The rock chosen for this study was Stanstead granite. This heterogeneous, hard, brittle crystalline rock consists of approximately 71% feldspar, 21% quartz, and 8% biotite.
3.1. Micro-scale Characterization
 Grid micro-indentation was performed by a CSM Instruments Micro indentation tester (MHT) with a Vickers indenter on a 1 mm thick polished section of the rock (Figure 2a). A grid of 15 × 14 indentations (in X and Y directions, respectively) with an equal spacing of 300 μm covered an area of 4.2 × 3.9 mm. Considering the average grain size of the rock (i.e., 0.6–1.5 mm), this area covered approximately 13 grains and should therefore, be statistically sufficient to distinguish between the different phases. The indentation tests were under load control, with all testing parameters identical for each individual indent in the grid. The maximum indentation load was 1000 mN with a linear loading and unloading rate of 2000 mN/min. A pause of 5 s was applied at the maximum load during which the load was kept constant. Results of the grid indentation tests are shown in Table 1 and Figure 2. As expected, the values of the indentation modulus tabulated in Table 1 demonstrate that quartz is the stiffest mineral, while biotite shows the lowest stiffness. These results compare closely to the data cited by Mavko et al. . Figure 2b illustrates the indentation modulus map of the indented area of Figure 2a. The modulus map coincided with the mineral distribution of the sample, meaning that grid indentation directly links mechanical properties with physical chemistry. For instance, the darker pixels in the map, that had the lowest modulus values, corresponded well with the locations of biotite grains (black phase in Figure 2a). Representative load vs. penetration depth curves for the three phases are shown in Figure 2c. Considering that the average grain size of the granitic rock used for this study was 0.6–1.6 mm [Nasseri and Mohanty, 2008], the maximum penetration depth below 8 μm conformed to the h/D < 0.1 criterion (in particular, h/D = 0.005–0.013). Therefore, indentation depths were small enough to accurately resolve the mechanical properties of individual phases, which was also confirmed with the modulus map. Following deconvolution, the histograms of the indentation modulus (Figure 2d) exhibited three distinct peaks corresponding to the three constituent minerals of the rock (i.e., biotite, feldspar, and quartz). These peak values are also presented in Table 1. The areas below the histograms were calculated to obtain volume fractions of the three minerals (Table 1). These values corresponded well with the percentages found by analyzing the CT images and the results of Nasseri and Mohanty .
Figure 2. (a) Surface of the specimen showing the location for grid indentation; (b) corresponding indentation modulus map; (c) representative load-penetration depth curves for the three phases; and (d) modulus histogram.
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Table 1. Summary of Indentation and Scratch Test Results
|Indentation modulus [GPa]||17.21||56.37||83.09|
|Volume fraction [%]||5||73||22|
|Fracture toughness [MPa ]||3.21 ± 0.05||4.18 ± 0.09||8.68 ± 0.18|
 The frequency plots of the harder minerals (quartz and feldspar) displayed a wider spread in their Gaussian distributions than the softer biotite mineral. This phenomenon, which is commonly observed in materials comprising phases with significantly different hardness/stiffness [Randall et al., 2009], is caused by the elastic response of the underlying material, implying that the h/D criterion might not be sufficient for these cases (even with h/D = 0.005–0.013). The histogram of the bulk or homogenized indentation modulus is also shown in Figure 2d with a peak value of 57 GPa, which corresponds well with the Young's modulus of the rock (52 GPa, see Table 2). Since no pile-up or sink-in was observed during the indentation tests, the indentation modulus should be comparable with the Young's modulus, also supported by the above values [International Organization for Standardization, 2002].
Table 2. Summary of Laboratory Test Results
|E [GPa]||ν [−]||ρ [kg/m3]||c [MPa]||ϕ [°]||σt [MPa]||KIc [MPa ]||Gf [J/m2]|
 Next, a CSM Instruments Micro Scratch Tester (MST) with a Rockwell diamond conical probe (tip radius = 200 μm, Figure 3a) was used to conduct micro-scratch tests. During this test, the vertical load,FV, was linearly increased, leading to an increase in scratch depth and groove size, while the tangential force, FT, was recorded. The parameters used for scratch testing were progressive loading from 0.02–30 N with a loading rate of 30 N/min and a speed of 1 mm/min. The resulting scratch paths for three tests on biotite, quartz, and feldspar are presented in Figures 3b–3d, respectively. A typical frictional force-scratch depth curve for biotite is illustrated inFigure 3e. The corresponding scaling of FT/d3/2 cos θ/(2 ) vs. d/R is presented in Figure 3f. As an emerging property of the scratching phenomenon, FT/d3/2 cos θ/(2 ) converges to a horizontal line as fracture processes occur. According to equation (1), fracture toughness of the scratched material can be calculated (in the figure, Kc = 3.21 ± 0.05 MPa ). The values for the other phases are summarized in Table 1.
Figure 3. (a) Schematic diagram of the conical scratch device; photographs of the scratched surface showing the residual groove for (b) biotite, (c) quartz, and (d) feldspar; (e) frictional force (FT) vs. scratch depth (d) for a scratch in biotite; and (f) the corresponding fracture scaling.
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3.2. Macro-scale Characterization
 In accordance with the applicable ISRM suggested methods, rock specimens were prepared and tested with the following nominal sizes for each test: four Brazilian tests, 37.57 mm in diameter and 18.94 mm in thickness, three uniaxial and nine triaxial compression tests, 53.70 mm in diameter and 107.43 mm in height, and four Chevron-notched semicircular bend tests, 37.76 mm in width and 18.82 mm in thickness. Triaxial tests were conducted at 5, 10, and 30 MPa confining pressures (three at each). Loading was applied by a stiff, servo-controlled frame at a constant rate of 0.060 mm/min. Strain gauges were attached to the uniaxial and triaxial samples to measure diametric and axial strains. The measured values for Young's elastic modulus calculated as tangent modulus at 50% of ultimate strength (E), Poisson's ratio (ν), density (ρ), cohesion (c), friction angle (ϕ), tensile strength (σt), Mode I fracture toughness (KIc), and fracture energy (Gf) are summarized in Table 2. Note that fracture energy and Mode I fracture toughness are related through Young's elastic modulus as, Gf = KIC2/E.
 A Brazilian disc specimen was CT scanned pre-loading and post-loading using a Phoenix|X-ray v|tome|x s microCT system. To obtain a high quality reconstruction of the object, 1440 images were acquired with a sensor timing of 400 ms at 1/4° rotation increments (i.e., 360°/1440). Voltage and current were set to 150 kV and 220μA to generate sufficient X-ray energy to penetrate the sample. The resulting X-ray energy combined with a voxel size of 41.97μm ensured that sufficient micro-structural detail was captured to distinguish between the constituent mineral phases.
 The 2D projections were reconstructed to obtain a 3D structure of the specimen. A cross-section of this 3D reconstruction at the center of the disc (midpoint of thickness) was chosen as the geometry for the numerical model (Figure 4a). Image analysis techniques were used to segment the grey-scale, cross-section image based on its mineral composition. The histogram of grey values is presented inFigure 4b, both in linear (black) and logarithmic scale (grey). Segmentation via manual thresholding was used to establish the range of the grey values corresponding to biotite, quartz, and feldspar grains as 0–65, 66–150, and 151–254, respectively. To automate the segmentation process, the Trainable Segmentation plug-in of ImageJ version 1.44 m (http://rsb.info.nih.gov/ij/) was applied on a slightly diffused image of the cross-section. Diffusion was used to reduce the noise in the segmented image.
Figure 4. (a) Two-dimensional MicroCT image cross-section at the center of the disc; (b) linear (black) and logarithmic (grey) histogram of grey values; Y-axis shows linear counts; (c) segmented image; (d) segmentation applied to the mesh; (e) simulated fracture patterns of heterogeneous and equivalent homogeneous models superimposed on the MicroCT scan of the failed specimen.
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3.4. Case Study
 To show the application of the proposed modeling procedure, one MicroCT scanned Brazilian disc specimen was chosen as the numerical case study. The numerical code used for this study was a combined finite-discrete element method (FEM/DEM), based on the Y-code developed byMunjiza , and known as Y-Geo (O. K. Mahabadi et al., Y-Geo: A new combined finite-discrete element numerical code for geomechanical applications, submitted toInternational Journal of Geomechanics, 2011). FEM/DEM is an innovative numerical method that combines the advantages of the continuum-based modeling techniques and the discrete element techniques to overcome the inability of these methods to capture progressive damage and failure processes in rocks. In particular, FEM/DEM offers the ability to explicitly model the transition from continuous to discontinuous behavior by fracture and fragmentation processes. This technique was previously shown to be applicable to the modeling of brittle failure in rocks, including Brazilian disc tests [Mahabadi et al., 2010a]. Pre-processing was done using the Y-GUI Graphical User Interface developed by the authors [Mahabadi et al., 2010b].
 The segmented MicroCT image of the Brazilian disc, as shown in Figure 4c, was used to map mineral geometries to the mesh using the material mapping function of Y-GUI V2.8 [Mahabadi et al., 2010b]. The result of this mineral mapping is shown in Figure 4d. The model used a one-to-one mapping of the actual rock specimen. For an accurate mapping, the mesh was refined around the vertical center line (element sizes = 0.2 mm), while coarser elements (1.0 mm) were used in the outer regions to ensure faster simulation run times. Therefore, in the areas of major fracturing, the fine mesh ensured that each mineral grain was composed of various elements, allowing both intergranular and intragranular cracking to be simulated. The mesh contained a total of 18303 elements and 9255 nodes. The mineral and rock properties obtained from the micro-mechanical and laboratory experiments, as given inTables 1 and 2, were used as input parameters for the model. The elements were critically damped, calculated as 2h , where h is the element size [Munjiza, 2004]. An optimal time step size of 9 × 10−7 ms was used in the simulations.
 The indirect tensile strength was calculated as σt = Pmax/πRt where Pmax is the load at failure, R is the disc radius, and tis the disc thickness. Preliminary simulations revealed that, due to the absence of pre-existing material flaws, the model overestimated the indirect tensile strength of the rock. To overcome this limitation, thin sections of the rock were studied, from which the microcrack density was found as approximately 4.5 mm/mm2. Based on a discrete Poisson distribution, this microcrack density was stochastically generated and introduced into the models using a built-in function of Y-GUI [Mahabadi et al., 2010b]. These microcracks coincided with the boundaries of finite elements, and hence, their lengths varied with the element sizes. The indirect tensile strength for the heterogeneous model with microcracks was found as 6.05 MPa, while a value of 6.15 MPa was obtained in the experiment. The error in the estimated strength was approximately 1.6%. In contrast, for an equivalent homogeneous model with no microcracks, tensile strength was estimated as 13.4 MPa which was almost twice the experimental value.
 The modeling results show that material heterogeneity disturbed the otherwise homogeneous stress field. Since the three phases had different elastic properties, their differential deformation created zones of local tensile stress, in particular on the boundaries between different phases. For the granitic rock used in this study, biotite, being the weakest material with very weak cleavage planes, played a major role in stress distribution, and fracture formation and propagation. As presented in Figure 4e, the tensile stress fields induced by biotite grains diverged the crack propagation paths. The simulated fracture pattern corresponded well with the one observed experimentally. Although there were divergences in the areas closer to the loading platens, the major tensile splitting fractures almost matched perfectly. The divergences at the platens can be an artifact of loading conditions, i.e., differences between loading in the laboratory and in the model, or specimen imperfections. In both the experiment and simulation, the mineral grains and grain boundaries highly influenced the fracture trajectories. Compared to the homogeneous model with average rock properties (Figure 4e), in which the fractures closely aligned with the loading direction located near the center line of the disc, the fracture patterns of the heterogeneous model were rougher and more irregular.