Department of Physics, Yale University, New Haven, Connecticut 06520, USA.
Department of Mechanics and Engineering Science, Peking University, Beijing 100871, China.
Phys Rev E. 2018 Jan;97(1-1):012909. doi: 10.1103/PhysRevE.97.012909.
We perform computational studies of static packings of a variety of nonspherical particles including circulo-lines, circulo-polygons, ellipses, asymmetric dimers, dumbbells, and others to determine which shapes form packings with fewer contacts than degrees of freedom (hypostatic packings) and which have equal numbers of contacts and degrees of freedom (isostatic packings), and to understand why hypostatic packings of nonspherical particles can be mechanically stable despite having fewer contacts than that predicted from naive constraint counting. To generate highly accurate force- and torque-balanced packings of circulo-lines and cir-polygons, we developed an interparticle potential that gives continuous forces and torques as a function of the particle coordinates. We show that the packing fraction and coordination number at jamming onset obey a masterlike form for all of the nonspherical particle packings we studied when plotted versus the particle asphericity A, which is proportional to the ratio of the squared perimeter to the area of the particle. Further, the eigenvalue spectra of the dynamical matrix for packings of different particle shapes collapse when plotted at the same A. For hypostatic packings of nonspherical particles, we verify that the number of "quartic" modes along which the potential energy increases as the fourth power of the perturbation amplitude matches the number of missing contacts relative to the isostatic value. We show that the fourth derivatives of the total potential energy in the directions of the quartic modes remain nonzero as the pressure of the packings is decreased to zero. In addition, we calculate the principal curvatures of the inequality constraints for each contact in circulo-line packings and identify specific types of contacts with inequality constraints that possess convex curvature. These contacts can constrain multiple degrees of freedom and allow hypostatic packings of nonspherical particles to be mechanically stable.
我们对各种非球形颗粒（包括圆形线、圆形多边形、椭圆、不对称二聚体、哑铃等）的静态堆积进行了计算研究，以确定哪些形状的堆积具有比自由度更少的接触（超静定堆积），哪些形状的堆积具有与自由度相等的接触数，并了解为什么尽管非球形颗粒的超静定堆积比基于简单约束计数预测的接触数少，但仍能保持机械稳定。为了生成高度精确的圆形线和 cir-polygons 的力和扭矩平衡堆积，我们开发了一种粒子间势，该势作为粒子坐标的函数给出连续的力和扭矩。我们表明，当所有非球形颗粒堆积的堆积分数和阻塞起始时的配位数相对于颗粒各向异性 A 绘制时，它们服从一种主形，A 与颗粒周长的平方与面积的比值成正比。此外，当在相同的 A 处绘制时，不同颗粒形状的堆积的动力学矩阵的特征值谱会崩溃。对于非球形颗粒的超静定堆积，我们验证了势能随着微扰幅度的四次方增加的“四次”模式的数量与相对于等静压值缺失的接触数量相对应。我们表明，随着堆积压力降低到零，总势能在四次模式方向上的四阶导数仍然不为零。此外，我们计算了圆形线堆积中每个接触的不等式约束的主曲率，并确定了具有凸曲率的具有不等式约束的特定类型的接触。这些接触可以约束多个自由度，并允许非球形颗粒的超静定堆积保持机械稳定。
