractVideo=null, citation=null, cebUrl=null, magXmlContent=null, mapNumber=null, authorCompany=null, fund=null, authors=null, authorsList=Shun Yan), CN=ArticleExt(id=1246028567503552849, articleId=1246028558989116395, tenantId=1146029695717560320, journalId=1241755870837649424, language=CN, title=基于离散元方法的颗粒系统休止角数值模拟研究, columnId=1241831201896469478, journalTitle=固体力学学报, columnName=研究论文, runingTitle=null, highlight=null, articleAbstract=

颗粒系统堆积的休止角是颗粒学中一个基础性的科学问题，深入认识其影响因素和变化规律对于土木、化工等工程优化具有重要意义. 目前基于实验测量的研究受限于可用的颗粒种类和测量手段，难以完备地揭示各种物理参数对休止角的影响规律. 离散元（Discrete element method，DEM）是一种可直接对复杂的大型颗粒系统的运动进行仿真的数值方法. 本文借助DEM方法对颗粒系统的休止角问题开展了高精度数值研究，揭示了影响颗粒堆积休止角的重要颗粒物性参数. 在DEM建模方面，为了表征非球形度对仿真的影响，本研究采用了滚动摩擦系数，并利用既有的实验数据进行了模型验证. 同时，本文利用堆积体在垂直平面内投影的二值化图像来合理计算休止角. 仿真结果表明，滑动和滚动摩擦系数均与休止角的大小呈正相关，滑动摩擦系数的增加可将休止角提升一倍，而滚动摩擦系数对休止角的影响存在一个上限阈值，达到该阈值后将不能继续增加休止角的大小；颗粒的杨氏模量、回弹系数对休止角的影响不显著. 此外，本文利用DEM方法研究了堆积体的传热特性，发现在加热过程中，堆积体内部具有最高热流率的区域从堆积体的底部迁移到上部.

, correspAuthors=阎顺, authorNote=null, correspAuthorsNote=
** E-mail：.
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label=Fig.2, caption=Schematic of natural granular heaps simulation to obtain the angle of repose of granular heaps using DEM, figureFileSmall=/77H/NMj/AclG86/AgfYng==, figureFileBig=iBVhQfzlpwmvxhAI62btQg==, tableContent=null), ArticleFig(id=1246028569407766967, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246028558989116395, language=CN, label=图2, caption=利用离散元的方法对颗粒自然堆积仿真获取颗粒堆积的休止角示意图, figureFileSmall=/77H/NMj/AclG86/AgfYng==, figureFileBig=iBVhQfzlpwmvxhAI62btQg==, tableContent=null), ArticleFig(id=1246028569487458749, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246028558989116395, language=EN, label=Fig.3, caption=Distribution of force chains between particles in granular heaps (wheat), figureFileSmall=plbLuMsi17YO4e/XDiMQwQ==, figureFileBig=uS1cFZZ4gtd43Uq7T8XXag==, tableContent=null), ArticleFig(id=1246028569567150528, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246028558989116395, language=CN, 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simulations, figureFileSmall=QT6JYaKQogYFmABErxTuBQ==, figureFileBig=g1lesEgr691jTIID8tDeYg==, tableContent=null), ArticleFig(id=1246028569860751824, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246028558989116395, language=CN, label=图5, caption=休止角的实验测量值与仿真计算值对比, figureFileSmall=QT6JYaKQogYFmABErxTuBQ==, figureFileBig=g1lesEgr691jTIID8tDeYg==, tableContent=null), ArticleFig(id=1246028569936249299, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246028558989116395, language=EN, label=Fig.6, caption=Effect of chassis diameter on the angle of repose, figureFileSmall=b9Rjpft2prV4nm+lZC1CXQ==, figureFileBig=EfdqipfbEfBEm8pBVAWhfQ==, tableContent=null), ArticleFig(id=1246028570028523991, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246028558989116395, language=CN, label=图6, caption=底盘直径对休止角的影响, figureFileSmall=b9Rjpft2prV4nm+lZC1CXQ==, figureFileBig=EfdqipfbEfBEm8pBVAWhfQ==, tableContent=null), 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Input parameters in DEM simulation (initial default values)

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输入参数单位值
粒径，dmm4
杨氏模量，YGPa0.1
泊松比，ν—0.25
回弹系数，e—0.3
颗粒真密度，ρkg/m31137
滚动摩擦系数，μr—0.15
滑动摩擦系数，μfrac—0.35
比热容，CpJ/（kg·K）1260
热导率，kW/（m·K）3.0
), ArticleFig(id=1246028571194540544, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246028558989116395, language=CN, label=表1, caption=

DEM仿真中的输入参数（初始默认值）

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输入参数单位值
粒径，dmm4
杨氏模量，YGPa0.1
泊松比，ν—0.25
回弹系数，e—0.3
颗粒真密度，ρkg/m31137
滚动摩擦系数，μr—0.15
滑动摩擦系数，μfrac—0.35
比热容，CpJ/（kg·K）1260
热导率，kW/（m·K）3.0
), ArticleFig(id=1246028571316175362, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246028558989116395, language=EN, label=Table 2, caption=

Measured values of the angle of repose of several seeds[33]

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种子类型密度（kg/m3）等效粒径（mm）休止角（°）滚动摩擦系数（-）
油菜籽12301.8263.0
黑麦12102.5211.5
小麦13003.4273.1
蚕豆11308.6283.3
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几种作物种子的休止角的实验测量结果[33]

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种子类型密度（kg/m3）等效粒径（mm）休止角（°）滚动摩擦系数（-）
油菜籽12301.8263.0
黑麦12102.5211.5
小麦13003.4273.1
蚕豆11308.6283.3
), ArticleFig(id=1246028571500724745, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246028558989116395, language=EN, label=Table 3, caption=

The variation of temperature and heat flux distribution over time in granular heaps

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温度分布热流率分布
t=20 s
t=200 s
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不同物理时间下颗粒堆积体的温度及热流率分布

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温度分布热流率分布
t=20 s
t=200 s
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基于离散元方法的颗粒系统休止角数值模拟研究
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固体力学学报 | 研究论文 2024,45(3): 416-426
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固体力学学报 | 研究论文 2024, 45(3): 416-426
基于离散元方法的颗粒系统休止角数值模拟研究
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阎顺**
作者信息
  • 中铁第四勘察设计院集团有限公司电化院，武汉，430070

通讯作者:

** E-mail：.
Numerical Analysis of the Angle of Repose in Particle Systems Using the Discrete Element Method
Shun Yan**
Affiliations
  • Electrified Design Institute, China Railway Siyuan Survey and Design Group Co, Ltd, Wuhan, 430070
出版时间: 2024-06-25 doi: 10.19636/j.cnki.cjsm42-1250/o3.2023.059
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摘要
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颗粒系统堆积的休止角是颗粒学中一个基础性的科学问题，深入认识其影响因素和变化规律对于土木、化工等工程优化具有重要意义. 目前基于实验测量的研究受限于可用的颗粒种类和测量手段，难以完备地揭示各种物理参数对休止角的影响规律. 离散元（Discrete element method，DEM）是一种可直接对复杂的大型颗粒系统的运动进行仿真的数值方法. 本文借助DEM方法对颗粒系统的休止角问题开展了高精度数值研究，揭示了影响颗粒堆积休止角的重要颗粒物性参数. 在DEM建模方面，为了表征非球形度对仿真的影响，本研究采用了滚动摩擦系数，并利用既有的实验数据进行了模型验证. 同时，本文利用堆积体在垂直平面内投影的二值化图像来合理计算休止角. 仿真结果表明，滑动和滚动摩擦系数均与休止角的大小呈正相关，滑动摩擦系数的增加可将休止角提升一倍，而滚动摩擦系数对休止角的影响存在一个上限阈值，达到该阈值后将不能继续增加休止角的大小；颗粒的杨氏模量、回弹系数对休止角的影响不显著. 此外，本文利用DEM方法研究了堆积体的传热特性，发现在加热过程中，堆积体内部具有最高热流率的区域从堆积体的底部迁移到上部.

离散元方法  /  休止角  /  控制变量法  /  颗粒堆积体

The angle of repose in particle systems is a fundamental scientific problem in particle science. A deep understanding of its influencing factors and patterns of variation is of great significance for optimization in fields such as civil and chemical engineering. However, existing research based on experiments is limited by available types of particles and measurement methods, making it difficult to comprehensively reveal the impact regulations of various physical parameters on the angle of repose. This paper conducts a high-precision numerical study on the angle of repose in particle systems using the discrete element method (DEM), uncovering the most important particle property parameters that affect the angle of repose. DEM is a numerical method that directly simulates the motion of complex particle systems. Specifically, this study adopts a rolling friction coefficient to characterize the influence of non-sphericity on the simulation and validates the established model with existing experimental data. In addition, binary images of the heap projection in the vertical plane are utilized to calculate the angle of repose reasonably. Simulation results show that both sliding and rolling friction coefficients are positively correlated with the angle of repose. Increasing the sliding friction coefficient can double the angle of repose, while increasing the angle of repose caused by the rolling friction coefficient will reach an upper threshold. Once reaching this value, the angle of repose cannot be further increased. The particle's Young's modulus and restitution coefficient have a relatively small impact on the angle of repose. At the same time, this study employs the DEM to investigate the heat transfer characteristics of granular heaps. It is observed that during heating, the region with the highest heat flux within granular heaps migrates from the bottom to the top. These results reveal the most important factors affecting the angle of repose and can guide engineering optimization.

discrete element method  /  angle of repose  /  control variable method  /  granular heaps
阎顺. 基于离散元方法的颗粒系统休止角数值模拟研究. 固体力学学报, 2024 , 45 (3) : 416 -426 . DOI: 10.19636/j.cnki.cjsm42-1250/o3.2023.059
Shun Yan. Numerical Analysis of the Angle of Repose in Particle Systems Using the Discrete Element Method[J]. Chinese Journal of Solid Mechanics, 2024 , 45 (3) : 416 -426 . DOI: 10.19636/j.cnki.cjsm42-1250/o3.2023.059
正文
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0　引言
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颗粒堆积的休止角（Angle of repose）是一个重要的物理参数，了解休止角的大小在很多工业和工程领域具有广泛和深远的意义[1]. 在岩土工程中，土体的休止角对于土坡稳定性、隧道爆破施工对既有高压电力铁塔安全性等方面具有关键作用[2]. 在矿石处理和存储中，准确预估矿石休止角能帮助设计更有效的料仓和传输系统. 了解谷物或其他农作物颗粒的休止角有助于储存和运输过程的优化. 在铁路隧道爆破施工中，了解土体的休止角可以提供山体滑坡、高压电力铁塔下沉和倾覆等灾害预防和预测的重要信息[1,3]. 可见，颗粒堆积的休止角在各种实际应用和科学研究中具有重要的参考价值和实用价值[3]. 从目前已发表的颗粒系统休止角相关的文献来看，研究方法大多基于实验测量[4]. 相比于实验测量，数值模拟的方法具有灵活、成本低的特点，且利用计算机可同时开展大量的模拟实验，从更加细观的尺度研究休止角的影响因素和变化规律.
离散元方法（Discrete element method，DEM）是一种基于单颗粒分辨率建模的数值模拟技术[5]，主要用于分析颗粒物料在离散状态下的力学行为和相互作用. 基于DEM，颗粒间的宏观行为是通过模拟微观颗粒之间相互作用的集合来得出的. DEM已广泛应用于土木、化工等领域的学术研究以及工程优化[6-9]. 在涉及颗粒堆积体的研究领域，张庆武等[10]研究了颗粒堆积体的应力凹陷问题，他们认为，底面粗糙度较大时切向应力较大，相应的法向应力“中心凹陷”现象越明显；孙其诚等[11]从颗粒接触力链分布的角度考察了颗粒摩擦系数对颗粒堆特性的影响，对重力作用下12000个球心共面的二维等径颗粒静态堆积进行了离散动力学模拟，对力链分布特征、接触力规律等做了量化分析. 然而，目前针对于堆积体的休止角问题的DEM数值研究仍然较少，尤其颗粒的物性参数对休止角问题，尚未见相关文献发表. 出现这种情况的一个重要原因是，实际应用中堆积体大多是由非球形颗粒构成，而具有复杂不规则外形的颗粒运动较难直接利用DEM方法直接计算.
目前存在多种直接表示非球形颗粒外形的DEM方法[12]，包括超椭球[13,14]、多球体[15]和多面体方法[16,17]. 其中，超椭球方法可通过调整角块参数和三个轴来直接创建一些特定的复杂颗粒形状[18]；相比之下，多球体方法的思路是利用多个刚性连接的球体构建非球形颗粒，从而形成单一的非球形实体；在多面体方法中，多边形/多面体颗粒的外形是通过角点、边和面来描述的，其适用于具有锐利边缘和/或平坦表面的颗粒[12]. 然而，这三种方法的缺点是计算量大且颗粒碰撞接触的检测步骤繁琐[12]. 近些年已出现了一些非球形颗粒建模的新方法，如引入滚动摩擦阻力的来表征非球形颗粒外形的影响[12,19-21]. 虽然颗粒的几何效应不能严格归类为滚动摩擦，但它确实是滚动阻力的重要来源. 这种方法已被用来研究沙堆的形成过程[22]，故适用于颗粒堆积体休止角的研究. 此外，这种通过对球形颗粒引入滚动摩擦来表征任意非球形颗粒的方法可以显著减少模拟计算时间[23]，这对工业建模至关重要.
本文研究目的在于利用DEM方法对颗粒堆积体的休止角进行了系统性的数值研究，探索了可影响休止角的重要的颗粒物性参数. 为研究颗粒外形因素对休止角的影响，在DEM仿真中将重点考察休止角随着颗粒的滚动摩擦阻力系数的变化趋势. 本研究通过揭示堆积颗粒系统休止角的影响因素，为工程应用领域颗粒堆积系统的参数优化提供了参考.
1　数学模型
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1.1　DEM控制方程
DEM将每个颗粒视为单独的实体，通过模拟颗粒之间的碰撞、接触、滑动以及分离等相互作用来输出颗粒体系的动力学和变形特性. 单个颗粒的平移和旋转运动由牛顿第二定律定律求解：
式中，j是与颗粒i接触的颗粒的编号，Fn，ij和Ft，ij是接触力Fc的法向和切向分量，Mr，ij是阻碍颗粒滚动的扭矩的大小. Fc采用最初由Tsuji等[24]提出的简化Hertz-Medlin接触力模型来计算接触力：
其中，Reff，ij和meff，ij分别是接触颗粒的有效半径和质量，kn、kt是颗粒表面法线和切向方向的弹性常数，γn和γt是颗粒表面法线和切向方向的粘弹性阻尼常数. 参数kn、kt、cn和ct通过颗粒材料的性质（如杨氏模量（Y）、泊松比（ν）、恢复系数（e）和摩擦系数（μfri）等计算得到，相应参数的关系式见参考文献[25].
如前所述，在本模型中滚动阻力被视为一种形状相关的参数[12]，并利用弹簧-阻尼模型[26]来计算滚动阻力. 滚动阻力（颗粒-颗粒或颗粒-墙壁）等于弹簧扭矩与阻尼扭矩之和，即：
弹簧扭矩类似于塑性材料的加载-卸载应力-应变曲线[26]，它是根据颗粒（或颗粒-壁）在接触中的旋转角度（θr）和滚动刚度（kr）的增量方式计算的，即：
的值需满足小于滚动角度（θm）对应的满载滚动力矩的条件，由以下公式得出：
其中，Req是等体积球的半径. 项与滚动角速度成正比：
其中，Cr是阻尼系数. 在完全滚动的情况下，阻尼系数将会被忽略，即.
上述DEM的计算模型基于GPU加速的开源软件blazedemGPU[27]部署，以实现高效的并行计算. 此外，根据Rayleigh准则[28]，DEM计算的最大时间步长. 本研究中，仿真的时间步长设置为10-6 s，以实现可靠的仿真计算.
在以上颗粒运动计算的基础上，本研究考虑了颗粒间的接触传热. 为了计算逐颗粒的温度，颗粒温度与每个颗粒的运动一同被追踪. 颗粒i的能量守恒方程可表达为[29]：
其中，Cp，i和Ti为颗粒材料的比热容以及颗粒的温度，j是与颗粒i相接触的颗粒的编号，qpp，ij和qpw，ij是颗粒i与颗粒j之间以及颗粒i与壁面间的接触传热，可以根据两者的温差直接计算：
其中hij为传热系数，可以通过两个颗粒各自的导热率（ki和kj）得到，Ac，ij是两个颗粒的碰撞接触面积.
1.2　建模过程
图2展示了利用DEM实现颗粒系统仿真并获取系统堆积的休止角的流程. 具体的步骤可简述为：（1）在柱形容器中注入一定体积的颗粒；（2）移除柱形容器的侧壁面，但保留底面，让颗粒堆积系统自由散落；（3）待颗粒系统稳定后，测量剩余在底面上的颗粒的堆积底角即为休止角. 底角的确定方法是先提取存留在底面上方的颗粒堆积区域的投影，并将投影图像二值化（颗粒占据的区域的像素值为1）；设定一个与二值化后为1的像素区域面积相同的等腰三角形，取等腰三角形的底角作为堆积体的休止角.
为了研究不同的输入参数对于堆积颗粒的休止角的影响，本文利用控制变量法开展了数值仿真研究. 改变的输入变量包括杨氏模量、回弹系数、滑动摩擦系数、滚动摩擦系数，研究这些因素对于颗粒堆积的休止角的影响. 表1列出了DEM仿真中使用的默认的输入参数，在下文的仿真研究中，若未单独提及，则采用默认的输入参数.
2　结果与讨论
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2.1　模型校核及尺寸无关性测试
DEM仿真都是基于理想的输入参数下，而实际颗粒的颗粒物性很难直接测量得到. 为了校核DEM模型的可靠性，表2中列出了文献中已有的几种典型的植物种子（非球形颗粒）的休止角. 由于滚动摩擦系数用来模拟非球形颗粒的影响，是一个重要的输入参数. 因此，本研究通过经验公式[21,30]估算滚动摩擦系数的值：
其中，<e>代表了所有可能接触中接触点离心率的平均值，可根据颗粒的几何尺寸直接估算[30,31]，Req是与非球形颗粒体积相同的等效球形颗粒的半径[21].
图3为DEM计算得到的一个典型的颗粒堆积体的外形轮廓和力链分布，根据该数据，进一步提取了基底表面颗粒间的应力沿径向的分布，如图4所示. 可以看出，堆积体均呈现出典型的应力“中心凹陷”现象[11,32]，即应力在堆积体的中央并非最大值，而是从堆积体区域中心到边缘先增加后减小的现象. 图5对比了通过实验测量和DEM计算两种方法得到的休止角的值. 可以看出，针对不同的颗粒类型，DEM准确地预测了休止角的值，并呈现出相同的变化趋势. 这说明了本文的DEM模型的可靠性.
DEM仿真计算中输入计算颗粒的总数量直接决定了整体的计算量. 为了排除颗粒总数量对于计算结果的影响，开展了仿真尺寸的无关性测试. 计算颗粒的粒径为4 mm，仿真模型中的柱体底盘的直径变化范围为40 mm～160 mm. 图6给出了休止角随着柱体底面直径的变化趋势，可以看出，粒径和底盘尺寸的比率对休止角的变化影响不大，而且最大值（底盘直径=160 mm）和最小值（底盘直径=40 mm）的误差在各自的偏差值的范围内. 为了减少计算量，本研究后续的计算均基于底盘直径为40 mm的仿真工况开展.
2.2　杨氏模量的影响
常规的颗粒材料的杨氏模量的变化范围是5×10-3 GPa～1 GPa，故将其作为DEM仿真中采用杨氏模量的范围. 从图7可以看出，杨氏模量在近3个数量级的变动范围内对休止角的影响微小，且不具有单调性的变化趋势. 出现这种现象的原因是，杨氏模量主要影响到颗粒碰撞的瞬态行为，而休止角是堆积颗粒的稳态性质，与之相关性不大.
2.3　回弹系数的影响
回弹系数是一个用于描述两个物体发生碰撞后能量恢复程度的物理量，故其取值范围在0和1之间. 当回弹系数等于1时，可称这种碰撞为“完全弹性碰撞”. 在这种情况下，两个物体碰撞后会分开，并且系统（两个物体）的总机械能（包括动能和势能）在碰撞前后保持不变. 如果回弹系数等于0，这样的碰撞被称为“完全非弹性碰撞”. 在这种情况下，两个物体碰撞后会粘在一起. 回弹系数大于0但小于1时，系统的机械能会有一定程度的损失，转化为其他形式的能量. 图8展示了回弹系数对休止角的影响，可以看出，随着回弹系数的增加，颗粒系统的休止角逐渐下降，但下降的值在3°的范围内. 造成这种趋势的原因是，在较大的回弹系数的情况下，颗粒的回弹运动比较明显，颗粒动能损耗较慢，颗粒体系很难快速稳定在某个堆积下，故降低了休止角.
2.4　滑动摩擦系数的影响
滑动摩擦系数是用于描述这种情况下摩擦力和正压力之间比例关系的量度. 当两个物体在接触面上相对滑动时，产生的摩擦力与它们之间的正压力（垂直于接触面的力）有关. 从图9可以看出，滑动摩擦系数的上升显著提升了休止角，在值为1时，休止角提升了近一倍到了40°. 滑动系数力上升的时候，由于颗粒间的摩擦力上升，上层颗粒滑落到底部的阻碍增加，颗粒间更不容易发生相对运动，从而造成了休止角的快速升高.
2.5　滚动摩擦系数的影响
滚动摩擦通常比滑动摩擦小，需要较小的力就能使物体滚动. 因此，堆积颗粒系统中颗粒由上到下的运动过程滚动现象占主要因素. 滚动摩擦系数的上升会使颗粒滚动的阻碍上升，使上层的颗粒滚动到底层更加困难，从而提升颗粒堆积的休止角. 从图10可见，休止角随着滚动摩擦系数的上升而上升，但当滚动摩擦系数的值大于7的时候，继续提升该系数的值对休止角的影响已经不再显著. 造成该趋势的原因是，当滚动摩擦系数大于7时，颗粒从上层落到底层的运动方式为颗粒间的滑动，滚动摩擦的影响成为次要影响因素. 如前所述，非球形颗粒在DEM仿真中可用球形颗粒的滚动摩擦系数来表征，故休止角会随着颗粒的非球形度的增加（滚动摩擦系数的增大）而升高.
由以上分析可见，影响堆积体系的休止角的外部变量主要包括滚动和滑动摩擦系数. 根据大量的DEM仿真结果，可采用曲面拟合的方法获得了堆积体的休止角计算公式. 图11为拟合曲面图，拟合得到的休止角（θ）和滚动摩擦系数（μr）、滑动摩擦系数（μf）的三次多项式的形式为：
2.6　堆积体传热特性研究
为了研究堆积体的传热特性，本研究提取了典型堆积体颗粒的温度以及热流密度分布，如表3所示. 在仿真模型中，堆积体的底层施加恒温边界（T=800 K），堆积体的初始温度设为300 K. 图12为利用表3的分布得到的颗粒堆积体内温度以及热流密度沿径向变化曲线. 从图中可以看出，堆积体的底面加热过程中边缘区域的颗粒相比于中心区域的颗粒温度升高更加快速，但是热流密度的分布呈现相反的趋势. 这是因为边缘区域的颗粒吸收热量后，由于其上方不存在与其他颗粒间的接触面，故不能向其上层传递热量，造成了较快的温升；待温度升高后，颗粒间的温差缩小，从而造成了热量密度的整体降低. 可见，颗粒的接触面对堆积体的导热速率有决定性的影响. 值得注意的是，随着加热的进行，热流密度的值逐渐降低，且具有最高热流率的区域从堆积体的底部迁移到堆积体的上部（见表3）. 这也反映了堆积体的温升并不是均匀的，存在一个自下到上的逐渐扩散的过程. 这种现象尚未见报于已发表的文献中.
3　结论
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本研究采用DEM方法深入探讨了颗粒学中的基础性问题——颗粒堆积的休止角. 为了全面揭示影响休止角的各种因素，采用了控制变量法开展了大量的仿真研究. 本研究的一个关键创新点在于，在DEM仿真中用球形颗粒的滚动摩擦系数来表征非球形颗粒运动. 本研究主要得到了以下主要结论：
（1）杨氏模量对于休止角没有显著的影响，而回弹系数的增加可以一定程度上降低休止角，但是影响较小；滑动摩擦系数的提升可显著增加休止角的大小，对于本项研究中的堆积颗粒系统，最高可提升1倍；
（2）滚动摩擦系数的上升会增加颗粒发生滚动阻力，使上层的颗粒滚动到底层更加困难，从而提升颗粒堆积的休止角. 但当滚动摩擦系数的值大于7的时候，休止角的变化已经不明显. 造成这种趋势的原因是，颗粒从上层落到底层的运动方式转换为颗粒间的滑动，颗粒的滚动摩擦变为次要因素；
（3）针对堆积体的热传导仿真结果揭示了随着加热的进行，整体的颗粒间的热流密度值逐渐降低，但具有最高热流率的区域会从堆积体的底部逐渐向上迁移.
总之，本文采用DEM数值仿真方法实现了实验测量难以达到的控制参数研究，从而更为完备地揭示了决定颗粒堆积休止角的因素，同时也揭示了堆积体的一个重要的热传导规律，为工程和工业过程，如铁路隧道爆破施工对既有高压电力铁塔下沉和倾覆计算评估等，提供了参考.
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  • 精细爆破国家重点实验室开放基金项目(42172286)
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2024年第45卷第3期
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doi: 10.19636/j.cnki.cjsm42-1250/o3.2023.059
  • 接收时间：2023-12-21
  • 首发时间：2026-04-01
  • 出版时间：2024-06-25
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  • 收稿日期：2023-12-21
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精细爆破国家重点实验室开放基金项目(42172286)
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    中铁第四勘察设计院集团有限公司电化院，武汉，430070

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2种不同金属材料的力学参数
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Family		 属数
Number of
genus		 种数
Number of
species		 占总种数比例
Percentage of
total species (%)		 属
Genus		 种数
Number of
species		 占总种数比例
Percentage of total
species (%)
鹅膏菌科Amanitaceae 2 11 5.26 鹅膏菌属 Amanita 10 4.78
小菇科 Mycenaceae 2 12 5.74 丝盖伞属 Inocybe 5 2.39
多孔菌科 Polyporaceae 8 14 6.70 蜡蘑属 Laccaria 5 2.39
红菇科 Russulaceae 3 23 11.00 小皮伞属 Marasmius 6 2.87
小菇属 Mycena 11 5.26
光柄菇属 Pluteus 5 2.39
红菇属 Russula 17 8.13
栓菌属 Trametes 5 2.39
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