Rock as a natural material is heterogeneous. Rock material consists of minerals, crystals, cement, grains, and microcracks. Each component of rock has a different mechanical behavior under applied loading condition. Therefore, rock component distribution has an important effect on rock mechanical behavior, especially in the postpeak region. In this paper, the rock sample was studied by digital image processing (DIP), micromechanics, and statistical methods. Using image processing, volume fractions of the rock minerals composing the rock sample were evaluated precisely. The mechanical properties of the rock matrix were determined based on upscaling micromechanics. In order to consider the rock heterogeneities effect on mechanical behavior, the heterogeneity index was calculated in a framework of statistical method. A Weibull distribution function was fitted to the Young modulus distribution of minerals. Finally, statistical and Mohr–Coulomb strainsoftening models were used simultaneously as a constitutive model in DEM code. The acoustic emission, strain energy release, and the effect of rock heterogeneities on the postpeak behavior process were investigated. The numerical results are in good agreement with experimental data.
Rock consists of crystals, grains, and cementitious material. Rock materials are usually made up of several different minerals. These different individual minerals and components are usually distributed in the geomaterials. They usually have different physical and mechanical properties and responses under external loading. One of the most important factors affecting the mechanical behavior during the failure process is the inhomogeneities and internal microstructure of geomaterials. More realistic characterizations of the mechanical responses and failure of geomaterials under loading necessitate the consideration of the inhomogeneities and microstructures of the materials. In most of the mechanistic models, the composite geomaterials are always assumed to be homogeneous or piecewise homogeneous and their microstructure behavior is largely ignored [
In recent years, attempts have been made by many researchers to examine the behavior or response of geomaterials under loading by taking into account the effects of their material inhomogeneities and microstructures. The heterogeneity and microstructure of rock materials have been characterized by using statistical methods. In this method, the heterogeneity of rock is described by assigning different material properties to the simulated rock sample. These statistical tools can simulate numerically material inhomogeneities that are statistically equivalent to those of actual rock materials with known statistical parameters. Recently, Tang et al. [
It is usually difficult to adequately specify the statistical distribution parameters in order to reproduce real microstructures in rock. Some recent studies have shown that digital image processing (DIP) can be used to study and determine the rock heterogeneity [
The literature review indicates that the digital images have been used for the morphological features in many fields of sciences and engineering including biology, medical sciences, geography, civil engineering, and rock mechanics [
In order to determine the matrix elastic properties of the studied rock sample, a micromechanical modeling of the mechanical behavior in the elastic regime was necessary. Although some researchers such as Zaitsev and Wittmann [
The advantage compared with macroscopic approaches is that the homogenized approach is able to systematically take into account the mineralogical composition influences on the mechanical properties of rock material [
This paper is intended to present an incorporation of digital image processing, upscaling micromechanics, and statistical methods for the mechanical analysis of geomaterials by taking into account their actual inhomogeneities and microstructures.
The proposed statistical Mohr–Coulomb softening model was implemented into a DEM code. The rock behavior was simulated and the experimental stressstrain curve was reproduced numerically. Comparisons between numerical results and experimental data will be finally presented in order to show the capability of the proposed model to describe the main features of rock mechanical responses. The acoustic emission, strain energy release, and the effect of rock heterogeneities on postpeak behavior were investigated.
The material studied is an extrusive porphyritic igneous rock called rhyodacite tuff. The mineralogical compositions, initial porosity, and natural water content of samples were first investigated. In Figure
Petrographic microscopic thin section of the rhyodacite tuff sample.
At the mesoscopic scale
The digital image consists of a rectangular array of image elements or pixels. At each pixel, the image brightness is sensed and assigned with an integer value named as the gray level. Their gray levels have the integer interval from 0 to 255 and from 0 to 1, respectively. As a result, the digital image can be expressed as a discrete function
As an alternative to the RGB color space, the hue, saturation, intensity (HSI) color space may be used, as it is close to how humans perceive colors. The hue component (
The commonly used image enhancement method called histogram equalization transformation and noise removal methods are adopted here. In Figure
The original image with the jcoordinates at
Variation of the intensity component (
For this original image in Figure
Figure
Figure
The intensity component (
The intensity component (
At each brightness level, the number in the vertical axis shows the number of the pixels that has the same brightness value in the image. We can divide the whole image pixels into four groups. Normally, the matrices in the image have low brightness levels and the feldspar minerals in the image have high brightness levels. The threshold value is a brightness level which is a boundary between two kinds of minerals. A trialanderror method is used to adjust the threshold values so that the best results are obtained. Thresholds of the intensity component (
Threshold values and volume fractions of the rock.
Mineral  Threshold value  Number of pixels  Volume fraction (%) 

Matrix 

29201  12 
Calcite 

129292  44 
Quartz 

18471  8 
Feldspar 

51120  35 
Micromechanics investigates the behavior of the heterogeneities as well as their effects on the overall properties and performance of a material. An important task of micromechanics is to link mechanical relations on different length scales. The entire behavior of the microstructure is interpreted as the mechanical state of a material point on the macroscopic level which thereby is ascribed effective material properties. Such a microtomacro transition formally proceeds by appropriate averaging processes and is called homogenization as shown in Figure
Homogenization and characteristic length scales [
Inhomogeneous material can be described by an equivalent homogeneous material. Based on Eshelby solution [
In the Mori–Tanaka scheme (1973), therefore, the strain or stress field in the matrix is, in a sufficient distance from a defect, approximated by the constant field. The loading of each defect then depends on the existence of further defects via the average matrix strain or the average matrix stress. Fluctuations of the local fields, however, are neglected in this approximation of defect interaction. It follows that the localization tensor is then given by [
This leads to the following estimate of the effective stiffness tensor [
Because of the isotropy of the constituents and the spherical shape assumption of inclusions, we have [
The mineralogical compositions of the rhyodacite tuff contain four main phases: calcite, feldspar, quartz, and matrix. It is organized in grains spread in a siliceous matrix. The first stage of the homogenization procedure is the definition of a representative elementary volume (r.e.v.). The observations led us to consider the rhyodacite tuff as a fourphase composite of the inclusion/matrix type in which we discern the calcite, feldspar, and quartz phases, assumed to be distributed individually in a siliceous matrix. The rhyodacite tuff sample can be represented by a fourphase composite with distinct mechanical properties. This material has a matrix/inclusion morphology with the phases randomly distributed, and the calcite, feldspar, and quartz minerals being embedded in the siliceous matrix. It is assumed a representative elementary volume containing four phases as shown in Figure
Representative elementary volume (r.e.v.) of the rhyodacite tuff.
The elastic properties of the crystalline siliceous matrix (
Rock is a heterogeneous material. This heterogeneity causes rock in compression to fracture via the formation, extension, and coalescence of microcracks. Studies showed that the variation of mechanical properties can be explained statistically. In a general study on rock fracture, the Weibull distribution function was considered for heterogeneity description.
In this study, the rock is assumed to be composed of many elements of identical size, with the mechanical properties such as bulk and shear modulus of elements to conform to the Weibull distribution, so the mechanical parameters of every element are specified stochastically according to the given Weibull distribution defined in the following probability density function [
Fitting the Weibull distribution function to the probability densityYoung modulus diagram.
According to Figure
The plastic rock behavior is represented by the Mohr–Coulomb model with strain softening. The most wellknown failure criterion for rock is the Mohr–Coulomb criterion. The criterion is a linear envelope touching the Mohr’s circles representing the magnitude of the maximum and minimum stresses at the moment of rock failure. The criterion states that the failure occurs if the magnitude of the shear stress on a specific plane reaches a critical threshold. The critical threshold is associated with both the cohesion of the rock grains at the plane of failure and friction resistance between them. The friction resistance of the failure surface is dependent on the normal stress imposed on the plane. The strainsoftening behavior of rocks is governed by shrinking of the failure criterion with the advance of plastic deformation. The decline of rock strength with plastic strain is denoted as strainsoftening behavior. The strainsoftening model allows representation of material softening at postpeak behavior based on prescribed variations of the Mohr–Coulomb model properties (cohesion and friction) as functions of the deviatoric plastic strain after the onset of plastic yield [
In the plastic zone, it is supposed that strength parameters of rock mass decrease by bilinear function according to a softening parameter
In (
The proposed statistical Mohr–Coulomb strainsoftening model was programmed within the C++ environment and was implemented into a commercial DEM code. Using the Weibull probability distribution function in a numerical simulation of a medium composed of many elements with different elastic properties, one can produce a heterogeneous material numerically. The proposed statistical Mohr–Coulomb strainsoftening model used in the presented analysis was linked to a commercial DEM code as a separate constitutive model.
The studied rock is an extrusive porphyritic igneous rock called rhyodacite tuff. The mineralogical compositions and initial porosity of samples were first investigated. This rhyodacite tuff sample was cored at the depth of 113 m in order to site investigation of a civil underground project located in the northwest region of Tehran [
The complete stressstrain curve of the rhyodacite tuff tested in rock mechanics laboratory [
Hence, numerical simulation of the rhyodacite tuff uniaxial compression strength (UCS) test was performed with the proposed statistical Mohr–Coulomb strainsoftening model. With regard to the experimental test, a summary of input data used in the numerical analysis is given in Table
Mechanical parameters used as input data.







2600  62  0.25  6  75.25  7 
Uniaxial compressive strength (UCS) test is the most widely conducted standard test on rock samples. The main objective of this test is to determine the peak strength
Moreover, employing the sophisticated servocontrol testing machine, the complete stressstrain behavior of the rock can be determined in this test. Additionally, the shape of the stressstrain curve in the postpeak region is an indicative of rock breakage mechanism and its brittleness.
In order to verify the statistical Mohr–Coulomb strainsoftening model, it was attempted to simulate the test condition as closely as possible. The sample shape, dimension, input material properties, and loading condition were selected similar to the test condition. The main objective was to reproduce the tested rock stressstrain curve numerically and delve into the sample failure mechanism in the postelastic range.
A plane stress condition was assumed for the analysis. It is understood that the actual problem has a 3D nature. But with regard to the 2D nature of the selected code, a twodimensional plane slice was selected at the center of the sample and analyzed.
The complete fracture characteristic of a numerical specimen under uniaxial loading may be investigated only in a stable displacementcontrolled test. The load is applied in a sequence of steps in the vertical direction through incremental axial displacement control at one end of the numerical sample in a quasistatic fashion, while the other end is prevented from vertical movement. The sample uniaxial loading was simulated imposing a velocity field in the range of static loading in accordance with the ISRM standard at the top of the model, while a zero vertical displacement was applied at the base. There are no constraints on the sides of the sample and the specimen sides are allowed to move in the horizontal and vertical directions. A view of the model geometry and employed boundary condition for the test condition is shown in Figure
Geometry and boundary condition of the numerical specimen.
The numerical specimen was discretized with 4096 elements. The numerical specimen failure process takes place within it due to the heterogeneity of its properties.
As mentioned, the mean Young modulus for the entire numerical specimen is 62 GPa, but specified Young modulus of different elements is considerably different with this value and conform to the Weibull distribution function. Even with the same distribution parameters for the specimen, the spatial distribution of properties of elements may be stochastically different. Therefore, the spatial distribution of mechanical properties is shown in Figure
The spatial distribution of mechanical properties (
However, the effect of the randomness due to the spatial distribution of mechanical properties of elements was studied thoroughly by Zhu and Tang [
In order to assess the local behavior of the sample, series of horizontal and vertical measuring points were placed throughout the model. Important variables such as stress, strain, and displacement components were monitored at these locations. The local stressstrain curves of some elements with different stiffness are shown in Figure
The local stressstrain curves of the elements with different stiffness.
In Figure
The overall stressstrain curve of the entire numerical sample and its comparison with the experimental data.
To study the influence of homogeneity indices (
As the homogeneity index increases, mechanical material properties become more homogeneous and approach that of the homogeneous body. The total envelope of the stressstrain curves for three numerical specimens with different homogeneity indices and experimental result can be seen in Figure
The total envelope of the stressstrain curves for three numerical specimens with different homogeneity indices and experimental result.
The stressstrain curves of these numerical specimens are linear in the prepeak region and lose most of their loadcarrying capacity in the postpeak region. As
Because of rock heterogeneity, some elements of the numerical specimen under loading reach to the failure criterion earlier than others. Their released strain energy is the origin of the acoustic emission in the rockfracturing process. Acoustic emission (AE) can be used to detect the microscopic processes associated with heterogeneous rock fracture. Generally, AE events are not notable until the occurrence of nonlinearity in the stressstrain curve, and the rate at which the AE events appear changes with the development of fracture. The AE rate increases gradually with extension of the microcracks and increases rapidly as the microcracks link together. The rate maximizes when the final fracture planes form [
The simulated stressstrain curve and the released strain energy.
The stressstrain curve of this numerical specimen is almost linear in the prepeak region, and then the stressstrain curve begins to deviate from linearity at stress about 50 MPa. Further increases in axial strain lead to a rapid released strain energy, and this process continues up to the peak strength. Finally, the strain energy release at the constant rate in the postpeak region and the stressstrain curve approaches a residual strength. The stressstrain curve as well as the AE counts during the fracture process of the numerical rock specimen under uniaxial compression test is shown in Figure
The stressstrain curve and the AE counts during the fracture process.
It can be clearly seen that there are a few failed elements during the initial loading phase. But these failed elements release much less energy as shown in Figure
The release of strain energy versus axial loading curves for the numerical specimens with different homogeneity indices is illustrated and compared in Figure
The release of strain energy versus axial loading for the numerical specimens with different homogeneity indices.
The strain energy release of the heterogeneous specimen is less than the strain energy release of the homogeneous specimen. The heterogeneous numerical specimen releases its strain energy gradually and in a controlled manner. However, the homogeneous numerical specimen releases its strain energy abruptly at a stress level about the peak strength. The AE accumulation counts versus axial loading curves for the numerical specimens with different homogeneity indices are shown and compared in Figure
The AE accumulation counts versus axial loading for the numerical specimens with different homogeneity indices.
Based on Figure
The volume fractions of minerals in the microscopic thin section of the rhyodacite tuff sample were calculated precisely by digital image processing. The unknown matrix properties were determined based on Mori–Tanaka scheme in the framework of micromechanics. Then, the Weibull distribution function was fitted to the distribution of minerals’ Young modulus, and the homogeneity index was determined. Using the statistical Mohr–Coulomb strainsoftening model, the rock behavior was simulated and the experimental stressstrain curve was reproduced numerically. From the numerical results, we can conclude that the homogeneity index in this model controls the strength and brittleness. The simulated AE and released strain energy during the loading process are dependent on the homogeneity index. The more the homogeneous numerical specimen, the more the AE and release of strain energy under UCS test condition.
The authors declare that they have no conflicts of interest.