Rock as a natural material is heterogeneous. Rock material consists of minerals, crystals, cement, grains, and microcracks. Eachcomponent of rock has a different mechanical behavior under applied loading condition.*erefore, rock component distributionhas an important effect on rock mechanical behavior, especially in the postpeak region. In this paper, the rock sample was studiedby digital image processing (DIP), micromechanics, and statistical methods. Using image processing, volume fractions of the rockminerals composing the rock sample were evaluated precisely. *e mechanical properties of the rock matrix were determinedbased on upscaling micromechanics. In order to consider the rock heterogeneities effect on mechanical behavior, the hetero-geneity index was calculated in a framework of statistical method. A Weibull distribution function was fitted to the Youngmodulus distribution of minerals. Finally, statistical and Mohr–Coulomb strain-softening models were used simultaneously asa constitutive model in DEM code. *e acoustic emission, strain energy release, and the effect of rock heterogeneities on thepostpeak behavior process were investigated. *e numerical results are in good agreement with experimental data.
1. Introduction
Rock consists of crystals, grains, and cementitious material.Rock materials are usually made up of several differentminerals. *ese different individual minerals and compo-nents are usually distributed in the geomaterials. *eyusually have different physical and mechanical propertiesand responses under external loading. One of the mostimportant factors affecting the mechanical behavior duringthe failure process is the inhomogeneities and internalmicrostructure of geomaterials. More realistic character-izations of the mechanical responses and failure of geo-materials under loading necessitate the consideration of theinhomogeneities and microstructures of the materials. Inmost of the mechanistic models, the composite geomaterialsare always assumed to be homogeneous or piecewise ho-mogeneous and their microstructure behavior is largelyignored [1].
In recent years, attempts have been made by many re-searchers to examine the behavior or response of geomaterialsunder loading by taking into account the effects of their
material inhomogeneities and microstructures. *e heteroge-neity and microstructure of rock materials have been char-acterized by using statistical methods. In this method, theheterogeneity of rock is described by assigning different ma-terial properties to the simulated rock sample. *ese statisticaltools can simulate numerically material inhomogeneities thatare statistically equivalent to those of actual rock materials withknown statistical parameters. Recently, Tang et al. [2, 3] carriedout numerical investigations on micro-macro relationship ofrock failure under uniaxial compression by taking into accountthe statistical material inhomogeneity. Using the method inTang et al. [2, 3], Li et al. [4] further investigated the failureprocess of Hong Kong granite. Fang and Harrison [5, 6] de-veloped a new technique to simulate the brittle fracture inheterogeneous rocks under compressive loading using com-mercial finite difference software.
It is usually difficult to adequately specify the statisticaldistribution parameters inorder to reproduce realmicrostructuresin rock. Some recent studies have shown that digital imageprocessing (DIP) can be used to study and determine the rockheterogeneity [7].
HindawiAdvances in Civil EngineeringVolume 2018, Article ID 7010817, 10 pageshttps://doi.org/10.1155/2018/7010817
�e literature review indicates that the digital imageshave been used for the morphological features in many eldsof sciences and engineering including biology, medicalsciences, geography, civil engineering, and rock mechanics[8–15]. In particular, Yue et al. [1, 15] developed a DIPtechnique to establish the actual microstructures of geo-materials, which is called DIP-based nite element method.
In order to determine the matrix elastic properties of thestudied rock sample, a micromechanical modeling of themechanical behavior in the elastic regime was necessary.Although some researchers such as Zaitsev and Wittmann[16],Wittmann et al. [17], Bazant et al. [18], and Schlangen andvan Mier [19] have proposed micromechanical models to takethe inhomogeneity into account, these models are generallybased on the fracture mechanics background without thesound and rigorous upscaling methods. In upscaling micro-mechanical theory, the rock material is considered as a com-posite material comprising di�erent individual components.In the upscaling micromechanical model, a micro-to-macrotransition called homogenization leads to evaluate the overall(e�ective) elastic properties [20, 21].
�e advantage compared with macroscopic approachesis that the homogenized approach is able to systematicallytake into account the mineralogical composition in�uenceson the mechanical properties of rock material [22]. �emicromechanical approach tries to relate the physicalmechanisms involved in the microstructure evolution andmacroscopic behaviors observed in laboratory. �e het-erogeneous material is considered as a matrix-inclusioncomposite. �e e�ective properties of materials are ob-tained by an upscaling method based on the Eshelby in-homogeneous inclusion solution [20, 22].
�is paper is intended to present an incorporation of digitalimage processing, upscaling micromechanics, and statisticalmethods for the mechanical analysis of geomaterials by takinginto account their actual inhomogeneities andmicrostructures.
�e proposed statistical Mohr–Coulomb softening modelwas implemented into a DEM code. �e rock behavior wassimulated and the experimental stress-strain curve wasreproduced numerically. Comparisons between numericalresults and experimental data will be nally presented in orderto show the capability of the proposed model to describe themain features of rock mechanical responses. �e acousticemission, strain energy release, and the e�ect of rock het-erogeneities on postpeak behavior were investigated.
2. Rock Heterogeneity Investigation
�ematerial studied is an extrusive porphyritic igneous rockcalled rhyodacite tu�. �e mineralogical compositions,initial porosity, and natural water content of samples wererst investigated. In Figure 1, the petrographic microscopicthin section of a rhyodacite tu� sample is shown. �isrhyodacite tu� sample was cored at the depth of 113m inorder to site investigation of a civil underground projectlocated in the north-west region of Tehran.
At the mesoscopic scale (μm− cm), the rhyodacite tu�sample is composed of the ne grains of calcite, feldspar, andquartz embedded in the crystalline siliceous matrix. At the
macroscopic scale (cm− dm), the rhyodacite tu� consti-tuted by the assembly of mineral grains and the crystallinesiliceous matrix is considered as a homogeneous continuum.
2.1. Digital Image Processing. �e digital image consists ofa rectangular array of image elements or pixels. At eachpixel, the image brightness is sensed and assigned with aninteger value named as the gray level. �eir gray levels havethe integer interval from 0 to 255 and from 0 to 1, re-spectively. As a result, the digital image can be expressed asa discrete function f(i, j) in the i and j Cartesian coordinatesystem [1].
As an alternative to the RGB color space, the hue, sat-uration, intensity (HSI) color space may be used, as it is closeto how humans perceive colors. �e hue component (H)represents repression related to the dominant wavelength ofthe color stimulus. �erefore, hue is the domain color per-ceived by human beings. �e saturation component (S)represents how strongly the color is polluted with white. �eintensity component (I) stands for brightness or lightness andis irrelevant to colors. In general, hue, saturation, and in-tensity are obtained by di�erent transformation formulate byconverting numerical values of R, G, and B in the RGB colorspace to the HSI color space. �e values of S and I vary fromzero to one. But the value ofH varies from 0 to 360, which canalso be normalized to be from 0 to 1. Distinct microstructures(such as fractures and minerals) with di�erent perceivedcolors in the rock sample are acquired according to the valuesof H, S, or I of individual pixels, and the di�erent materialproperties (such as Young modulus) are specied for eachpixel according to its catalog of minerals or colors. In theory,the material properties of di�erent minerals or structuresmust be known based on mineralogical analysis of therock sample, by this means, the relation between values of I(H or S) of the digital image pixels and their materialproperties can be uniquely established [7]. �e HSI color
Mineral SymbolFeldspar FQuartz QCalcite CaMatrix M
M
Q
CaF
Figure 1: Petrographic microscopic thin section of the rhyodacitetu� sample.
2 Advances in Civil Engineering
space was used to investigate the rhyodacite tu� rock samplemicrostructure characteristics.
�e commonly used image enhancement method calledhistogram equalization transformation and noise removalmethods are adopted here. In Figure 2, an RGB image of therhyodacite tu� rock sample with the AA section at i� 150 isshown. �e variation of intensity component (I) with thej-coordinates at i� 150 (across the AA section) is illustratedin Figure 3.
For this original image in Figure 2, the numbers of thescanning lines along the i-axis and the j-axis are 555 and 416,respectively. Using the above image, we extract brightnesslevels along the line at i� 150 in Figure 2 and draw the var-iation of the brightness levels with the j-coordinate in Figure 3.
Figure 3 shows that there exist two major interface pointsalong the i� 150 line.We nd that the brightness level changesabruptly at the corresponding two positions in Figure 3.
Figure 3 shows that a majority of the feldspar mineralshave higher brightness levels than those of the calciteminerals and matrices. Furthermore, the brightness levels ofthe calcite minerals are higher than those of the matrices.
�e intensity component (I) variations diagram repre-sents the change in mineral composition and heterogeneity inthe rock microscopic section. A histogram of an image is usedto display the distribution of brightness values in the image. Itis a function to show, for each brightness level, the number ofpixels in the image that have that brightness level. Figure 4shows the histogram of the image brightness levels in Figure 1.
At each brightness level, the number in the vertical axisshows the number of the pixels that has the same brightnessvalue in the image. We can divide the whole image pixelsinto four groups. Normally, the matrices in the image havelow brightness levels and the feldspar minerals in the imagehave high brightness levels. �e threshold value isa brightness level which is a boundary between two kinds ofminerals. A trial-and-error method is used to adjust thethreshold values so that the best results are obtained.�resholds of the intensity component (I) values for thesefour components of the rock and their volume fractionsbased on the trial-error process are listed in Table 1.
2.2. Matrix Properties Determination Based onMicromechanics. Micromechanics investigates the behavior
of the heterogeneities as well as their e�ects on the overallproperties and performance of a material. An important taskof micromechanics is to link mechanical relations on di�erentlength scales. �e entire behavior of the microstructure isinterpreted as the mechanical state of a material point on themacroscopic level which thereby is ascribed e�ective materialproperties. Such a micro-to-macro transition formally pro-ceeds by appropriate averaging processes and is called ho-mogenization as shown in Figure 5 [21].
100 200 300Number of pixels (555 scanning lines)
RGB imageN
umbe
r of p
ixels
(416
scan
ning
line
s)
400 500400350300250200A A
10050
Figure 2: �e original image with the j-coordinates at i� 150(section AA).
0.1100
C
F
M
200 300 400 500 600
0.2
0.3
0.4
0.5
0.6
0.7
0.8
I val
ue
Number of pixels
Plot(1(150))
Mineral SymbolFeldspar FCalcite CMatrix M
Figure 3: Variation of the intensity component (I) with the j-coordinates at i� 150 (section AA in Figure 2).
0
500
1000
0 0.1 0.2 0.3 0.4 0.5I value
Histogram I
Num
ber o
f pix
els
0.6 0.7 0.8 0.9 1
1500
2000
2500
3000
Figure 4: �e intensity component (I) histogram of the image inFigure 1.
Table 1: �reshold values and volume fractions of the rock.
Inhomogeneous material can be described by an equiv-alent homogeneous material. Based on Eshelby solution [23]of equivalent homogeneous material, the concentrationtensor of each phase (Ar) is constant. �erefore, the sti�nesstensor of the heterogeneous media can be expressed as [20]
Chom � C
s +∑N
c�1φc C
r −Cs( ) : Ar, (1)
where φr and Ar are, respectively, the volume fraction andthe concentration tensor of the rth inclusion family. �eRVE is composed of an isotropic linear elastic matrix withsti�ness tensorCs and of a random distribution of sphericalinclusions with sti�ness tensor Cr. To evaluate the ho-mogenized sti�ness tensor (Chom), the inclusion concen-tration tensor (Ar)must be determined. It is generally usedfrom the Eshelby solution [23] and analytical schemes suchas dilute, Mori–Tanaka [24], and self-consistent schemes toevaluate the inclusion concentration tensor Ar in (1) formaterials with random microstructure. It is shown that,unlike the dilute and the self-consistent schemes, theMori–Tanaka scheme describes the in situ experimental datawell [22]. Considering the previous results, the Mori–Tanaka scheme is the most suitable for a composite withmatrix-inclusion morphology which is the case of therhyodacite tu�.
In the Mori–Tanaka scheme (1973), therefore, the strainor stress eld in the matrix is, in a su¦cient distance froma defect, approximated by the constant eld. �e loading ofeach defect then depends on the existence of further defectsvia the average matrix strain or the average matrix stress.Fluctuations of the local elds, however, are neglected in thisapproximation of defect interaction. It follows that the lo-calization tensor is then given by [22]
Ar � I + P
0 : Cr −C0( )[ ]
−1: ∑
N
s�0φs I + P
0 : Cs −C0( )( )
−1 −1
. (2)
�is leads to the following estimate of the e�ectivesti�ness tensor [22]:
Chom � ∑
r�0φrCr : I + P
0 : Cr −C0( )[ ]
−1: ∑
N
s�0φs I + P
0 : Cs −C0( )( )
−1 −1
. (3)
Because of the isotropy of the constituents and thespherical shape assumption of inclusions, we have [22]
where μr and kr are the shear and bulk moduli, respectively,of the phase r and μhom and khom are the homogenized shearand bulk moduli. fr and fs are the volume fractions of thephases.
�e mineralogical compositions of the rhyodacite tu�contain four main phases: calcite, feldspar, quartz, andmatrix. It is organized in grains spread in a siliceous matrix.�e rst stage of the homogenization procedure is thedenition of a representative elementary volume (r.e.v.). �eobservations led us to consider the rhyodacite tu� as a four-phase composite of the inclusion/matrix type in which we
discern the calcite, feldspar, and quartz phases, assumed to bedistributed individually in a siliceous matrix. �e rhyodacitetu� sample can be represented by a four-phase composite withdistinct mechanical properties. �is material has a matrix/inclusion morphology with the phases randomly distributed,and the calcite, feldspar, and quartz minerals being embeddedin the siliceous matrix. It is assumed a representative ele-mentary volume containing four phases as shown in Figure 6.
�e elastic properties of the crystalline siliceous matrix(K0; μ0) are not precisely known, and there is no directmeasurement available. An “inverse method” is thereforeused to determine these elastic properties from those whichare known for the composite: the macroscopic elasticproperties of the rhyodacite tu� were determined experi-mentally as Khom � 16.6GPa and μhom � 15.7GPa, and theelastic properties of calcite, feldspar, and quartz grains aredetermined from the existing data found in literature[22, 25]. For the calcite, k1 � 70.8GPa and μ1 � 32.7GPa, forthe feldspar k2 � 54.2GPa and μ2 � 25GPa, and for thequartz k3 � 36.8GPa and μ3 � 44.4GPa. �e elastic
x2
V
l
d (RVE)
(Homogenization)
(Microstructure)
Cijkl(x)
C*ijkl
x1
x3
∂V
L
L
x1macro
x2macro
x3macro
xmacro
Figure 5: Homogenization and characteristic length scales [21].
4 Advances in Civil Engineering
properties of the matrix are calculated by solving thenonlinear (4) as E0 � 25GPa and ϑ0 � 0.3.
2.3. Statistical Distribution. Rock is a heterogeneous mate-rial. �is heterogeneity causes rock in compression tofracture via the formation, extension, and coalescence ofmicrocracks. Studies showed that the variation of me-chanical properties can be explained statistically. In a generalstudy on rock fracture, theWeibull distribution function wasconsidered for heterogeneity description.
In this study, the rock is assumed to be composed ofmany elements of identical size, with the mechanicalproperties such as bulk and shear modulus of elements toconform to the Weibull distribution, so the mechanicalparameters of every element are specied stochasticallyaccording to the given Weibull distribution dened in thefollowing probability density function [21]:
f(u) �m
u0
u
u0( )
m−1
exp −u
u0( )
m
, (5)
where u is the variable that follows the Weibull distributionand applied here to both bulk and shear modulus. m is theshape parameter describing the scatter of u and u0 is a scaleparameter expressing the average of the considered me-chanical properties. In theWeibull distribution function, theshape parameter m plays a signicant role. Generally, thehigher the value for m, the smaller is the amount of het-erogeneity in the model. Following the determination of theYoung modulus of each mineral composing the rock sampleand its volume fraction in the rock sample, the Weibulldistribution function was tted to the calculated probabilitydensity-Young modulus diagram as shown in Figure 7.
According to Figure 7, the homogeneity index (m) of therhyodacite tu� sample was calculated to be equal to 6. Also,the mean value of minerals’ Young modulus was obtained62GPa.
3. The Numerical Simulation
3.1. �e Proposed Model and Input Data. �e plastic rockbehavior is represented by the Mohr–Coulomb model withstrain softening. �e most well-known failure criterion forrock is theMohr–Coulomb criterion.�e criterion is a linearenvelope touching the Mohr’s circles representing the
magnitude of the maximum and minimum stresses at themoment of rock failure. �e criterion states that the failureoccurs if the magnitude of the shear stress on a specic planereaches a critical threshold. �e critical threshold is asso-ciated with both the cohesion of the rock grains at the planeof failure and friction resistance between them. �e frictionresistance of the failure surface is dependent on the normalstress imposed on the plane.�e strain-softening behavior ofrocks is governed by shrinking of the failure criterion withthe advance of plastic deformation. �e decline of rockstrength with plastic strain is denoted as strain-softeningbehavior. �e strain-softening model allows representationof material softening at postpeak behavior based on pre-scribed variations of the Mohr–Coulomb model properties(cohesion and friction) as functions of the deviatoric plasticstrain after the onset of plastic yield [26].
In the plastic zone, it is supposed that strength pa-rameters of rock mass decrease by bilinear functionaccording to a softening parameter (cp) in comparison withthe critical softening parameter (cp∗) in the softening regionand it reaches a minimum constant value in the residualregion. �e critical softening parameter controls the tran-sition between the softening and residual stages.
μ cp( ) �μp − μp − μr( )
cp
cp∗, 0< cp < cp∗
μr, cp ≥ cp∗
. (6)
In (6), μ represents one of the strength parameters φ, andC and the subscripts p and r denote the peak and residualvalues, respectively [27]. �e rock is assumed to be a het-erogeneous material, and its mechanical properties areconsidered to conform to the Weibull distribution function.�e mean values of bulk and shear modulus are speciedaccording to real values obtained from laboratory tests. Forsimplication, it is assumed that the bulk and shear modulushave the same homogeneity index.
�e proposed statistical Mohr–Coulomb strain-softeningmodel was programmed within the C++ environment andwas implemented into a commercial DEM code. Using theWeibull probability distribution function in a numerical
5
Matrix
0.050.1
0.150.2
0.250.3
0.350.4
15 25 45 55 65 85
Prob
abili
ty d
ensit
y
95 1057535Elasticity modulus (GPa)
Feldspar
Calcite
Quartz
Figure 7: Fitting the Weibull distribution function to the prob-ability density-Young modulus diagram.
Matrix
Calcite
Feldspar
Quartz
Figure 6: Representative elementary volume (r.e.v.) of the rhyo-dacite tu�.
Advances in Civil Engineering 5
simulation of a medium composed of many elements withdi�erent elastic properties, one can produce a heterogeneousmaterial numerically. �e proposed statistical Mohr–Coulombstrain-softening model used in the presented analysis was linkedto a commercial DEM code as a separate constitutive model.
�e studied rock is an extrusive porphyritic igneous rockcalled rhyodacite tu�. �e mineralogical compositions andinitial porosity of samples were rst investigated. �isrhyodacite tu� sample was cored at the depth of 113m inorder to site investigation of a civil underground projectlocated in the north-west region of Tehran [28]. �e com-plete stress-strain curve of the rhyodacite tu� under the UCStest condition is shown in Figure 8.
Hence, numerical simulation of the rhyodacite tu� uni-axial compression strength (UCS) test was performed with theproposed statistical Mohr–Coulomb strain-softening model.With regard to the experimental test, a summary of input dataused in the numerical analysis is given in Table 2.
3.2. Geometry and Boundary Condition. Uniaxial compres-sive strength (UCS) test is the most widely conductedstandard test on rock samples. �emain objective of this testis to determine the peak strength (σc), modulus of elasticity(E), and Poisson’s ratio (]) of the rock material.
Moreover, employing the sophisticated servo-controltesting machine, the complete stress-strain behavior of therock can be determined in this test. Additionally, the shapeof the stress-strain curve in the postpeak region is an in-dicative of rock breakage mechanism and its brittleness.
In order to verify the statistical Mohr–Coulomb strain-softening model, it was attempted to simulate the test con-dition as closely as possible. �e sample shape, dimension,input material properties, and loading condition were selectedsimilar to the test condition. �e main objective was to re-produce the tested rock stress-strain curve numerically and delveinto the sample failure mechanism in the postelastic range.
A plane stress condition was assumed for the analysis. Itis understood that the actual problem has a 3D nature. Butwith regard to the 2D nature of the selected code, a two-dimensional plane slice was selected at the center of thesample and analyzed.
�e complete fracture characteristic of a numericalspecimen under uniaxial loading may be investigated only ina stable displacement-controlled test. �e load is applied ina sequence of steps in the vertical direction through in-cremental axial displacement control at one end of thenumerical sample in a quasi-static fashion, while the otherend is prevented from vertical movement. �e sampleuniaxial loading was simulated imposing a velocity eld inthe range of static loading in accordance with the ISRMstandard at the top of the model, while a zero verticaldisplacement was applied at the base. �ere are no con-straints on the sides of the sample and the specimen sides areallowed to move in the horizontal and vertical directions. Aview of the model geometry and employed boundary con-dition for the test condition is shown in Figure 9.
�e numerical specimen was discretized with 4096 el-ements. �e numerical specimen failure process takes placewithin it due to the heterogeneity of its properties.
As mentioned, the mean Young modulus for the entirenumerical specimen is 62GPa, but specied Young modulusof di�erent elements is considerably di�erent with this value
0
10
20
30
40
50
60
70
80
0 0.1 0.2 0.3Axial strain (%)
Figure 8: �e complete stress-strain curve of the rhyodacite tu�tested in rock mechanics laboratory [24].
Table 2: Mechanical parameters used as input data.
c(kg/m3) E (GPa) ] m σc(MPa) σT(MPa)2600 62 0.25 6 75.25 7
10
5 cm
o
y
x
Figure 9: Geometry and boundary condition of the numericalspecimen.
6 Advances in Civil Engineering
and conform to theWeibull distribution function. Even withthe same distribution parameters for the specimen, thespatial distribution of properties of elements may be sto-chastically di�erent. �erefore, the spatial distribution ofmechanical properties is shown in Figure 10.
However, the e�ect of the randomness due to the spatialdistribution of mechanical properties of elements wasstudied thoroughly by Zhu and Tang [29].
3.3. �e Simulation Results. In order to assess the localbehavior of the sample, series of horizontal and verticalmeasuring points were placed throughout the model. Im-portant variables such as stress, strain, and displacementcomponents were monitored at these locations. �e localstress-strain curves of some elements with di�erent sti�nessare shown in Figure 11.
In Figure 11, the local stress-strain behavior of elementscomposing the numerical sample with dissimilar sti�ness issignicantly di�erent. �e elements with higher sti�ness bearmore stress, so their stress states approach to the yield surfaceearlier. However, the elements with lower sti�ness bear lessstress, so their stress states approach to the yield surface later.�e overall stress-strain curve of the entire numerical samplewas shown and compared to the experimental result as shownin Figure 12. �e comparison of numerical and experimentalresults shows that they are in agreement.
To study the in�uence of homogeneity indices (m) onmacroscopic mechanical behavior, especially the postpeakregion, three numerical specimens with homogeneity indices of1, 3, and 6, respectively, are subjected to uniaxial compressionloading while other input data remain as specied in Table 2.
As the homogeneity index increases, mechanical materialproperties become more homogeneous and approach that ofthe homogeneous body. �e total envelope of the stress-strain
curves for three numerical specimens with di�erent homoge-neity indices and experimental result can be seen in Figure 13.
�e stress-strain curves of these numerical specimens arelinear in the prepeak region and lose most of their load-carryingcapacity in the postpeak region. As m increases, the strengthand brittleness of the resulting stress-strain curve increases.�us, the higher m is, the more brittle the stress-strain curve.�e lowerm is, themore ductile themodel behavior. From theabove results, we can conclude that the homogeneity index inthis model controls the strength and ductility. �is suggeststhat hard (brittle) rock has a higher homogeneity index thansoft (ductile) rock.
Figure 10: �e spatial distribution of mechanical properties(m� 6).
80
90
70
60
50
40
30
20
10
00 0.1 0.2 0.3 0.4 0.5
Axi
al st
ress
(MPa
)
Axial strain (%)
E = 38 GpaE = 36 Gpa
E = 52 Gpa
E = 40 Gpa
Δσ = 16 MPa
Figure 11: �e local stress-strain curves of the elements withdi�erent sti�ness.
80.00
70.00
60.00
50.00
40.00
30.00
20.00
10.00
0.000.00 0.05 0.10 0.15 0.20 0.25 0.30
Axi
al st
ress
(MPa
)
Axial strain (%)m = 6Experimental data
Figure 12: �e overall stress-strain curve of the entire numericalsample and its comparison with the experimental data.
Advances in Civil Engineering 7
3.4. AE and Released Strain Energy. Because of rock het-erogeneity, some elements of the numerical specimen underloading reach to the failure criterion earlier than others. �eirreleased strain energy is the origin of the acoustic emission inthe rock-fracturing process. Acoustic emission (AE) can beused to detect the microscopic processes associated withheterogeneous rock fracture. Generally, AE events are notnotable until the occurrence of nonlinearity in the stress-straincurve, and the rate at which the AE events appear changeswith the development of fracture. �e AE rate increasesgradually with extension of the microcracks and increasesrapidly as the microcracks link together. �e rate maximizeswhen the nal fracture planes form [9]. Monitoring acousticemission (AE) event rates seems to be a good way to identifythe initiation and propagation of microcracks in heteroge-neous rock. In quasi-brittle materials such as heterogeneousrock, AE is predominantly related to the release of strainenergy. �erefore, as an approximation, it is reasonable toassume that the AE counts are proportional to the number offailed elements and that the released strain energy by failedelements is all in the form of acoustic emissions [30]. By thismeans, in this model, each AE event corresponds to failure ofan element. �erefore, the AE counts are accounted by thenumber of failed elements and the released strain energy iscalculated based on energy computation of the entire nu-merical specimen. �e released strain energy of the entirenumerical specimen which is the di�erence between the workdone at the boundary of the model and the total stored andplastic dissipated strain energies can be written as [31]
Wr � σijεij − Uc + Ub +Wp( ), (7)
where Uc is the total stored strain energy, Ub is the totalchange in potential energy, and Wp is the plastic dissipatedwork. �ese energies described in (7) are calculated in anincremental fashion at each timestep from the stress, force,displacement, and strain changes. Based on the above as-sumptions, the cumulative AE counts and cumulative re-leased strain energy can be realistically simulated with theabovementioned numerical model. �e simulated stress-strain curve and the released strain energy for the numer-ical specimen withm� 6 under UCS test condition is shownin Figure 14.
�e stress-strain curve of this numerical specimen isalmost linear in the prepeak region, and then the stress-strain curve begins to deviate from linearity at stress about50MPa. Further increases in axial strain lead to a rapidreleased strain energy, and this process continues up to thepeak strength. Finally, the strain energy release at theconstant rate in the postpeak region and the stress-straincurve approaches a residual strength. �e stress-strain curveas well as the AE counts during the fracture process of thenumerical rock specimen under uniaxial compression test isshown in Figure 15.
80
70
60
50
40
30
20
10
00 0.05 0.1 0.15 0.2 0.25 0.3
Axi
al st
ress
(MPa
)
Axial strain (%)m = 3m = 1m = 6Experimental data
Figure 13: �e total envelope of the stress-strain curves for threenumerical specimens with di�erent homogeneity indices and ex-perimental result.
80.0070.0060.0050.0040.0030.0020.0010.00
0.00
160140120100806040200
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35
Axi
al st
ress
(Mpa
)
Ener
gy re
leas
e acc
umul
atio
n (j)
Axial strain (%)
EnergyStress
Figure 14:�e simulated stress-strain curve and the released strainenergy.
80.0070.0060.0050.0040.0030.0020.0010.00
0.00
400350300250200150100500
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35
Axi
al st
ress
(Mpa
)
AE
accu
mul
atio
n (c
ount
)
Axial strain (%)
AEStress
Figure 15: �e stress-strain curve and the AE counts during thefracture process.
8 Advances in Civil Engineering
It can be clearly seen that there are a few failed elementsduring the initial loading phase. But these failed elementsrelease much less energy as shown in Figure 14. *erefore,the curve is nearly linear up to approximately 60% of thepeak strength. Localization of deformations may appear dueto cracking caused by these failed elements. After reachingthe peak strength, the load-carrying capacity of rock dropsconsiderably, followed by a long tail until fracture of thespecimen.
*e release of strain energy versus axial loading curvesfor the numerical specimens with different homogeneityindices is illustrated and compared in Figure 16.
*e strain energy release of the heterogeneous specimenis less than the strain energy release of the homogeneousspecimen. *e heterogeneous numerical specimen releasesits strain energy gradually and in a controlled manner.However, the homogeneous numerical specimen releases itsstrain energy abruptly at a stress level about the peakstrength. *e AE accumulation counts versus axial loadingcurves for the numerical specimens with different homo-geneity indices are shown and compared in Figure 17.
Based on Figure 17, the more the heterogeneous nu-merical specimen, the sooner and more gradual and ductilethe failure process was done. *e more homogeneous rockfails abruptly because of its more brittleness.
4. Conclusion
*e volume fractions of minerals in the microscopic thinsection of the rhyodacite tuff sample were calculated pre-cisely by digital image processing. *e unknown matrixproperties were determined based on Mori–Tanaka schemein the framework of micromechanics. *en, the Weibulldistribution function was fitted to the distribution ofminerals’ Young modulus, and the homogeneity index wasdetermined. Using the statistical Mohr–Coulomb strain-
softening model, the rock behavior was simulated and theexperimental stress-strain curve was reproduced numeri-cally. From the numerical results, we can conclude that thehomogeneity index in this model controls the strength andbrittleness. *e simulated AE and released strain energyduring the loading process are dependent on the homoge-neity index. *e more the homogeneous numerical speci-men, the more the AE and release of strain energy underUCS test condition.
Conflicts of Interest
*e authors declare that they have no conflicts of interest.
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0.350.30.250.2Axial strain (%)
0.150.10.0500
140
120
100
80
Ener
gy re
leas
es ac
cum
ulat
ion
(j)
60
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0.4
m = 1m = 3m = 6
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0.350.30.250.2Axial strain (%)
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