The governing equations of stress tensor fields are significant to understand the mechanical behaviors of static granular matter. A multiscale method based on the algebra of translation operators is proposed to derive the governing equations of stress transmission in rhombic lattice packings of disks with friction. Except for the two well-known equations, i.e. the symmetry of stress tensor and the stress equilibrium equation, we also obtain a stress-structure equation as the constitutive relation of the lattice packings. A key variable called friction-geometry tensor is introduced for the expression of the constitutive equation and the Janssen coefficient is given from the eigenvalues of this tensor. DEM (Discrete Element Method) simulations are performed to calculate the numerical values of Janssen coefficient and the results are in good agreement well with theoretical predictions.

应力张量场的控制方程对于理解静态颗粒物的力学行为具有重要意义。提出了一种基于平移算子代数的多尺度方法，推导了具有摩擦力的盘形菱形盘根中的应力传递控制方程。除了应力张量的对称性和应力平衡方程这两个众所周知的方程式之外，我们还获得了应力结构方程式作为晶格堆积的本构关系。引入了一个称为摩擦几何张量的关键变量来表达本构方程，并根据该张量的特征值给出了詹森系数。