Stress propagation through frictionless granular material.
Stress propagation through frictionless granular material.
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应力通过无摩擦颗粒材料传播。
DOI：
10.1103/physreve.60.687
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发表时间：
1998
期刊：
影响因子：
--
通讯作者：
T. A. Witten
中科院分区：
文献类型：
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作者：
Alexei V. Tkachenko;T. A. Witten
我们研究了由任意大小的坚硬、无摩擦的球形珠子组成的静态集合中所期望的力网络，例如胶体玻璃。这样的部件是最小连接的：对于大型部件，约束方程与接触力的比率接近1。然而，在组件的有限个子区域中的珠子位置确定不足。因此，为了保持平衡，一半的外部接触力由另一半决定。我们认为力的传递可以被认为是单向的，而不是弹性材料中的力的传递。专门针对顺序沉积的珠子，我们展示了给定埋藏珠子上的力可以唯一地指定为涉及更多新近添加的珠子的力。我们推导出在比单个珠子大得多的尺度上平均的应力传递方程。这一推导要求力的统计涨落与接触几何的涨落无关。在此条件下，d(d+1)/2分量应力场可表示为d分量矢量场。这一过程可以推广到非顺序的填料。在两个维度上，应力按照波动方程传播，就像最近在其他地方的工作中所假设的那样。在假设堆积几何具有单轴对称性的情况下，我们证明了类似的高维波状传播。在宏观颗粒材料中，我们认为我们的方法可能是有用的，即使颗粒具有摩擦力并且不是按顺序堆积的。
We examine the network of forces to be expected in a static assembly of hard, frictionless spherical beads of random sizes, such as a colloidal glass. Such an assembly is minimally connected: the ratio of constraint equations to contact forces approaches unity for a large assembly. However, the bead positions in a finite subregion of the assembly are underdetermined. Thus to maintain equilibrium, half of the exterior contact forces are determined by the other half. We argue that the transmission of force may be regarded as unidirectional, in contrast to the transmission of force in an elastic material. Specializing to sequentially deposited beads, we show that forces on a given buried bead can be uniquely specified in terms of forces involving more recently added beads. We derive equations for the transmission of stress averaged over scales much larger than a single bead. This derivation requires the ansatz that statistical fluctuations of the forces are independent of fluctuations of the contact geometry. Under this ansatz, the d(d+1)/2-component stress field can be expressed in terms of a d-component vector field. The procedure may be generalized to nonsequential packings. In two dimensions, the stress propagates according to a wave equation, as postulated in recent work elsewhere. We demonstrate similar wave-like propagation in higher dimensions, assuming that the packing geometry has uniaxial symmetry. In macroscopic granular materials we argue that our approach may be useful even though grains have friction and are not packed sequentially.