International Journal of Rock Mechanics and Mining Sciences
Microcrack modelling of Brazilian tensile tests with the boundary element method
Introduction
The Brazilian test, or the splitting tension test, is widely used in engineering practice to indirectly obtain the tensile strength (TS) of intact rock samples obtained from drill cores. The standards (e.g., [1], [2]) suggest using the value of the tensile stress at the centre of the sample given by the analytical solution for a continuous, homogenous, isotropic and elastic medium (e.g., [3]). In this study, the TS obtained from Brazilian tests is addressed as “indirect” in contrast to the “direct” TS determined from uniaxial tensile tests (e.g., [4]). Both TSs have to be considered “macroscopic” in the sense that intensive local stress concentrations inside the samples may overcome the value of the macroscopic strength at the tips of the cracks.
Recent studies have shown that the samples loaded in Brazilian test conditions undergo cracking or activation of pre-existing flaws before the failure of the specimens (e.g., [5], [6], [7], [8], [9]). Thus, the hypothesis of continuity does not hold, and the stresses inside the samples may differ very much from those given by the closed-form solutions. This is particularly true for the tensile stress along the loading diameter of the sample, which is responsible for crack initiation. Despite this, several recent publications still approach the problem of TS determination from Brazilian testing by using continuum analytical solutions (e.g., [10]) and finite-element analyses (e.g., [11], [12]). These analyses give results that only marginally differ from the closed-form solutions given in the textbooks and do not address the issue of the stress concentrations induced by crack initiation and propagation inside the specimens or of the overestimation of the TS obtained from Brazilian testing compared to direct tensile testing.
Several authors (e.g., [13], [14]) have suggested performing Brazilian tests on samples with artificially drilled holes in different positions. This allows controlling the point at which the crack propagation starts and enhances the local opening stress. These testing setups were also used to determine the toughness of the material by reloading a sample where crack propagation had been induced, stopped and measured. However, very seldom have numerical models been able to successfully replicate the magnitude of the indirect TS from Brazilian testing in laboratory.
The objective of this study is to improve the interpretation of the results of Brazilian tests by studying how newly generated cracks affect the stress field in the sample during loading. This is achieved by modelling the Brazilian tests by means of a boundary element method (BEM) code (FRACOD2D by FRACOM Ltd, Finland, [15]) that implements the displacement discontinuity method (DDM), crack initiation and propagation algorithms. The code has been successfully applied to solve different problems involving fracture propagation in rocks (e.g., [16], [17], [18]). This study does not present a calibration of FRACOD models based on Brazilian tests. Rather, we aim to (a) test the predictive capabilities of the code, keeping most of the input parameters unchanged; (b) study the stress distribution inside the model induced by the initiated cracks; (c) analyse the deformations of the model and propose a possible experimental method for estimating the direct TS of the samples from indirect Brazilian tests; and (d) focus on the fact that the models can explain the difference between the input or direct and Brazilian or indirect TS; (e) show why continuum numerical models cannot be applied to describe failure during Brazilian tests due to the need of explicitly take into account the cracks.
In this paper, experimental results on samples collected at the sites of Shobasama and Mizunami Underground Research Laboratory (MIU), Gifu Prefecture, Central Japan, are used to obtain all the necessary input parameters for the intact rock and cracks of the Toki granite for modelling Brazilian tests.
Section snippets
Experimental results
Laboratory tests were carried out on samples of the Toki granite (Tono Area), a medium to coarse biotite granite, for determining the physical and mechanical properties: effective porosity, elastic wave velocity, density, uniaxial compressive strength (UCS), triaxial compressive strength and indirect TS from Brazilian testing, fracture stiffness and toughness. Samples were collected at rather uniform spacing along boreholes MIU-1 to 4 and AN-1 (Shobasama), MIZ-1 (MIU Construction Site) and
FRACOD2D
FRACOD2D is a Windows based program that features the initiation of newly created fractures or the propagation of pre-existing fractures in a continuous, homogeneous, elastic and isotropic medium in plane-strain conditions [15], [24]. To achieve this, the BEM and the DDM are used. Although microcracking cannot be directly simulated, the code can simulate the process that takes place when microcrack coalescence develops into meso- and macro-scale fracture propagation.
Numerical models
The DDM numerical models in this study represent cylindrical samples of 35 mm diameter (Fig. 3a). The models consist of 60 circumferential elements and an internal grid of 33×33 points. The continuum medium (e.g., intact rock) is characterised by means of elasticity properties (Young's modulus and Poisson's ratio), TS and Coulomb's failure criterion parameters (cohesion and friction angle). The initiated and propagating cracks are characterised by stiffness properties (normal and shear
Discussion
The modelling presented in this contribution shows that the stress distribution in the samples during Brazilian testing might not be as uniform as assumed when applying the closed-form solution [3]. Despite this, the formula is usually used in engineering practice to obtain the TS from Brazilian testing results.
If flaws exist or new cracks develop during loading, the distribution of the stresses perpendicular to the loading direction becomes very ragged. High stress gradients develop at the tip
Conclusions
The numerical modelling in this contribution shows that the formula for calculating the indirect TS from the failure load obtained by Brazilian testing might not be applicable for rocks containing minerals with different strength, flaws and cracks. Firstly, the theoretical closed-form solution does not consider the fact that pre-existing or newly initiated flaws and cracks affect the stress distribution in the samples to such extent that it violates the hypothesis of continuity and homogeneity
Acknowledgements
The authors would like to acknowledge Mr. A. Yamada (Hazama Corporation, Japan) and Mr. S. Nakama (JAEA, Japan) for making the laboratory results available for this study and for supporting with all practicalities. The comments by the Journal's Reviewers have helped to greatly improve the manuscript and are, therefore, very much appreciated.
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