Deformation of solid rock when a concentrated charge is exploded, with allowance for the screening surface

Abstract

In the ore mining industry, large-scale (bulk) blasting into the compensation and worked-out spaces is used to break down ore and for other special purposes [1,2]. The behavior of the rock during an explosion was investigated by precise analytical methods in [3][4][5][6], and on this basis we can obtain the fundamental characteristics of the mechanical effects of an underground blast in the spherically symmetric case. In the cases of axially symmetric problems and of media with markedly nonlinear behavior, analytical solution of the problem involves great mathematical difficulties. The internal screening surfaces and the free boundaries of the rock further complicate the calculations. The use of computers extends the possibilities of solving dynamic problems in the spherically symmetric and axially symmetric cases. Various problems for blasting in various media have been solved by means of numerical methods [7][8][9][10][11]. With the aid of a two-dimensional program developed by the method suggested in [12], we have solved the problem of the propagation of stress waves during explosion of a concentrated charge, taking account of the influence of the undercut slot and the free surface.The problem is formulated as follows: at a depth Z 0 in the solid rock with known mechanical properties lies a charge of explosives. Coaxially with the charge, at a depth Z 1 (Z 1 > Z 0) is a disk-shaDed slot of radius R 1 and thickness h I (Fig. 1). We wish to determine the displacement and velocity, to discussthe interaction of the direct and reflected waves on the contour of the undercut slot, and to assess the influence of undercutting on the effectiveness of the blast.The method used to calculate pulsed actions in continuous bodies was devised by Korotkikh [12], and enables us to solve two-dimensional problems of axially symmetric and plane deformation and plane states of stress.The method is based on approximation of the determinant differential equations by finite-difference equations with the aid of Noch's determinationofthe partial derivatives [13]. The numerical scheme involves the use of Lagrange polynomials~ In this case the position of the boundaries of the region is determined automatically. Inasmuch as the subdivision net moves together with the medium, the region under investigation is approximated by the same number of nodes at any time with preservation of the initial accuracy of approximation. Numericalschemes in Lagrange polynomials have high accuracy, and the number of nodes necessary for good approximation of the region can be comparatively small (in our case we used a net of 400 elements). To approximate the time derivatives we used a "cross" scheme [13]. The program was compiled for the B~SM-4 computer. We used a nonuniform Lagrange net with quadrangular cells (see Fig. 1). The memory storage necessary for the work includes two operative memory blocks each of 4096 words and two magnetic drums each with 16,000 words. With the 400-element net, the time to calculate one step is 30 sec. The time step is determine...