Геометрия блочных сред

Чебышевский сборник. 2024. Т. 25. № 2. С. 102–126.

Канель-Белов А., Kirova V.

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
 

Keywords: эргодическая теорема, Эргодический подход, ergodic theorem