A vertical surface load acting on a half-space made of discrete and elastic particles is supported by a network of force chains that changes with the specific realization of the packing. These force chains can be transformed into equivalent stress fields, but the obtained values are usually different from those predicted by the unique solution of the corresponding boundary value problem. In this research the relationship between discrete and continuum approaches to Boussinesq-like problems is explored in the light of classical statistical mechanics. In the principal directions of the stress established by the continuum-based approach, the probability distribution functions of the extensive normal and shear stresses of particles are anticipated to be exponential and Laplace distributions, respectively. The extensive stress is the product of the volumetric average of the stress field within a region by the volume of that region. The parameters locating and scaling these probability distribution functions (PDFs) are such that the expected values of the extensive stresses match the solution to the corresponding boundary value problem: zero extensive shear stress and extensive normal stresses equal to the principal ones. The continuum-based approach is still needed to know the expected values, but this research article presents a powerful method for quantifying their expected variability. The theory has been validated through massive numerical simulation with the discrete element method. These results could be of interest in highly fragmented, faulted or heterogeneous media or on small length scales (with particular applications for laboratory testing).

中文翻译：

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离散介质中类 Boussinesq 问题的随机求解方法

作用在由离散和弹性粒子组成的半空间上的垂直表面载荷由力链网络支持，该网络随填料的具体实现而变化。这些力链可以转化为等效应力场，但得到的值通常与相应边值问题的唯一解所预测的值不同。在这项研究中，根据经典统计力学探讨了类 Boussinesq 问题的离散和连续方法之间的关系。在基于连续介质的方法建立的应力主方向上，粒子的广泛法向应力和剪切应力的概率分布函数预计分别为指数分布和拉普拉斯分布。扩展应力是区域内应力场的体积平均值与该区域体积的乘积。定位和缩放这些概率分布函数 (PDF) 的参数使得扩展应力的预期值与相应边界值问题的解相匹配：零扩展剪切应力和扩展法向应力等于主应力。仍然需要基于连续体的方法来了解预期值，但这篇研究文章提出了一种量化其预期可变性的强大方法。该理论已通过离散元方法的大量数值模拟得到验证。这些结果可能对高度分散的、