Expansion of a blasting cavity in a material with internal friction and dilation

Abstract

Development of modern technologies for mining economic minerals requires solution of the problem of the effect of blasting on both strong rock and soil. A key point in solving problems of rock mechanics including dynamic problems is selection of a mathematical rock model which adequately describes its behavior under real conditions.As is well known, for rocks the effects of internal friction and dilation are characteristic features of their deformation.Currently several mathematical models which describe this property are accepted. For example in [I, 2] the dynamic problem is solved for expansion of a blasting cavity in a boundless rock mass within the scope of the Prandtl elastoplastic model.In this paper, solution of this problem is considered within the scope of the elastoplastic model of a solid suggested by Mosinetz and Shemyakin [3].In this model the main relationships which describe material behavior for a rigidly plastic arrangement have the form( 1 )where FP is plastic shear, T is maximum tangential stress, e is first invariant of the strain tensor, G is linear strengthening modulus, f is internal friction coefficient, A is a rate of dilation, k is adhesion. The rate of dilation A and internal friction coefficient f in a condition of total plasticity with ~ = 1 (~ is the Lode-Nadai parameter), which occurs in the case of the unidimensional spherical problem, are connected by the simple relationship A = 4H(3 -/).At time t = 0 a cavity of radius the action of gas pressure a0 starts to expand in an unbounded rock mass under po = poo (ao/a (t) )s:where P00 is initial pressure, ~ is adiabatic index, a(t) is current cavity radius.Simultaneously with the start of movement a shock wave propagates from the surface of the cavity into the plastic flow of soil observed behind this front. Ahead of the wave front the material is assumed to be im~nobile.The problem is spherically symmetrical.Movement of the material in the plastic region is described by an equation of motionand acontinuity condition tgtl . t dp O, a-7+27 + p dt where o~ = o 8 > o r are principal stresses, u is a mass velocity, p is density.At the boundary of the expanding cavity and in the shock wave front fulfillment of boundary conditions is required: a) at the cavity boundary with r --a(t)