Stochastic Approach to the Solution of Boussinesq-Like Problems in Discrete Media
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Stochastic Approach to the Solution of Boussinesq-Like Problems in Discrete Media Ignacio G. Tejada1
Received: 13 December 2019 © Springer Nature B.V. 2020
Abstract 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 continuumbased 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). Keywords Theoretical analysis · Statistical analysis · Statistical mechanics · Discrete-element modelling · Elasticity Mathematics Subject Classification 74A25 · 74A40 · 74A60 · 74E20 · 82B21
B I. G. Tejada
[email protected]
1
Universidad Politécnica de Madrid, Madrid, Spain
I. G. Tejada
1 Introduction The estimation of the stresses caused by surface loads in a continuous, homogeneous and linear elastic half-space is one of the most known problems in elasticity, with considerable applications in geotechnics [23, 24, 36]. The solution to the case of a vertical point force was given by Boussinesq [3]. The two-dimensional version (i.e., a vertical line load acting on the free surface of the half-space) was solved a few years later by Flamant [10]. When point forces are replaced by surface loads, solutions can be obtained from the superposition of infinitesimal loading states. The situations in which there is a vertical load acting on the free surface of a half-space are referred to herein as Boussinesq-like problems. From the point of view of continuum mechanics, the stress field in Boussinesq-like conditions is obtained by solving the equations governing the corresponding linear elastic boundary va
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