ISSN 1062-7391, Journal of Mining Science, 2017, Vol. 53, No. 5, pp. 824–830. © Pleiades Publishing, Ltd., 2017.
Original Russian Text © N.I. Aleksandrova, 2017, published in Fiziko-Tekhnicheskie Problemy Razrabotki Poleznykh Iskopaemykh, 2017, No. 5, pp. 29–36.
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Propagation of Pendulum Waves under Deep-Seated Cord
Charge Blasting in Blocky Rock Mass
N. I. Aleksandrova
Chinakal Institute of Mining, Siberian Branch, Russian Academy of Sciences,
Novosibirsk, 630091 Russia
Received April 21, 2017
Abstract—The object of the numerical study is the travel of pendulum waves in a blocky medium under
nonstationary impact of deep-seated charge blasting on the surface of the expansion chamber. The blocky
model is simulated by a two-dimensional lattice of masses connected by elastic springs along the axes and
diagonals. The displacements and velocities of the masses at different half-space points are calculated using
the finite-difference method.
Keywords: Deep-seated cavity, blocky medium, half-space, advancing wave, Rayleigh wave, numerical
Understanding transient deformation of nonuniform media is required in creating models and
methods to calculate and prove estimates of failure of ground and underground structures under
seismic waves of earthquakes, mine explosions and vibration treatments.
The recently accomplished researches point at the necessity to include mathematical models of
geomechanics and seismicity with the block structure of rocks [1, 2]. A rock mass is considered in this
case as a system of nested blocks of different scales with in-between layers composed of weaker
jointed rocks. The dynamic behavior of a block-structure medium can be approximately described as
movement of rigid blocks owing to pliancy of interlayer . The analytical model in this case is
a lattice of masses connected by springs and dampers. Such models can be two-dimensional [3–6] and
three-dimensional [6, 7].
This study uses a two-dimensional model to obtain a finite-difference solution of a dynamic
problem on impact of deep-seated cord-like explosion on the surface of the explosion cavity in
a block-structure medium.
1. PROBLEM FORMULATION
In focus is the load exerted by an expansion center on the surface of a cylindrical cavity in a block
half-space. The problem is plane. The cavity generators are parallel to the stress-free surface of the half-
space. The block medium is modeled as a uniform 2D lattice of masses connected by springs along axes
and diagonals (Fig. 1). Here, x, y are the horizontal and vertical axes; n, m are the numbers of masses in
the directions of the axes x, y. The free surface relates with the value
, the half-space—with
. The cavity inside the half-space is modeled as a zone without links. The cavity symmetry
center is at the point О at a depth
(l is the length of the springs in the directions x, y). The axis
y is the axis of symmetry of the problem. The forces of the same amplitude and different orientation
are applied at four points on the cavity surface. The time dependence of loading is described by the
)(tH is the Heaviside function;
is the load amplitude.