NON-LINEAR CONTACT MODEL FOR SUPER-QUADRIC ELEMENT CONSIDERING THE EQUIVALENT RADIUS OF CURVATURE1)
WangSiqiang, JiShunying2), State Key Laboratory of Structure Analysis of Industrial Equipment, Dalian University of Technology, Dalian 116023, China 中图分类号:O347.7,O373 文献标识码:A
关键词:离散单元法;超二次曲面单元;等效曲率半径;非线性黏弹性;接触模型 Abstract The super-quadric elements based on the continuous function representation can effectively describe the non-spherical particles in nature and industrial applications, and accurately calculate the contact force between the elements through a non-linear iterative method. For the super-quadric elements with complex geometric shapes, the linear contact force model cannot precisely calculate the contact force under various contact patterns. Considering different shapes and surface curvatures between non-spherical elements, a corresponding non-linear viscoelastic contact force model is developed. In this model, the equivalent radius of curvature is introduced to calculate the elastic contact stiffness and viscous force in normal direction. Meanwhile, the elastic force and the viscous force in tangential direction are simplified based on the contact force model of spherical element. To validate the super-quadric algorithms and the contact force model, the normal collision between spherical particles, the oblique contact between ellipsoidal elements, the static packing of cylinders and the dynamic hopper discharge of ellipsoids are simulated with the super-quadric elements. The proposed method is well verified by finite element numerical results and physical experimental data. The non-linear contact force model of super-quadric element with considering the equivalent radius of curvature can accurately calculate the inelastic collision, so as to reasonably reflect the motion law of the non-spherical particle system. Based on the aforementioned method, the effects of aspect ratio and blockiness on the flow characteristics in the discharging process are further analyzed. The results show spherical particles have the fastest flow rate while cube-like particles have the slowest flow rate. Meanwhile, the flow rate of ellipsoids and cylinder-like particles decreases with increasing or decreasing the aspect ratio. In addition, cube-like particles are more likely to form face-face contacts and have a lower flow rate. The super-quadric element with non-linear contact force model can provide an effective numerical approach to simulate the flow characteristics of non-spherical granular materials.
Keywords:discrete element method;super-quadric element;equivalent radius of curvature;non-linear viscoelastic;contact force model -->0 PDF (13016KB)元数据多维度评价相关文章收藏文章 本文引用格式导出EndNoteRisBibtex收藏本文--> 王嗣强, 季顺迎. 考虑等效曲率的超二次曲面单元非线性接触模型1)[J]. 力学学报, 2018, 50(5): 1081-1092 https://doi.org/10.6052/0459-1879-18-103 WangSiqiang, JiShunying. NON-LINEAR CONTACT MODEL FOR SUPER-QUADRIC ELEMENT CONSIDERING THE EQUIVALENT RADIUS OF CURVATURE1)[J]. Acta Mechanica Sinica, 2018, 50(5): 1081-1092 https://doi.org/10.6052/0459-1879-18-103
采用超二次曲面单元构造10 500个椭球体颗粒, 在长方体容器中考虑单元的随机位置和角度并在重力作用下实现堆积. 超二次曲面方程中函数参数满足: $2a=13.56~\text{mm}$, $2b=2c=7.19~\text{mm}$, $n_1=n_2=2$, 主要的离散元计算参数列于表2中. 容器的长为290 mm, 宽为55 mm, 并在底部设置孔径分别为$D_0=54$和63 mm的卸料口. 当容器内颗粒无相对运动时, 卸料口打开并将离散元模拟的数值结果和文献[49]中的试验结果进行对比. Table 2 表2 表2椭球体动态卸料的主要计算参数 Table 2Major computational parameters of dynamic discharge of ellipsoids
Definition
Value
particle density/(kg • m3)
1338
Young’s modulus/GPa
1.0
Poisson’s ratio
0.29
normal damping coefficient
0.3
tangential damping coefficient
0.3
sliding friction coefficient
0.4
time step/s
5 x 106
新窗口打开 图8给出了$t=0$, 0.9, 1.8, 2.7, 3.6, 4.5, 6.6 s 时超二次曲面颗粒的流动过程和试验结果[49] 的对比. 颗粒在重力作用下呈现V形流动图案. 随着颗粒数量的减小, V 形图案逐渐消失并最终在容器两侧产生堆积. 图9给出了孔径分别为54 mm和63 mm时容器内剩余颗粒百分比随时间的变化曲线. 当孔径$D_0=54~\text{mm}$时, 离散元的数值结果与试验结果基本吻合; 当$D_0=63~\text{mm}$时, 数值结果略小于试验结果. 这主要是由于当孔径变大且颗粒间的相对流速变快时, 摩擦系数和阻尼系数对颗粒流动产生显著影响. 显示原图|下载原图ZIP|生成PPT 图8不同时刻下颗粒流动过程的试验结果[49](上) 和离散元数值结果(下) 的对比 -->Fig.8Snapshots of the flow pattern between experiment[49] (top) and DEM simulation (bottom) at different moments -->
显示原图|下载原图ZIP|生成PPT 图9对于孔径$D_0=54$ 和63 mm, 漏斗内颗粒百分比随时间的变化关系 -->Fig.9Percentage of particles remaining in the hopper as a function of time: $D_0=54$ and 63 mm -->
为对比分析颗粒单元长宽比对卸料过程的影响, 这里分别取长宽比$\sigma=0.5$, 0.75, 1, 1.25 和1.5, 对椭球体和圆柱体颗粒的流动过程进行离散元分析, 其从漏斗中流出体积百分比随时间的变化过程如图11所示. 此外, 图中标出了不同形态的颗粒单元及其所对应的长宽比 . 图11(a) 和图11(b)结果表明, 椭球体和圆柱体颗粒在长宽比$\sigma=1$时流动速率最快, 长宽比增大或减小都会使流动速率减慢. 这主要是由于长宽比增加了颗粒流动过程中的碰撞次数, 同时产生的咬合互锁效应显著影响颗粒的流动性能. 将长宽比为1的球体、圆柱体和正方体颗粒的流出体积百分比随时间的变化关系进行对比, 如图11(c)所示. 可以发现, 球形颗粒具有最快的流动速率, 而块体颗粒的流动速率是最慢的. 对于块体颗粒, 颗粒单元间的面接触会导致颗粒在容器两侧形成堆积, 同时约有59\%的颗粒流出. 该数值小于文献[52]中65\% 块体颗粒流出的试验结果. 这主要是文献[52]中采用的是固定边界, 而本文采用的周期性边界条件消除了固定边界对颗粒堆积的影响, 从而使得颗粒在容器两侧更容易形成堆积以及产生更低的流出量. 显示原图|下载原图ZIP|生成PPT 图11不同长宽比及颗粒形态影响下流出体积百分比随时间的变化关系 -->Fig.11Percentage of discharged volume as a function of time for different aspect ratios and particle shapes -->
3.3 颗粒单元尖锐度对流动特性的影响
为研究颗粒表面尖锐度对颗粒材料流动特性的影响, 这里通过调整超二次曲面方程中单元的形状参数来控制颗粒的表面形态. 为显示形状参数对颗粒表面几何形态的影响, 图12(a)给出了1/4个颗粒单元表面轮廓随形状参数$n_1$和$n_2$从2到7变化过程. 同时, 通过函数形状参数的连续变化可实现颗粒形状从球体到圆柱体、球体到正方体的两种形态转变, 如图12(b)所示. 显示原图|下载原图ZIP|生成PPT 图12函数形状参数对颗粒形状的影响 -->Fig.12The influence of blockiness parameters on the particle shapes -->
按如图12所示颗粒形态的构造方法, 对不同形态颗粒材料从漏斗中流出的卸料过程进行离散元分析. 当颗粒单元形态分别从球体到圆柱体、球体到正方体的变化过程中, 即单元表面尖锐度增加过程中, 颗粒材料从漏斗中流出体积百分比变化过程如图13所示. 从中可以发现, 随着颗粒尖锐度的增加, 颗粒的流动速率均呈减小趋势. 然而, 不同的是从球体到圆柱体变化过程中没有出现容器两侧的堆积, 即颗粒材料均能从漏斗中全部流出; 然而, 颗粒形状从球体到正方体变化过程中, 随着尖锐度的增加, 容器两侧颗粒的堆积数量显著增加. 这主要是由于球体和圆柱体颗粒具有更加光滑的几何表面, 而正方形颗粒随着尖锐度的增加, 颗粒间的面接触可增加系统的稳定性从而降低了颗粒的流动性能. 显示原图|下载原图ZIP|生成PPT 图13单元形态变化过程中颗粒材料从漏斗中流出体积百分数随时间的变化过程 -->Fig.13The time series of percentage of discharged granular materials from hopper with the changing of particle shapes -->
4 结论
本文在基于连续函数包络的超二次曲面方程构造非球形颗粒形态中, 考虑了单元形状及表面曲率对接触模式的影响, 并在传统非线性接触模型的基础上引入局部接触点处的曲率半径, 发展了相应的非线性黏弹性接触模型. 为验证超二次曲面单元的数值算法和力学模型的有效性, 对圆柱体的静态堆积和椭球体的动态卸料过程进行了离散元分析, 计算值与试验结果对比良好. 由此表明基于接触点处等效曲率的超二次曲面模型能合理地描述非球形颗粒形态及运动规律. 在此基础之上, 进一步采用超二次曲面单元数值分析了颗粒长宽比和表面尖锐度对卸料过程中颗粒流动特性的影响. 计算结果表明: 球形颗粒具有最快的流动速率, 而块体颗粒的流动速率最慢. 同时, 增加或减小椭球体和圆柱体颗粒的长宽比都会增加颗粒间的碰撞次数和咬合互锁效应, 从而产生更低的流动速率. 此外, 表面尖锐度会增加颗粒间的面接触, 不利于颗粒间的相对滑动和转动, 进而增强颗粒系统的稳定性并降低颗粒的流动速率. 针对非球形颗粒间复杂的接触模式, 还需要进一步考虑恢复系数与单元间相对速度的变化关系. 同时, 在后续工作中需考虑能量准则的加载和卸载过程并建立基于超二次曲面单元的最大重叠量与塑性变形相关的理论模型. The authors have declared that no competing interests exist.
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