Crack interaction mechanism of pre-split rock blasting

The paper investigates and models the mechanism of pre-split cutting, a technique used in excavations in hard rock and based on creating large splits by simultaneous blasting or slow pressurising of closely placed holes. Only the case of initially intact rock (no discontinuities comparable with the dimension of pre-split fracture) is considered. The critical spacing between the holes is found separating the case of splitting from the case of bulk fracturing of the rock. The determination of the critical spacing is based on the comparison between the hole pressure required to start the pre-existing cracks with the pressure required to bring the cracks to the length at which the interaction between them will make their propagation unstable. If the initial cracks are relatively large such that the pressure of the crack start is less than the pressure of unstable growth, other cracks will appear fracturing the bulk of material. If the initial cracks are small enough, the starting pressure will be high, hence the cracks will propagate dynamically until the coalescence with the neighbours, which produces splitting. For the case of pre-splitting prepared parallel to a free boundary, the influence of the latter is minor, so the dominant mechanism of pre-splitting is the crack interaction. INTRODUCTION Pre-split or smooth blasting is used to produce a fine cut surface in hard rock by simultaneous blasting of usually small, closely placed blastholes, e.g., [1-4]. Currently the parameters of this method are chosen based on experience or physical modelling on rock samples or artificial materials, e.g.,[1], [5], rather than on an understanding of the fracture mechanics. A common belief in analysing this technology is that the splitting cracks are driven by gases penetrating into the crack. Both the formation of the pre-split fracture and the rock damage near the blastholes are processes based on crack formation. The damage in this case mainly consists of radial cracks growing from the blastholes well beyond the initial crushed zone (e.g., [1], [6]). Therefore, it is necessary to address the fracture mechanics of the process in order to be able to optimise the technological parameters. The present paper attempts to investigate and model the fracture mechanics of pre-splitting in initially intact rocks when no discontinuities (comparable with the dimension of pre-split fracture) are present. The pressure holes will be assumed to be aligned. The aim is to determine the critical spacing between the pressure holes separating the case of splitting formation from the case of bulk fracturing of the material. Since the pre-splitting occurs through the formation of fractures connecting the holes, it is natural to base the investigation of its mechanism on the consideration of crack growth form the pressurised holes. Following [6] the consideration will be quasi-static i.e. based on the equations of static fracture mechanics. This will provide a simple and reasonable approximation for the extent of crack growth. Also, this static modelling is important per se, since it is relevant to fracturing using the non-explosive rock breaking systems, e.g., [7]. An example of such a system is given by the expanding grout in which case the pre-splitting is formed slowly, over a few weeks time, e.g., [3]. MECHANISM OF FORMATION OF THE SPLITTING FRACTURE Let a chain of holes of equal radius, r, be drilled at an equal distance, d, along a line. Each hole is subjected to internal pressure p. The influence of the in-situ stress will be neglected at this stage. In the case of a single hole, the pressurising will create cracks initiating from pre-existing flaws near the hole surface and growing from the surface in different directions. When the influence of remote stress field is neglected, the only mechanism of the preferential direction of the crack growth and the formation of a single splitting fracture can be the interaction between the holes in the chain. It is also natural to assume that the formation of the splitting fracture is caused by the coalescence of cracks growing toward each other from the adjacent holes. The coalescence will occur in the form of overlapping due to the tendency of the tensile cracks to avoid each other [8]. This is consistent with the results of physical modelling of the pre-split blasting, e.g., [1]. Consider the case when the distance between the holes is considerably greater than their radius, d>>r. Then the cracks forming the splitting fracture have to grow at the length much greater than the hole radius. Therefore, in order to investigate the crack interaction, the hole with collinear cracks will be modelled in 2-D as a single crack opened by a pair of concentrated forces at its centre. The magnitude of these forces will be taken equal to the force per unit length of the hole (in the direction perpendicular to the drawing) created by the internal pressure on the upper/lower half of the hole. This allows the following approximation for the mode I stress intensity factor (e.g., [9]): pr d r p P l P KI 2 sin , 0 = φ φ = π = ∫ π , (1) The influence of the gas flow into the crack is neglected. Figure 1 demonstrates the accuracy of such a modelling by comparison of the approximation presented by Eqn 1 with the numerical solution for two radial crack emanating from the hole [9]. It is seen that as long as the cracks become greater than the hole radius ( l>2r) the representation becomes quite accurate. Therefore, this approximation is sufficient for the purpose of the present analysis. It is also seen that the growth of small cracks is unstable. It however becomes stable when the length of the sprouted cracks reaches half of the hole radius (l>1.5r). This approximation facilitates the analysis of the interaction of cracks originated from holes arranged in a periodical chain modelling pre-split blasting. The modelling of crack interaction will be based upon the consideration of a periodic array of equally inclined cracks loaded by concentrated forces at their centres. The consideration of inclined cracks in the chain is necessary since the cracks from the holes can in principle, be initiated at arbitrary angle. The influence of a free surface is to be taken into account if the holes were drilled close to the surface. Thus the general problem to be considered can be formulated for the half-plane containing the array of parallel cracks as it is shown in the Figure 2. The following notations are in use: l i half length of the cracks, d is spacing between the centres of cracks, h is distance from the crack centre to the boundary of semi-plane and α is crack inclination to the boundary. The problem has been reduced to solving the singular integral equation of a certain form [10] with respect to the density of the displacement discontinuity across the cracks. The solutions of the integral equations for both cases of periodical arrays in the plane and semi-plane have been found by the numerical method [11]. All calculations have been performed using the package MATHCAD 6PLUS. In the case of collinear cracks (α=0°) in entire plane, calculated KI coincide with analytical results [9]. For parallel cracks ( α=90°) KI is in good agreement with an approximate analytical solution obtained in [10].