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Granular materials are ubiquitous in nature and industrial production. Investigating the structure of packing is crucial for understanding the physical properties of granular materials. Owing to their symmetry and simple geometry, spherical particles have long served as an ideal model for studying granular packing, yielding numerous research outcomes. In recent years, the influence of particle shape on packing structures has drawn considerable attention. Non-spherical particles, characterized by complex shapes, tend to interlock and form stable structures. Their significant geometric cohesion notably affects the stability and porosity of granular packing. To investigate the structural evolution and compaction mechanisms of three-dimensional concave particles (hexapod-shaped) under external tapping, focusing on the role of geometric cohesion in enhancing mechanical stability, we employ hexapod-shaped particles that are composed of three mutually orthogonal spherocylinders in this study. The granular system subjected to consecutive tapping can reach a stationary state. In the densifying process of the system, packing structures with different volume fractions will be formed. Meanwhile, by combining with X-ray tomography, we can obtain the microstructure. The findings reveal that the volume fraction of “hexapod” particle packing is significantly lower than that of hard-sphere systems. The compaction curves of “hexapod” particles across varying tapping intensities are accurately described by the Kohlrausch-Williams-Watt (KWW) law, which is consistent with hard-sphere system, suggesting a relaxation process governed by heterogeneous modes. Furthermore, both the volume fraction of the steady-state granular packing and the average contact number exhibit an inverse relationship with tapping intensity, increasing as the intensity decreases. A detailed statistical analysis of contact points indicates that the compaction process of “hexapod” particles is predominantly influenced by two factors: the augmentation in the number of neighboring contacting particles and the modification of contact forms. These factors collectively enhance the degree of interlocking among hexapods within the system. Specifically, the compaction process is primarily propelled by the escalation in neighboring contacts and the refinement of contact types, particularly the increase in cylinder-cylinder (cc) contact. This rise in cc contact significantly enhances mechanical stability through strengthening geometric interlocking. This study reveals the structural evolution characteristics of non-spherical particles in the compaction process and provide important experimental support for understanding the unique mechanical and dynamic properties of concave particle packing. This research not only enriches the experimental data of granular packing structures but also offers a new perspective for exploring the universal laws of packing for particles of different shapes. This study is to lay a more solid foundation for the theoretical research and industrial applications of granular materials, thereby promoting technological progress and innovation in related fields. 
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Keywords:
- packing of “hexapod” concave particles /
- compaction process /
- contact structure /
- X-ray tomography technology
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图 1 (a) “六足体”颗粒示意图, 其中紫色部分为圆柱体, 粉色部分为半球帽; (b)实验装置图; (c) CT拍摄示意图; (d)“六足体”颗粒堆积的 CT 切片图
Figure 1. (a) Schematic diagram of “hexapod” particles (the purple part is a cylinder, the pink part is a hemispherical cap); (b) diagram of the experimental setup; (c) schematic diagram of CT scan; (d) CT slice of “hexapod” particle packing.
图 2 (a)不同振动加速度$\varGamma $条件下, 颗粒堆积的压实曲线及KWW拟合曲线(实线); (b)弛豫时间$\tau $与振动加速度$\varGamma $的关系图; (c)稳态体积分数$\varPhi {\text{f}}$与振动加速度$\varGamma $的关系图
Figure 2. (a) Compaction curves of various tapping intensities $\varGamma $and the corresponding KWW fits (lines); (b) relaxation time $\tau $ as a function of tapping intensities $\varGamma $; (c) steady-state packing fraction $\varPhi {\text{f}}$ as a function of tapping intensities $\varGamma $.
图 3 (a)近邻“六足体”颗粒臂之间的表面间距$\Delta r{\text{arm}}$的概率分布及高斯拟合曲线(实线); (b)接触数$Z$与接触阈值$\delta {\text{th}}$的函数关系图
Figure 3. (a) Probability distribution function (PDF) of the surface distance $\Delta r{\text{arm}}$ between the arms of neighboring hexapods and the Gaussian fitting curve (solid line); (b) the contact number $Z$ as a function of the contact threshold $\delta {\text{th}}$.
图 4 (a)不同振动加速度$\varGamma $时, 接触数$Z$随振动次数$t$的变化曲线; (b)不同振动加速度$\varGamma $时, 接触数$Z$随体积分数$\varPhi $的变化曲线
Figure 4. (a) The contact $Z$ number as a function of the tapping number $t$ under different tapping intensities $\varGamma $; (b) the contact number $Z$ as a function of the volume fraction $\varPhi $ under different tapping intensities $\varGamma $.
图 5 (a)接触距离$d{\text{con}}$示意图. 图中白色实心圆表示颗粒中心, 绿色实心圆表示颗粒接触位置, 黑色实线表示过颗粒中心的颗粒臂轴线; (b)接触距离$d{\text{con}}$的概率分布图; (c)不同振动加速度$\varGamma $下, 平均接触距离$\left\langle {d{\text{con}}} \right\rangle $随振动次数$t$的变化曲线; (d)不同振动加速度$\varGamma $下, 平均接触距离$\left\langle {d{\text{con}}} \right\rangle $随体积分数$\varPhi $的变化曲线
Figure 5. (a) Schematic diagram of the contact distance $d{\text{con}}$. The white solid circle represents the center of the particle, the green solid circle represents the contact position of the particles, and the black solid line represents the axis of the particle arm passing through the particle center; (b) probability distribution function (PDF) of the average contact distance $\left\langle {d{\text{con}}} \right\rangle $; (c) the contact distance $d{\text{con}}$ as a function of the tapping number $t$ under different tapping intensities $\varGamma $; (d) the average contact distance $\left\langle {d{\text{con}}} \right\rangle $ as a function of the packing fraction $\varPhi $ under different tapping intensities $\varGamma $.
图 6 (a) 3种接触(cs接触、cc接触、ss接触)类型示意图. 紫色部分为圆柱体, 粉色部分为半球帽, 绿色实心圆表示接触点; (b)不同振动加速度$\varGamma $下, 3种接触类型所占比例随振动次数$t$的变化曲线; (c)不同振动加速度$\varGamma $下, 3种接触类型所占比例随稳态体积分数$\varPhi $的变化曲线
Figure 6. (a) Schematic diagram of three contact types (cs contacts, cc contacts, ss contacts), the purple part is the cylinder, the pink part is the hemispherical cap, and the green solid circle represents the contact point; (b) ratios of three contact types as a function of the tapping number $t$ under different tapping intensities $\varGamma $; (c) ratios of three contact types as a function of the steady-state packing fraction $\varPhi $ under different tapping intensities $\varGamma $.
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