The mechanical response of geomaterials is highly influenced by geometrical and material heterogeneity. To date, most modeling practices consider heterogeneity qualitatively and their choice of input parameters can be subjective. In this study, a novel approach to combine a detailed micro-scale characterization with modeling of heterogeneous geomaterials is presented. By conducting grid micro-indentation and micro-scratch tests, the instrumented indentation modulus and fracture toughness of the constituent phases of a crystalline rock were obtained and used as accurate input parameters for the numerical models. Additionally, X-ray micro Computed Tomography (CT) was used to obtain the spatial distribution of minerals, and thin section analysis was performed to quantify the microcrack density. Finally, a Brazilian disc test was modeled using a Combined Finite-Discrete element method (FEM/DEM) code. Compared with the laboratory results of a sample that was initially CT scanned, the simulation results showed that by incorporating accurate micromechanical input parameters and the intrinsic rock geometric features such as spatial phase heterogeneity and microcracks, the numerical simulation could more accurately predict the mechanical response of the specimen, including the fracture patterns and tensile strength. It is believed that the proposed micromechanical approach for evaluating the material properties and the sample geometry can be readily applied to other problems to accurately model the mechanical behavior of heterogeneous geomaterials.
 It is well known that geomaterials including rock and concrete are heterogeneous and, at the grain scale, are characterized by the presence of different materials (i.e., minerals) and micro-structural defects (i.e., microcracks). When the material is loaded, these features influence the stress distribution, creating zones of local tensile stress at the grain scale, thus affecting the initiation and propagation of fractures. Hence, heterogeneity should be considered when modeling the mechanical behavior of geomaterials. The most common approach used in practice today is to incorporate heterogeneity through a stochastic distribution of properties (e.g., Weibull distribution) both in terms of elastic properties and strength parameters [e.g.,Wong et al., 2006; Fang and Harrison, 2002]. Although these studies highlighted the influence of heterogeneity on the global mechanical response of rocks, the choice of input parameters can be subjective and highly dependent on the statistical distribution parameters, and therefore, the simulation results are qualitative. Another approach is to reproduce the presence of mineral grains as part of the initial mesh. Following this approach, Lan et al. used meshes based on Voronoi tessellation as initial geometry for the models. Also, digital image-based techniques have been suggested as a means to incorporate the actual rock heterogeneity into numerical models [Chen et al., 2007]. Although the numerical models and the initial geometries used in these studies are relatively sophisticated in trying to reproduce the exact rock structure, the choice of input parameters remains questionable.
 The purpose of this paper is to illustrate a novel approach to combine established techniques (e.g., X-Ray microCT, micro-indentation, and micro-scratch) in an innovative way to quantitatively model geomaterials by incorporating their real microscopic heterogeneity, that is, the spatial distribution and material parameters of the constituent minerals. The actual heterogeneity of the rock (i.e., phase mapping) was obtained using the X-ray micro Computed Tomography (CT). For this purpose, rock specimens were CT scanned prior to laboratory testing, and the resulting reconstructed image was used to map mineral locations to the model. To obtain accurate mechanical properties for the minerals, the rock was characterized at the micro-scale and macro-scale. Micro-scale characterization was achieved by the grid micro-indentation technique [Randall et al., 2009] and micro-scratch test technique [Akono et al., 2011], which resulted in elastic properties and fracture toughness of the constituent phases, respectively. Macro-scale characterization was performed by conducting standard rock mechanics experiments in the laboratory. Also, thin sections were analyzed, from which the microcrack density of the rock was found. Simulation results were verified against laboratory data of a Brazilian disc test (i.e., indirect tensile strength) and the fracture trajectory obtained from CT scans of the specimen after laboratory testing. The results were in good agreement with the experimental findings.
2. Material and Methods
 To obtain reliable input parameters for the numerical model, the rock was characterized at the micro-scale (grain scale) and macro-scale.
2.1. Micro-scale Characterization
 State-of-the-art micromechanical testing techniques were utilized to characterize rock minerals. These techniques include micro-indentation and micro-scratch tests.
2.1.1. Micro-indentation Testing
 Micro-indentation and nano-indentation are established methods for analyzing mechanical properties of thin films and bulk materials at low loads and shallow depths. A depth-sensing indentation produces a load-displacement curve from which mechanical properties, including indentation modulus, can be quantified. Although these techniques can be easily used for homogeneous materials, their application to heterogeneous media such as most geomaterials has certain disadvantages [Randall et al., 2009], including: (a) the difficulty in measuring intragranular (i.e., grain-only) properties while ensuring sufficient indentation depth to overcome surface roughness; (b) uncertainty in locating single grains by microscopy and the time required to do so; and (c) visibility of phases by microscopy. Moreover, geomaterials cannot be easily polished flat on the order of microns because of the significant mismatch in properties of their constituent phases. The resulting surface roughness variations lead to a wide scatter of indentation results.
 To overcome the above limitations, grid indentation technique was proposed [Constantinides et al., 2006] and successfully applied to a wide range of materials [Randall et al., 2009]. This method can yield accurate mechanical properties of individual phases of the material. A grid indentation consists of an array of indentations at a given spacing on the surface of the material. As illustrated in Figure 1, to ensure that only the intrinsic properties of individual phases and not of the composite material are measured, indentation depths (h) should be shallow enough to satisfy the criterion h/D < 0.1, where D is the characteristic size of heterogeneity (here, grain size) [Constantinides et al., 2006]. Indentation results can be plotted as histograms (or frequency plots) of the measured properties (e.g., indentation modulus). The mean value of each peak corresponds to the mean property of that phase. The frequency plots should be analyzed using the so-called deconvolution technique [Randall et al., 2009], a process that should be automated to ensure repeatability of the method. The area under a frequency curve is correlated to the possibility of indenting a specific phase and thus is a measure of surface fraction of that phase. For a material with random mineral grain shapes, sizes, orientations, and spatial distributions, surface fractions can be indicative of volume fractions. Thus, an estimate of the volume fractions of constituent phases can be obtained by grid indentation.
2.1.2. Micro-scratch Testing
 Scratch test is a common means of characterizing the surface mechanical properties of thin films and coatings (e.g., adhesion, fracture, and deformation). For this test, a controlled scratch is generated on the surface of the material by drawing a sharp tip (typically diamond) across the surface under constant, incremental, or progressive load. The load at which the surface material fails is known as the critical load. As recently proposed by Akono et al. , scratching can be regarded as a fracture process, from which fracture toughness of the surface can be quantified. For a conical probe (Figure 3a), which was used in this study, Akono et al.  demonstrated that the general fracture criterion, written as FT/ = Kc, where FT is the frictional force in tangential direction (in Figure 3a), p = 2d/cos θ, A = d2 tan θ, and d is the scratch depth, yields,
where θis the half-apex angle andKc is the fracture toughness, which corresponds to the straight line from the convergence of FT/d3/2 (see Figure 3f). Acoustic emission measurements demonstrate that this phenomenon is accompanied by a fracture process [Akono et al., 2011].
 Standard rock mechanics laboratory testing, including uniaxial and triaxial compression tests, and Brazilian tests were performed conforming to the applicable ISRM (International Society for Rock Mechanics) suggested methods for rock characterization and testing. Also, Chevron-notched semicircular bend tests were conducted as suggested byChang et al. . These tests provided the rock properties, which were later used as inputs for the homogeneous model, and also as the data for verifying the simulations.
 To assess the heterogeneity of rock and perform phase mapping, micro X-ray Computed Tomography (CT) was used. CT is a non-destructive imaging technology, for which the three-dimensional internal structure of an object is reconstructed from a series of two-dimensional X-ray projections. In the simplest sense, the object is irradiated with an X-ray source with a known intensity, while the X-ray intensity transmitted through the object is measured by a detector. The extent by which X-rays are attenuated depends on the path they follow through the object (i.e., thickness and attenuation characteristics of the material). In this study, microCT was used at two stages: first to reveal the internal micro-structural heterogeneity of the rock specimens prior to laboratory testing, which was used to map the resulting phase distribution to the model, and second to obtain the internal fracture pattern of the samples after testing, which was used to verify the simulation results.
3. Results and Discussion
 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 .
Table 1. Summary of Indentation and Scratch Test Results
Indentation modulus [GPa]
Volume fraction [%]
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
KIc [MPa ]
 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.
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.
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.
 In this study, a novel approach was presented for characterizing and modeling heterogeneous geomaterials. By conducting micro-mechanical tests, including grid micro-indentation and micro-scratch tests, the rock constituent phases were characterized, and accurate input parameters for the models were obtained. Grid indentation proved to be an efficient means for mapping minerals and obtaining their indentation modulus, and volume fractions. Micro-scratch test was successfully applied to derive the fracture toughness of the phases. In addition, X-ray MicroCT scanning was used to obtain the phase mapping (spatial distribution of minerals) and standard rock mechanics laboratory testing was used to acquire average rock properties and data for model verification. It was shown that, by incorporating accurate input parameters and the intrinsic rock geometric features such as the spatial mineral heterogeneity and microcracks, the models can accurately predict the mechanical response of the specimen, including the fracture patterns and tensile strength. It is envisaged that, by combining new and established techniques in an innovative way, the proposed approach can be readily applied to other problems to accurately model the mechanical behavior of rock.
 This work has been supported by NSERC/Discovery grant 341275, CFI-LOF grant 18285, and an OGSST held by O. K. Mahabadi. The authors would like to thank Ange-Therese Akono at MIT for performing the fracture toughness calculations and Dr. A. Munjiza for providing the original Y-code.