Keywords: rock fracture, elastodynamics, blasting, adaptive analysis, wave propagation, finite element method.

Rock blasting is generally carried out by drilling into a rock mass, charging the blast holes and firing the igniters located at the centre of the cylindrical charges. The blast-induced P waves provoke high compressive stresses in the radial direction as well as high tensile stresses in the circumferential directions. Since the problem geometry is generally bounded by a free surface (a rock-air interface) a multiple reflection mechanism, involving the free surface as well as the discontinuities introduced during fracturing, will render the stress analysis a quite difficult task.

In this work the fracturing of a sound granite rock mass by blast-induced stress waves is numerically simulated by the finite element method. The rock is admitted as an isotropic and homogeneous medium that remains linear elastic until the moment of breakage, a brittle material assumption that closely corresponds to the behaviour of many sound rocks containing a high percentage of quartz. Most of the research works related to dynamic rock fragmentation considers an instantaneous fracturing mechanism determined solely on basis of the maximum stress fields generated by the explosion. This approach does not take into account some important elements that influence the fracturing process, such as the stress redistribution due to changes in the problem geometry as fractures grow, open or close in the time of analysis.

The stress components in the crack tip region can be described by single-term parameters, known as stress intensity factors, whose analytical expressions are available from the linear elastic fracture mechanics for some simple cases. This work considers fracture propagation under the mixed mode I-II, provoked by P and S waves generated by the explosion itself and further wave reflections.

The fractures are assumed to grow at constant velocity less than half the shear wave velocity [1], and along a direction determined from the direction of the maximum tensile circumferential stress.

The finite element mesh, composed of quadratic triangular elements is generated through an adaptive mesh generation procedure. The algorithm for mesh generation is based on the "quadtree" scheme combined with a boundary triangulation technique [2]. Once a new mesh is generated, the state variables should be transferred from the old finite element mesh at time to the current mesh at time t. The updating process basically consists of identifying the element in the previous mesh that contains the node n of the present mesh, followed by the determination, in the old mesh, of the parametric coordinates of the corresponding point and computation of the displacement, velocity and accelerations components from the nodal values through interpolation with the shape functions.

The elastic stress singularity at crack tip is approximated using a special mesh configuration (rosette) formed by eight triangular quarter-point elements [3]. The stress intensity factors for mixed mode I-II are obtained by the modified crack-closure method [4]. By fracture closure the condition of non penetration must be guaranteed by both fracture walls at each time step. In this research, the contact condition is approximately satisfied by the use of a penalty formulation [5].

The numerical model developed in this research presents some special features in order to simulate such a complex dynamic problem: quarter-point quadratic elements around the fracture tips, an efficient algorithm for mesh generation as fracture growth is detected, silent boundaries to represent the stress wave radiation condition, determination of the stress intensity factors and the corresponding direction of fracture propagation considering the mixed mode I-II, a penalty based contact algorithm to control fracture wall penetration, etc.

1

D.E. Grady & M.E. Kipp, "The micromechanics of impact fracture of rock", International Journal of Rock Mechanics Science & Geomechanics Abstract, n. 16, vol 5, pp. 293-302, 1979. doi:10.1016/0148-9062(79)90240-7

2

T.D. Araújo, J.B. Cavalcante Neto, M.T.M. Carvalho, T.N. Bittencourt & L.F. Martha, "Adaptive simulation of fracture processes based on spatial enumeration techniques", Int. J. Rock Mech. Mining Sc., 3/4, 551 (Abstract) and CD_ROM (full paper), 1997.

3

R.S. Barsoum, "On the use of isoparametric finite elements in linear fracture mechanics", International Journal for Numerical Methods in Engineering, vol. 10, pp. 25-37, 1976. doi:10.1002/nme.1620100103

4

I.S. Raju, "Calculation of strain-energy release rates with higher order and singular finite elements", Engineering Fracture Mechanics, vol. 28, n. 3, 1987. doi:10.1016/0013-7944(87)90220-7

5

P. Papadopoulos, R.E. Jones & J.M. Solberg, "A novel finite element formulation for frictionless contact problems", Int. J. Numerical. Methods. Eng., vol.28, pp. 2603-2617, 1995. doi:10.1002/nme.1620381507

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