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Angle of repose and contact-force distribution in granular pile under variable g
CHEN Hui, ZHAO Xianqiong, LIU Yilun
College of Mechanical and Electrical Engineering, Central South University, Changsha 410083, China
Abstract:Taking the discontinuity and random character of granular system into account, a kinematic model of particles was established by three-dimensional discrete element method in order to investigate the relationship between angle of repose and gravity. Accumulation process of granular pile was simulated using the model under variable gravity. The angle of repose and probabilistic distribution of contact forces were obtained in the model. Results show that the force-chain formed from contact forces within particle pile has the characteristics of structure of irregular mesh. Contact forces within particle pile are approximately log-normal distributed and there are about 65% of contacts carrying a force lower than the mean. Moreover, there are about 70% of contacts whose friction force are fully mobilized and the ratio of tangential force to normal force among the rest of contacts is uniformly distributed. The distributions of contact force of granular piles in different gravity field have similar properties. The contact forces, which are normalized with respect to particle's gravity, of piles with variable gravity have nearly the same distribution. The angle of repose is not affected by the gravity, though granular pile has a randomness property of microscopic structure.
Key words: granular pile     angle of repose     gravity     contact force     discrete element method

 图 1 DEM颗粒模型示意图Fig. 1 Sketch of DEM model of particle

 参数 数值 颗粒半径R/mm 6 颗粒密度ρ/(kg·m-3) 2 090 颗粒-颗粒滑动摩擦系数μt-pp 0.154 颗粒-颗粒滚动摩擦系数μr-pp/mm 0.025 颗粒弹性模量Ep/MPa 57 颗粒-颗粒恢复系数epp 0.5 颗粒-底板恢复系数epb 0.3 颗粒泊松比ξp 0.25 底板泊松比ξb 0.3 底板弹性模量Eb/MPa 2 000 颗粒-底板滚动摩擦系数μr-pb/mm 0.3 颗粒-底板滑动摩擦系数μs-pb 1 底板直径D/mm 700 出料口直径d/mm 50 时间步长Δt/μs 43.4

 图 2 “点源式”颗粒堆积过程Fig. 2 Particle accumulation of “point source type”
2 结果及讨论

 图 3 颗粒堆截面上的力链Fig. 3 Force chain of granular pile in cross section

 图 4 接触力的概率分布Fig. 4 Probability distribution of contact-force

η为接触点切向力与法向力的比值,即η=Fijt/Fijn.图 5η的概率分布,由图可知:在0＜ημt-pp区域,P(η)为均匀分布;η=0.154=μt-pp处,P(η)出现脉冲峰值.η的均匀分布可能是由颗粒堆积过程的随机性导致的,这一特征与受压颗粒系统相同.当η=μt-pp时,表示接触点处于临界滑移状态,单轴实验中受压的颗粒系统约50%的接触点处于临界滑移[8],而图 5表明颗粒堆内约70%的接触点处于临界滑移状态.根据颗粒的堆积过程可以推断:由 于安息角是颗粒堆的临界稳定角,而颗粒堆是由颗粒沿着安息角一层一层堆积而形成,因此,与单轴实验的颗粒系统相比,颗粒堆内部有更多接触点处于临界滑移状态.

 图 5 η的概率分布Fig. 5 Probability distribution of η
2.2 不同重力场中的颗粒堆

 图 6 不同重力场中颗粒堆的力链Fig. 6 Force chains of granular piles under variable g

 图 7 各重力场中颗粒堆的接触力概率分布Fig. 7 Probability distributions of contact-force of granular piles under variable g

 颗粒样本S R/mm μt-pp μr-pp/mm S1 6 0.254 0.05 S2 6 0.254 0.1 S3 6 0.5 0.2 S4 4 0.254 0.05 S5 4 0.254 0.1 S6 4 0.5 0.2

 颗粒样本S α 标准偏差 0.1g0 0.4g0 g0 2g0 S1 23.8 24.5 24.1 24.0 0.06 S2 29.7 29.8 30.1 29.5 0.06 S3 33.3 33.2 33.7 33.0 0.1 S4 24.3 25.0 24.8 24.6 0.1 S5 29.4 30.0 30.1 30.9 0.38 S6 35.5 35.8 36.2 36.0 0.07

 图 8 各样本颗粒堆的接触力概率分布Fig. 8 Probability distributions of contact-force within granular piles for all samples
2.3 讨 论 颗粒堆由力链网络支撑而保持平衡,宏观呈现的倾斜角即为安息角.不同重力场中,颗粒堆力链的微观结构存在随机性差异,但是接触力的统计分布规律具有相似性,即接触力的强度随重力成比例增长,且接触点的η分布规律不受重力大小的影响.根据经典力学原理可知,力链的强度成比例增大后不改变力链网络节点的力学平衡,虽然颗粒堆积后的微观结构存在差异,但是整体结构的稳定性不变,即安息角不受影响.

1) 散体颗粒堆积后,颗粒之间的接触力形成力链,力链结构呈非规则的网络状,且强、弱力链相互交替分布.其中,接触力强度呈对数正态分布,约65%接触力低于平均值,大于平均值的强力链呈指数递减.接触点中约70%处于临界滑移状态,其余接触点的切向力与法向力比值η在[0,μt-pp)区间为均匀分布.

2) 不同重力场中,散体颗粒堆接触力的概率分布具有相似性,即接触力相对颗粒重力无量刚化后的概率密度函数高度相符,且接触点η的概率分布不受重力大小的影响.接触力分布规律的相似性特征不受颗粒粒径大小、粒径级配和粗糙度的影响.颗粒堆的微观结构存在随机性差异,但宏观安息角与重力的大小无关.

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#### 文章信息

CHEN Hui, ZHAO Xianqiong, LIU Yilun

Angle of repose and contact-force distribution in granular pile under variable g

Journal of Beijing University of Aeronautics and Astronsutics, 2015, 41(6): 1141-1146.
http://dx.doi.org/10.13700/j.bh.1001-5965.2014.0468