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浙江大学学报(工学版)
Journal of ZheJiang University (Engineering Science)  2021, Vol. 55 Issue (2): 348-357    DOI: 10.3785/j.issn.1008-973X.2021.02.015
    
Statistical analysis of fragment shape of rock grain after crushing based on FDEM
Jian ZHOU1(),Gang MA1,*(),Wei ZHOU1,Yong-gang CHENG1,Quan-shui HUANG1,Xue-xing CAO2
1. State Key Laboratory of Water Resources and Hydropower Engineering Science, Wuhan University, Wuhan 430072, China
2. Huaneng Lancang River Hydropower Inc, Kunming 650214, China
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Abstract  

3D scanning technology was employed to obtain the surface point cloud data of real rock grains and these grains were reconstructed by using digital image processing technology, aiming at the influence of fragment size and shape on the strength and the deformation of crushing rock grain. The combined finite-discrete element method (FDEM) was used to simulate the fracture and fragmentation of single grain under compression of flat plates. The fragment shape was characterized and quantified, and then the relationships between fragment shape and grain shape and size were analyzed by identifying each fragment produced by single grain crushing. The sensitivity analysis of grain finite element mesh density was conducted to accurately describe the stress gradient and damage evolution along the crack tip. Results show that at least 5、6 cohesive interface elements are required in the fracture process zone to reduce the effects of the element size. The overall shape of fragments generated after grain crushing was focused on, while the local undulation and roughness changes of fragment surface caused by fracture were ignored. Although the grains shapes are different, there are some generic features of shape distribution of fragments after crushing. The sensitivity of sphericity, aspect ratio, Domokos shape descriptor, and convexity of fragments to grain shape gradually increases, and no significant correlation was found between the distribution of other shape descriptors and fragment size except sphericity. The sphericity distribution of fragments with different sizes shows that the larger size fragments tend to be more angular.

Key words： rock grain      single grain crushing      finite-discrete element method (FDEM)      cohesive zone model (CZM)      shape descriptor     
Received: 08 July 2020      Published: 09 March 2021
CLC:  TV 41  
Fund:  国家自然科学基金资助项目（51825905，U1865204）；华能集团科技资助项目（HNKJ18-H26）
Corresponding Authors: Gang MA     E-mail: 604036445@qq.com;magang630@whu.edu.cn
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Jian ZHOU
Gang MA
Wei ZHOU
Yong-gang CHENG
Quan-shui HUANG
Xue-xing CAO
Cite this article:

Jian ZHOU,Gang MA,Wei ZHOU,Yong-gang CHENG,Quan-shui HUANG,Xue-xing CAO. Statistical analysis of fragment shape of rock grain after crushing based on FDEM. Journal of ZheJiang University (Engineering Science), 2021, 55(2): 348-357.

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http://www.zjujournals.com/eng/10.3785/j.issn.1008-973X.2021.02.015     OR     http://www.zjujournals.com/eng/Y2021/V55/I2/348


基于FDEM的岩石颗粒破碎后碎片形状的统计分析

针对碎片尺寸和形状会影响岩石强度和变形的问题，采用三维扫描技术获得真实岩石颗粒的表面点云数据，然后通过数字图像处理技术重构数字颗粒，利用连续-离散耦合方法（FDEM）模拟单个颗粒在平板压缩下的断裂破碎. 识别颗粒破碎后所产生的碎片，并进行碎片形状的表征和量化，分析碎片形状与颗粒初始形状、碎片尺寸的关系. 为了准确描述裂纹尖端的应力梯度和损伤演化，进行颗粒有限元网格密度的敏感性分析，结果表明，断裂过程区至少需要5、6个界面单元以减弱网格尺寸的影响. 本研究关注颗粒破碎后所产生碎片的整体形态，忽略断裂引起的碎片表面局部起伏和粗糙变化. 尽管所研究颗粒的初始形状存在较大差异，仍发现颗粒破碎后所产生碎片的形状指标分布具有一些共性特征. 碎片的圆度、扁平率、Domokos因子和凸度对颗粒初始形状的敏感性逐渐增强，并且除了圆度外，其他形状指标分布与碎片尺寸之间并未发现显著的相关性. 不同粒径组碎片的圆度分布表明，较大的碎片棱角更明显.


关键词： 岩石颗粒,  单颗粒破碎,  连续-离散耦合方法（FDEM）,  内聚力模型,  形状指标 
Fig.1 Angular granite grains
颗粒编号 $L$/mm $I$/mm $S$/mm ${\rm{EI}}$ ${\rm{FI}}$ ${S_{\rm{f}}}$ ${\psi _{{\rm{3D}}}}$ ${C_{\rm{x}}}$
G1 64.725 55.034 35.883 0.850 0.554 3.274 0.847 0.904
G2 73.791 41.169 35.398 0.558 0.480 3.496 0.787 0.862
G3 84.285 54.624 37.843 0.648 0.449 3.507 0.811 0.928
G4 68.353 55.415 46.224 0.811 0.676 3.117 0.796 0.849
G5 62.311 57.968 48.701 0.930 0.782 3.048 0.808 0.858
G6 63.983 46.226 43.552 0.722 0.681 3.135 0.852 0.921
G7 77.504 48.492 37.44 0.626 0.483 3.436 0.800 0.921
G8 62.171 61.087 39.292 0.983 0.632 3.196 0.838 0.895
G9 61.726 58.954 40.825 0.955 0.661 3.150 0.820 0.884
Tab.1 Shape descriptors of angular granite grains
Fig.2 FDEM model of rock grain
Fig.3 Constitutive relations of cohesive interface element
参数 数值 单位
实体单元 $\; \rho$[1, 19, 25] 2 700 ${{{\rm{kg}}} / {{{\rm{m}}^{\rm{3}}}}}$
$E$[1, 19, 25] 40 ${\rm{GPa}}$
$\nu $[1, 19, 25] 0.2 ?
界面单元 ${k_{\rm{n}}}$ 4.8×1013 ${{\rm{N}} / {{{\rm{m}}^{\rm{3}}}}}$
${k_{\rm{s}}}$ 2.0×1013 ${{\rm{N}} / {{{\rm{m}}^{\rm{3}}}}}$
${f_{\rm{t}}}$[19, 25] 均值为11 ${\rm{MPa}}$,方差为0.5的对数正态分布 ${\rm{MPa}}$
${\varphi _i}$[1, 25] 40 （°）
${\varphi _{\rm{f}}}$[1, 25] 30 （°）
$c$[19, 25] $c = { {15{f_t}\left( {1 - {\rm{sin}}\;{\varphi _i} } \right)} / {\left( {2{\rm{cos}}\;{\varphi _i} } \right)} }$ ${\rm{MPa}}$
$ {G}_{{\rm{I}}}$[25] 100 ${{\rm{N}} / {\rm{m}}}$
$ {G}_{{\rm{II}}}$[25] 500 ${{\rm{N}} / {\rm{m}}}$
接触准则 $\;\mu$[1, 25] 0.577 ?
Tab.2 Input parameters used in FDEM model
Fig.4 Schematic diagram of four different meshes of grain
编号 ${l_{\rm{e}}}$/mm ${n_{\rm{n}}}$ ${n_{\rm{e}}}$ $n_{{\rm{CIE}}}^{\min }$
G-M1 5.18 21 476 13 671 4
G-M2 4.49 31 731 21 064 5
G-M3 3.10 91 751 63 761 7
G-M4 2.21 249 662 176 650 9
Tab.3 Four different meshes of grain
Fig.5 Typical load-displacement curve and external work
Fig.6 Box chart of external work in different meshes
Fig.7 Saw-toothed load-displacement curve
Fig.8 Number of fragments and accumulated fraction of broken cohesive interface element
Fig.9 Ten largest fragments after bulk fragmentation
Fig.10 Cumulative fragment size distribution
Fig.11 Probability distribution of Domokos shape descriptor at end of compression
Fig.12 Frequency distribution of aspect and sphericity at end of compression
Fig.13 Cumulative distribution of shape descriptors
Fig.14 Load-displacement curve and number of fragments
Fig.15 Cumulative distribution of four fragments shape descriptors at end of compression
颗粒 ${S_{\rm{f}}}$ ${S / L}$ ${\psi _{{\rm{3D}}}}$ ${C_{\rm{x}}}$
$q$ $a$ $b$ $a$ $b$ $a$ $b$
G1 2.345 0.506 5.248 0.617 12.330 0.828 3.854
G2 1.881 0.503 3.530 0.601 13.200 0.852 2.810
G3 2.402 0.527 4.618 0.604 10.020 0.834 4.289
G4 2.455 0.543 4.213 0.613 10.620 0.811 4.064
G5 2.193 0.518 4.280 0.606 12.730 0.842 4.056
G6 2.292 0.525 4.078 0.609 11.590 0.804 5.400
G7 2.242 0.525 4.300 0.607 9.881 0.809 3.736
G8 2.595 0.557 4.260 0.617 10.990 0.844 4.093
G9 1.907 0.487 4.327 0.596 12.220 0.871 5.366
Tab.4 Fitting parameters of cumulative distribution for different shape descriptors
Fig.16 Cumulative distribution and fitting of four shape descriptors at end of compression
Fig.17 Cumulative distribution and Kruskal - Wallis test for sphericity of fragments of different size groups
Fig.18 Zingg classification chart of different size groups
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