A fast computational code is presented that is dedicated for the elastic analysis of three-dimensional excavations and cracks in rocks. The problem is solved on the boundaries that are discretized with a new triangular leaf constant displacement discontinuity element with one collocation point. The creation of the new triangular element was inspired from Mindlin's special version of grade-2 or strain-gradient elasticity theory (second gradient of displacement, g2). This element is characterized by a much better measure of the average stress at the center of gravity of the triangular element compared to that of the classical elasticity element close to regions with stress or strain gradients (e.g. notches, cracks etc). In a verification stage, the accuracy of the computational algorithm for the pressurized penny-shaped and mixed-mode elliptical crack problems that have analytical solutions is demonstrated. More specifically, it is shown that the average error of the crack tip Stress Intensity Factor predicted by the gradient modified method for nine discretizations of varying density is around 3.5 % with a maximum error of 5 %, while the constant displacement discontinuity element displays errors varying around 14 %. Moreover, the new method preserves the simplicity and hence the high speed of the constant displacement discontinuity with only one collocation point per element, but it is far more efficient compared to it, especially close to the crack tips and corners of excavations where the displacement and stress gradients are highest.

1. Introduction

It is almost certain that any planned underground excavation in the scale of 101 m or more will transect a fault or persistent joint. The problem in the design phase is to examine the effect of the faults on the behavior of the rock mass during excavation and then to optimize the design of the underground openings and pillars. In Geophysics there remains the reasonable trend to explain earthquake mechanisms by means of dislocation or fracture mechanics models. Also, in petroleum engineering as well as in Rock Mechanics, the hydraulic fracturing technique where a pressurized mode-I crack propagates from a shallow or deep borehole, is widely used for permeability enhancement and measurement of in situ stresses, respectively. There many more problems involving threedimensional excavations and fractures that should be attacked with computational methods capable to tackle in a formal and accurate manner crack tip or corner singularities.

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