Geometry of block environs

A. Ya. Kanel-Belov ; V. O. Kirova

Čebyševskij sbornik, Tome 25 (2024) no. 2, pp. 102-126.

Résumé

To study a block array, it is important to be able to determine the relative number of blocks that satisfy a given property. Thus, when developing a deposit of facing stone, it becomes necessary to determine the distribution of blocks by volume based on data on fracturing. We will assume (unless otherwise stated) that the cracks are modeled by unbounded planes and are grouped into systems of approximately parallel cracks. Below we consider the model of equidistant cracks and the Poisson model, in which it is assumed that the intersections of each system of cracks with a generic line ${L}$ form a Poisson set of points, and in addition, the unions of any number of these sets of intersection points also form Poisson sets of points. For the model of equally spaced cracks (we will henceforth call it the equally spaced model), an ergodic theorem is proven that relates the averages over volume and over realizations for the number of blocks satisfying this property. A computer program based on this theorem has been developed. The problems of determining the average volume of a block, the distribution of blocks by volume and the yield of so-called tariff (i.e., having a certain size and shape) blocks when developing a deposit of facing stone using stone-cutting machines are also considered.

Détail

Keywords: Ergodic approach, ergodic theorem.

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