Research ArticleMechanical Properties and FailureModes of Rock Specimens withSpecific Joint Geometries in Triaxial Unloading Compressive Test

Guoyong Duan 123 Jianlin Li123 Jingyu Zhang3 Eleyas Assefa4 and Xushu Sun 3

1Key Laboratory of Ministry of Education for Geomechanics and Embankment Engineering Hohai UniversityNanjing 210098 China2Jiangsu Research Center for Geotechnical Engineering Technology Hohai University Nanjing 210098 China3Key Laboratory of Geological Hazards on +ree Gorges Reservoir Area Ministry of Education China +ree Gorges University8 Daxue Road Yichang 443002 China4College of Architecture and Civil Engineering Addis Ababa Science and Technology University Addis Ababa Ethiopia

Correspondence should be addressed to Xushu Sun sunxsctgueducn

Received 21 August 2018 Revised 26 January 2019 Accepted 14 February 2019 Published 25 March 2019

Academic Editor Jun Liu

Copyright copy 2019GuoyongDuan et alis is an open access article distributed under theCreative CommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

e effects of disconnected joints on the mechanical characteristics of rock masses are interesting and challenging aspects of rockmechanics e prime objective of this study is to investigate the effect of joint orientations and joint connectivity rates on thestrength deformation and failure mechanisms of rock specimens under unloading condition To establish the relationshipsbetween different factors (confining pressure joint orientation and joint connectivity) and failure mechanisms a series of triaxialunloading tests were performed e results showed that the joint orientation had a more considerable effect than the jointconnectivity on the strength and deformation of the specimens Generally three different types of failures were observed(ie shear mixed and split) Finally Griffithrsquos theory was utilized to analyze the maximum tensile stress around the crack efindings of this paper can also be used for practical engineering problems

1 Introduction

Understanding the mechanical behavior of jointed rock isvery important for stratum stability in gas and oil engi-neering [1] e stress release and redistribution around theexcavation surface generally results in deformation of rockmass together with expansion and extension of existeddiscontinuities and development of macroscopic fracturese discontinuities (such as joints fissures faults cleavagesand bedding planes) can significantly affect the mechanicalbehavior of jointed rock mass [2 3]

Several attempts [4ndash6] have been made in the past toinvestigate the influence of fractures on the macroscopicbehavior of the rock mass and to illustrate the crackpropagation mechanisms

Practically it is too difficult to find a homogeneous rockwith a single discontinuity erefore researchers have usedrock-like materials to investigate the effects of cracks on the

mechanical properties of rock masses Cao et al [7] con-ducted a series of uniaxial compression tests on similarmaterials to simulate the effect of preexisting joints andfissures of rock e variation of joint orientation draws theattention of researchers Cracks easily initiated at the tips ofdiscontinuity when the angles of inclination were 0deg 30deg and60deg under a series of uniaxial compression tests [8] Binget al [9] performed the test on gypsum specimens con-taining a single discontinuity with length 10mmndash30mmand different inclination angles 0degndash90deg e resulting peakstresses were affected by both factors Meanwhile the dis-continuity scale was studied as a critical factor by Zhang [10]through the acoustic emission monitoring system in theuniaxial compression test Similar patterns of crack prop-agation were observed through numerical simulation Mengand Liu [11] used RFPA software to simulate the relationshipbetween a rock mass with one discontinuity and differentconfining pressures According to their result the crack was

HindawiAdvances in Materials Science and EngineeringVolume 2019 Article ID 1340934 14 pageshttpsdoiorg10115520191340934

extended vertically at low confining pressure and traced ahorizontal trend at a high confining pressure

Based on the above facts the uniaxial compression testwas predominantly used in the last decades It has twoadvantages as shown below (1) uniaxial loading equipmentis conventional in the laboratory and convenient for mul-tiscale specimens and (2) one can efficiently use high-speedcameras and acoustic emission monitoring instrumentduring the experiment However a uniaxial compressiontest is merely applicable to stimulate free face in mining andexcavation Underground rock mass with a three-dimensional stress state is another research topic in prac-tice e mechanical behavior of rock material under thebiaxial and triaxial conditions is beneficial to explore thefailure mechanism of rock engineering

Sagong et al [12] conducted the biaxial compression teston specimens with different orientations (30deg 45deg and 60deg)ey discovered that tensile crack initiation and propagationplays a dominant role at a small joint dip angle In contrasttensile cracks exhibit abrupt initiation and propagationunder large joint angle Chen et al [13] conducted a series ofconventional triaxial compression tests to scrutinize theprefailure damage characteristics According to their studythe damage propagation rate varied distinctly before thereversal of the volumetric strain Yang et al [14] performedcyclic triaxial tests and categorized four stages based onPoissonrsquos ratio and Youngrsquos modulus parameters Erarslan[15] focused on micromechanical and microstructural dy-namics to study subcritical crack propagation Wang et al[16] studied the effects of the joint dip angle by using distinctelement-based numerical simulations More cracks de-veloped when the joint dip angles were small

Although the influences of rock joints on the strengthdeformation and failure mechanism have been studiedanalytically numerically and experimentally loading con-ditions are merely a glimpse of the problems in practiceisconcept is discussed with a practical engineering problembelow As it is known China ree Gorges Dam is thebiggest hydropower project in the world e reservoir is185meters deep with a storage capacity of about393 times 109 m3 Due to its large storage capacity and cyclicfluctuation the rock masses in the vicinity of the reeGorges Reservoir area have been repeatedly subjected byloading and unloading process with respect to its geologicalage (Figure 1) As a result of similar large-scale projectconstructions and long-term operation in the high-stressregion the unloading effect on the rock mass became afrontier topic

Laboratory investigations [17 18] and numerical sim-ulations [19] revealed the fact that the behavior of rock underunloading condition is different from that of loading con-dition Li et al [20] proposed a new micromacromethod topredict the shear strength of the brittle rock by relating crackpropagation with the axial strain e fractal values ofsandstone samples were used to predict the failure pre-cursors at low and high confining pressures and rock failureat low and high confining pressures respectively [21] eaxial circumferential and volumetric deformations at stresslevel close to the peak stress result in a considerable volume

expansion and a large deformation platform during theunloading process [22] Stress relaxation behavior wasstudied under loading and unloading conditions [23] eyfound that the rate of unloading is closely related to the stressrelaxation behavior In the stress relaxation test the failure isdue to tensile rheological cracks of smooth coalescenceHowever the fracture in the traditional compression test isrelated to the wing cracks with abrupt coalescence From theperspective of energy conversion Li et al [24] compared thereleased elastic strain energy with the absorbed energy in thetriaxial unloading tests e increased rate of energy dissi-pation widely varies between the triaxial loading andunloading conditions New models were proposed for thebrittle rocks by the researchers Zhou et al [19] presented aphase field model to study crack propagation branchingand concentration of coalescence on twenty parallel flawsand multiple echelon flaws Liu and Zhang [25] established amodel considering the coeffect of macroscopic and meso-scopic flaws based on the Lemaitre strain equivalence

Now the existing study on the crack damage of rockmechanics is often based on the uniaxial compression testswhich is well developed and understood However the studyon flaw propagation under the unloading condition is notrelatively enough since existing studies predominantlyconsidered a single flaw parameter to analyze the anisotropicbehavior of the rock mass with a consideration of the jointgeometrical properties such as its length and dip angleHowever its mechanical theory was not studied very wellHence there is a lack of theoretical analysis of the test results

erefore in this paper the results of a series of con-ventional triaxial loading and unloading tests on a rock-likematerial containing a single flaw under variable confiningpressures ranging between 0 and 10MPa were theoreticallyanalyzed and interpreted Based on the crack damage modesthe coeffect of the joint angle and joint connectivity on thestrength and deformation parameters was investigateden the effects of the confining pressure on the propa-gation and joint type were analyzed Finally the damagecharacteristics of the specimens were analyzed using Grif-fithrsquos strength theory in detail

2 Testing Specimens (Sample Preparation)

Considering the discreteness and the difficulty in acquiring anatural rock together with disconnected joint a cementmortar was prepared from a mix of ordinary Portland ce-ment (OPC) sand and water at the ratio of 1 173 04respectively by weight e strength parameters of the ce-ment mortar were similar to the weak weathering sandstonein China ree Gorges Reservoir Region e parametercomparisons between sandstone and cement mortar arelisted in Table 1 Due to the similarity-based parameters thecement mortar is reasonably selected as a rock-like material

Afterwards the cement mortar was poured into a cy-lindrical plastic mold (ie 50mm inner diameter by 105mmheight) A small size shaking table system was employed toeliminate the development of bubbles during samplepreparation As shown in Figure 2 symmetrical grooveswere established on the surface of cylindrical molds to create

2 Advances in Materials Science and Engineering

joints with thin steel sheets A release agent was presmearedinside the molds and the steel sheets to extrude the samplessmoothly after the initial set time Due to the presence andextraction of thin sheets the joints were developed duringsolidification en the cement mortar samples were kept ina curing room (at 18degC room temperature 99 relativehumidity and atmospheric pressure for 28 days) No fillerswere employed inside the joint openings (all of the openingwidths were less than 1mm) To get flat surfaces the tips ofthe specimens were trimmed off

21 Joint Geometries Created to Observe the Influence of JointConnectivity Rate e first set of triaxial compression testwas executed to investigate the effect of joint connectivityrate on the mechanical properties of the specimens withpartially cut-through joints Here the angle of inclinationwas kept constant It is defined as the ratio between the jointarea and the joint location plane across the specimen asshown in Figure 2 Meanwhile k stands for joint connec-tivity rate and three different values were considered (030045 and 060) However all the other joint geometrical

Lianziya hazardous rock mass

Yangtze river

Three gorges dam

Xiangxi town

Guojiaba town

Quyuan town

Taipingxi town

FIGURE 1 Lianziya hazardous rock mass in ree Gorges Reservoir area

Table 1 e parameter comparisons between sandstone and cement mortar

Sandstone Cement mortarCohesion (MPa) 162 107Internal friction angle (deg) 35 315Uniaxial compression strength (MPa) 3929 3265Elasticity modulus (GPa) 1025 924Poissonrsquos ratio 040 038Unit weight (kNm3) 220 206

(a)

The joint plane

(b)

Figure 2 Schematic diagram of specimens (a) Cylindrical plastic molds used for sample preparation (b) Joint location and jointconnectivity rate

Advances in Materials Science and Engineering 3

parameters were kept constant e joints were symmetri-cally located at the center of the specimen to minimize theconceivable effect of eccentricity

22 Joint Geometries Created to Observe the Influence of JointOrientation e other set of triaxial compression test wasperformed to investigate the effect of joint orientation on themechanical response of the specimens with partially cut-through joints In this case the joint connectivity rate waskept constante centroid of the joint was located at 50mmdistance from the tip of the specimen ree different valueswere adopted as joint orientations (ie 30 45 and 60deg) ejoint orientations were measured from the horizontal axisExcept the joint orientation the other geometrical param-eters were kept constant Based on the MohrndashCoulombfailure criterion the dip angle of the structural plane withrespect to the intact failure surface is slightly larger than 45degAs a result one can investigate the impact of joint orien-tation by altering the angles Figure 3 shows the geometricalspecifications used for preparing nine different specimens

3 Description of Laboratory Test

In this testing program a computerized triaxial compressivetesting apparatus was used as shown in Figure 4

is device (RMT-150C) was designed and manufac-tured by the Institute of Rock and Soil Mechanics (affiliatedto the Chinese Academy of Sciences) It can simulate theloading and the unloading conditions and perform differenttypes of tests (uniaxial test direct shear test Brazil disk splittest etc)

e PCI-2 AE screening systemwas used to eliminate thespecimens with defects

4 Experimental Methodology

To investigate the mechanical properties of rock mass withdisconnected joints especially under loading and unloadingconditions five different values of confining pressures anddifferent kinds of stress paths were adopted as listed inTable 1

e adopted testing procedures for the cylindrical rockspecimens are as follows

(1) Initially the ldquoforce-confining pressurerdquo mode wasselected and a predefined axial force and confiningpressures were applied on the specimens simulta-neously at the loading rate of 02 kNs and 01MPas respectively e axial load was increasing at therate of 05 kNs until failure occurred at a constantconfining pressure en the maximum triaxialcompression strength was determined

(2) e same type of specimen was chosen(3) e ldquoforce-confining pressurerdquo mode was selected

and a predefined axial force and confining pressureswere applied on the specimens simultaneously at theloading rate of 02 kNs and 01MPas respectively

(4) After setting the confining pressure constant theaxial load was applied at the rate of 05 kNs up to the70 of the maximum triaxial compressive strength

(5) en the confining pressure was unloaded to failureat a rate of 001MPas e predefined loadings forthe specimen with 30deg joint orientation and 03connectivity rates are shown in Table 2

5 Testing Results and Discussion

As it was mentioned in Section 4 the testing programme wasdesigned to analyze the effect of joint orientation andconnectivity rate on the mechanical properties of rockspecimens with partially cut-through joints eir effects onthe compressive strength and deformation characteristicsare discussed below

Brittle failure occurred frequently in the triaxialunloading tests e process of unloading was conducted bydecreasing the confining pressure Hence the percentage ofreduction in the confining pressure was a key factor whichaffected the sample failure During the unloading tests allthe parameters were recognized as equivalent variableswhich were different from the typical parameters

rough the analysis of the deformation modulus of thespecimens with different dip angles and connectivity ratiosthe variation of different types of deformation modulus withunloading is shown in Figure 5 e deformation modulus isrepresented as E50 and Δσ3 is the reduction percentage ofthe confining pressure e results show that the de-formation modulus of specimens with different connectivityratios is similar under the same confining pressure level As aresult the deformation modulus is less sensitive to thechange of joint connectivity As the inclination angle in-creases the average value of the deformation modulus en-largeserefore the deformation modulus is more sensitiveto the change of the inclination angle When the confiningpressure level varies the deformation modulus of the sametype of specimens fluctuates slightly It can be concluded thatthe confining pressures ranging from 0MPa to 10MPa havea little effect on the deformation modulus According to thevariation of the deformation modulus a large decrease in theamplitude arouses at the unloading percentages of 0sim20and 80sim100 Especially after the 80 reduction ofconfining pressure the deformation modulus declinedrapidly However when the unloading loss ranges from20ndash80 the decreasing amplitude is stable relativelyAccelerated variation crops up at the early stage and the latestage of unloading During the triaxial unloading processthe internal strain of the specimen is accumulated by theconfining pressure constraint When the confining pressureis declining accumulated energy release initiates graduallye internal stress and strain are adjusted while maintainingthe residual bearing capacity Accordingly the middle stageof deformation modulus alteration is relatively flat At theend of unloading process the residual confining pressure isinsufficient to restrain the energy inside the specimen andthereupon failure occurs Hence the development ofunloading procedure can be divided into two stages in

4 Advances in Materials Science and Engineering

consonance with the decrease rate of deformation modulusere are 0sim80 and 80sim100 respectively

6 Failure Modes

e failure modes of jointed samples were identified by thelocation and the size of cracks e influence of joint pa-rameters on the crack propagation was discussed As shownin Figure 6 the failure modes were categorized into threegroups

(1) e shear failure occurred frequently under com-pression condition It can be recognized by the shearplane Ignoring stress concentrations on theboundaries of samples the shear plane was relativelyflat is may be induced due to the conversion fromdefects to weak structural planes Based on the as-sumption of sliding mostly occurs along the weak

planes in rockmass the joints are considered as weakstructural planes However rupture did not tran-spired along the crack orientation completely duringthe test In other words the prefabricated cracks didnot thoroughly affect the crack propagation inseveral cases (Figure 7(a)) e condition of a sampleseparated into two parts along the joints accountedfor two third percentage Joint geometries and thedegree of coincidence between the joint and therupture trajectory can be used to characterize thefailure modes (Figure 3)

(2) Generally splitting failure arises at lower confiningpressures e lateral deformation increases muchmore rapidly in the circumstances And the splittingcracks were nearly parallel to themaximum principalstress of the sample Meanwhile the secondarycracks propagated from the crack tips during the

30deg

k = 03

Φ 50

100

(a)

30deg

k = 045

Φ 50

100

(b)30deg

k = 06

Φ 50

100

(c)

45deg

k = 03

Φ 50

100

(d)

45degk = 04

5

Φ 50

100

(e)

45degk = 06

Φ 50

100

(f )

60deg

k = 0

3

Φ 50

100

(g)

60degk = 0

45

Φ 50

100

(h)

60degk = 0

6Φ 50

100

(i)

Figure 3 Geometrical specifications

Advances in Materials Science and Engineering 5

unloading tests It is noteworthy that splitting failureoccurs in the intact rock samples in the procedure ofuniaxial compression Nevertheless similar phe-nomena were observed on the specimens withsmaller joint angles and connectivity rates In otherwords the likelihood of splitting failure is high whenthe influence of joint is insignificant Meanwhile theprobability of splitting failure is also dependent onthe angle of the joint and the vertical axis of thesample Based upon the results samples with 30degjoint orientation were more inclined to splittingfailure

(3) A mixed type of failure (tension and shear) wasdeveloped at higher joint orientation and mediumjoint connectivity rate It is characterized by slidingof joints and development of tension cracks from thecrack tip Based on the rupture morphology theprecondition of mixed failure hinges on middlevalues of joint orientation and connectivity rate asshown in Figures 7(d)ndash7(f) Considering the ten-dency of shear along the joint plane tension cracksdeveloped by the reason of stress concentrations at

the tips of the cracks It is an important characteristicthat reveals the dislocation of sample pieces beforethe ultimate failure e two parallel tension frac-tures were normal to the original crack In mostcases the secondary cracks were connected to thetips of joints where stress concentrations are in-evitable erefore the propagation of cracks can bedetermined by analyzing the geometries of joints andloading conditions

61 Effect of Confining Pressure on the Failure of the SamplesIn the triaxial test the mechanical properties of rockspecimens can be affected by the confining pressure defi-nitely e effect of unloading was remarkable when theconfining pressures were 2 and 4megapascals (based uponthe analysis of 30 diagrams) As a result the specimensexhibited different levels of damages at different confiningpressures During the unloading test parts of the specimenswere shattering when the confining pressure was lower forexample 2 and 4megapascals In this case the fragments

Figure 4 Triaxial compressive testing apparatus

Table 2 Stress state for the triaxial compression and unloading compression

Stress state Minimum principal stress σ3 (MPa) Maximum principal stress σ1 (MPa) Remarks

σ1

σ1σ1 gt σ2 = σ3

σ3σ3

10 61526 Triaxial compression8 55468 Triaxial compression6 47028 Triaxial compression4 37239 Triaxial compression2 29256 Triaxial compression10 43106 Triaxial unloading compression8 38677 Triaxial unloading compression6 32110 Triaxial unloading compression4 25502 Triaxial unloading compression2 20060 Triaxial unloading compression

6 Advances in Materials Science and Engineering

0 20 40 60 80 1000

1

2

3

4

5

6

7

8E 5

0 (G

Pa)

∆σ3 ()

10 MPa8 MPa6 MPa

4 MPa2 MPa

(a)

E 50 (

GPa

)

∆σ3 ()0 20 40 60 80 100

0

1

2

3

4

5

6

7

8

10 MPa8 MPa6 MPa

4 MPa2 MPa

(b)

E 50 (

GPa

)

∆σ3 ()

10 MPa8 MPa6 MPa

4 MPa2 MPa

0 20 40 60 80 1003

4

5

6

7

8

(c)

E 50 (

GPa

)

∆σ3 ()

10 MPa8 MPa6 MPa

4 MPa2 MPa

0 20 40 60 80 1000

4

8

2

16

20

(d)

E 50 (

GPa

)

∆σ3 ()

10 MPa8 MPa6 MPa

4 MPa2 MPa

0 20 40 60 80 1000

2

4

6

8

10

(e)

E 50 (

GPa

)

∆σ3 ()

10 MPa8 MPa6 MPa

4 MPa2 MPa

0 20 40 60 80 1000

2

4

6

8

0

(f )

Figure 5 Continued

Advances in Materials Science and Engineering 7

E 50 (

GPa

)

∆σ3 ()

10 MPa8 MPa6 MPa

4 MPa2 MPa

0 20 40 60 80 1000

2

4

6

8

10

12

14

(g)

E 50 (

GPa

)

∆σ3 ()

10 MPa8 MPa6 MPa

4 MPa2 MPa

0 20 40 60 80 1000

2

4

6

(h)

E 50 (

GPa

)

∆σ3 ()

10 MPa8 MPa6 MPa

4 MPa2 MPa

0 20 40 60 80 1000

2

4

6

8

(i)

Figure 5 Relationship between deformation modulus and unloading percentage (joint orientation defined as β joint connectivity ratedefined as k) (a) β 30deg k 03 (b) β 30deg k 045 (c) β 30deg k 06 (d) β 45deg k 03 (e) β 45deg k 045 (f ) β 45deg k 06 (g) β 60degk 03 (h) β 60deg k 045 (i) β 60deg k 06

ndash

Failure modes

Shear failure

Splitting failure

Mixed failure (tension and shear)

Slide along the joint planeCompletely

Partly

Does not slide along the joint plane

ndash

Figure 6 Failure modes

8 Advances in Materials Science and Engineering

2 MPa 6 MPa4 MPa 8 MPa 10 MPa

(a)

2 MPa 6 MPa4 MPa 8 MPa 10 MPa

(b)

2 MPa 6 MPa4 MPa 8 MPa 10 MPa

(c)

2 MPa 6 MPa4 MPa 8 MPa 10 MPa

(d)

2 MPa 6 MPa4 MPa 8 MPa 10 MPa

(e)

Figure 7 Continued

Advances in Materials Science and Engineering 9

cannot be reconstructed to form intact specimens(Figure 7(d)) e amount of cracks initiated during the testwas also considered as a parameter to characterize the effectof confining pressures Note that the amount of crack isinversely proportional to the confining pressure Here theconfining pressure was designated by σ3 e fractureexhibited a continuous line-type crack at higher confiningpressures (ie 8 and 10megapascals) However the crackshowed a fork-like pattern when the confining pressure was

less than 4MPa At lower confining pressures samples weremore vulnerable for damages during the unloading test ephenomena were explained in terms of the restraint effect ofconfining pressure on the lateral deformation of the spec-imens According to MohrndashCoulomb failure criterion thedeviatoric stress (σ1 minus σ3) will increase when the confiningpressure (σ3) decreases and the failure envelop can be easilyreached Meanwhile internal defects were rapidly propa-gating as the deviatoric stress increased Lastly perforated

2 MPa 6 MPa4 MPa 8 MPa 10 MPa

(f )

2 MPa 6 MPa4 MPa 8 MPa 10 MPa

(g)

2 MPa 6 MPa4 MPa 8 MPa 10 MPa

(h)

2 MPa 6 MPa4 MPa 8 MPa 10 MPa

(i)

Figure 7 Sample failures under different pressures (joint orientation defined as β joint connectivity rate defined as k) (a) β 30deg k 03 (b)β 30deg k 045 (c) β 30deg k 06 (d) β 45deg k 03 (e) β 45deg k 045 (f ) β 45deg k 06 (g) β 60deg k 03 (h) β 60deg k 045 (i)β 60deg k 06

10 Advances in Materials Science and Engineering

cracks emerged On the contrary internal defects weresqueezed by the confining pressure Hence higher values ofconfining pressures can impose a significant restraint againstthe lateral deformations Similarly excavation can beregarded as unloading condition in rock slope engineering

62 Effect of Orientation on the Failure Modes In naturerocks have different orientations of joints which affect theirphysical and mechanical properties In this study the re-lationships between joint orientations and crack propaga-tions were studied Obviously the failure modes for differentjoint orientations were divergent As shown in Figure 6splitting failure was the predominant failure type forspecimens with 30deg joint orientations (Figures 7(a)ndash7(c))Hence joints with mild slopes have relatively small impacton the properties of rocks On the contrary mixed types offailures have been noticed on the specimens with 45deg jointsDuring testing the specimens were fragmented into piecesIn fact it was difficult to trace the failure paths However at60deg joint orientation shear failure was the predominant oneA flat surface was formed as a result of flaw cuttings Henceat 60deg joint orientation dislocations of pieces occurred dueto a noticeable sliding

63 Effect of Joint Connectivity Rate on the Failure ModesFor different joint angles the effect of connectivity rate (k) isdistinct It had a considerable effect when the joint angleswere 45deg or 60deg However at 30deg joint angle the effect of jointconnectivity was relatively small e path of crack prop-agation was twisted when the value of kwas 03 (Figures 7(a)7(d) and 7(g)) is is because of the restraint action ofuncracked section of the specimens against sliding erupture face was rough since it is a result of multiple factorsOn the one hand joint and rock bridge alter or replace eachother in accordance with the connectivity rate On the otherhand the nonlinear variation of tension stress at the tip offlaws exists As is shown in Figure 6 the crack path wasbecoming edge shaped when the value of k increasedMeanwhile the failure modes gradually changed by thejoints Higher values of k yielded the straight plane ofruptures Nonetheless the concentrations of stresses werenoticeable as shown in the development of secondary cracks(Figures 7(e) and 7(f )) e cracks were extended between

the tips of the joints and the edges of the specimens when thejoint angle and the connectivity rate were 45deg and 045respectively For small values of k the crack pattern becamepolygonal However the crack pattern was transforminginto an arc shape when the value of k increased Besides thesecondary cracks became normal to the edge of the speci-mens during shearing At 60deg joint orientation there was noremarkable effect on the failure modes due to the variationsof joint connectivity rate (k) All surfaces of failures werenearly flat Nonetheless there was a slight discrepancy be-tween them Similar failure modes were developed whenjoint orientation and connectivity rate were 45deg and 03respectively e significant effect of joint connectivity ratewas obvious Hence shear failures can easily develop andtheir probability of occurrence depends on the values of kShear failures were developed at higher values of k(Figures 7(h) and 7(i))

Table 3 summarizes different types of failure modes forspecimens with diverse joint orientations and connectivityrates e failure mode was governed by the joint param-eters Fundamentally the failure modes depend on the jointorientations initially For instance splitting failure is un-likely to occur when the joint angle is above 45deg Meanwhileconnectivity rate determines the extent of shear failuree kvalues control the transformation between mixed andcomplete shear failures At 60deg joint orientation both typesof failures were observed e critical value for k was 03 inthis case

7 Discussions of Crack Force Analysis Based onGriffithrsquos Theory

MohrndashCoulomb and Griffith criteria are conventionallyused to interpret the failure of rock specimens It is no-ticeable that the MohrndashCoulomb criterion cannot clarify theinfluence of inside flaws on rock sample failure Howevermaximum tensile stress in the perimeter of Griffithrsquos crack(Figure 8) can be analyzed according to Griffithrsquos theory forthe sake of exploring the effectiveness of the internal defects

Han et al [26] deduced equation (2) a function of α andc considering primary stress of equation (1) and Inglisrsquosformula

σy σ1 + σ3

2+σ1 minus σ3

2cos 2c

σx σ1 + σ3

2minusσ1 minus σ3

2cos 2c

τxy minusσ1 minus σ3

2sin 2c

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(1)

σb σ1

m2 + α2m 1 + α21113872 1113873(1 + λ) +(1 + m) mminus α21113872 1113873(1minus α)cos 2c + α(1 + m)

2(1minus λ)sin 2c1113966 1113967 (2)

Advances in Materials Science and Engineering 11

As shown in the specimen schematic diagram α and thecrack width can be valued as 0degand 1mm respectively sincethe crack is long and thin Hence the following equation isattained from equation (2)

σb σ1m

(1 + λ) +(1 + m)(1minus λ)cos 2c1113864 1113865 (3)

e stress extremum hinges on σ1 λ and m at the tips ofthe crack in the condition of c minus(π2) e equation isobtained as follows

σ0 σ1m

2λminusm(1minus λ) (4)

According to equation (4) the maximum tensile stress atthe tips of flaw was calculated Meanwhile the influences onthe maximum tensile stress were analyzed considering jointinclined angles joint connectivity and confining pressures(Figures 9 and 10) Obviously the maximum tensile stressincreases promptly in accordance with the confining pres-sures Simultaneously higher joint connectivity rates lead toa large stress extremum And the confining pressure enlargesthis trend As it is shown in Figure 10 the escalation of jointorientation boosts the maximum tensile stress at the tips offlaw erefore the alteration of confining pressure is ofmost significant on the variation of the tensile stress whilethe joint angle contributes the minimal impact

With confining pressure (ie λ) declining gradually inthe unloading stage the longitudinal crack may arise due toextremum of tensile stress on the tips and when it satisfiesthe fracture strength the rupture of the longitudinal crackappears along the plane of the crack and may propagatecontinuously to the specimenrsquos failure Considering the axialstress stops expansion in the unloading stage the specimencannot be broken by some slanting cracks or crossed cracks

directly during the compression erefore the splittingfailure coupled with shear failure emerges finally

8 Conclusions

In the preceding pages the crack propagation was studied onthe specimens with partially cut-through joints e primeobjective of this study was to examine the effect of jointorientations and joint connectivity rates on the mechanicalcharacteristics of specimens Even though nonlinear me-chanical characteristics were observed the macroscopicbehavior was studied based upon the different values of jointgeometries Furthermore nine types of specimens were usedto investigate the failure mechanisms at different confiningpressures e confining pressure joint orientation andconnectivity rate were considered as key parameters toanalyze different failure modes Finally Griffithrsquos theory wasused to analyze the maximum tensile stress Nevertheless afurther study is important to extend the application of thisstudy in real engineering problemsemain findings of thispaper are summarized below

A nonlinear correlation was observed among the elas-ticity modulus joint orientation and joint connectivity rateMeanwhile the confining pressure was arithmetically

Y

σy

σx

Xα

γ

σx

σy

σ3σ1

τxy τxyτyx

τyx

Figure 8 Stress state of the crack based on Griffithrsquos theory [26]

0

100

200

300

400

500

600

2 4 6 8 10

σ 0 (M

Pa)

σ3 (MPa)

β = 30deg k = 03β = 30deg k = 045β = 30deg k = 06

Figure 9 e stress extremum with 30deg joint angle under differentconfining pressures

0

200

400

600

800

1000

30 45 60

σ 0 (M

Pa)

β (deg)k = 03k = 045k = 06

Figure 10 e stress extremum under different joint orientationsand connectivity rates

Table 3 Failure modes for different types of samples

Connectivity rateOrientation 03 045 0630 Splitting Splitting Splitting45 Mixed Mixed Mixed60 Mixed Shear Shear

12 Advances in Materials Science and Engineering

increasing During the unloading stage the reduction ofdeformation modulus can be used to predict the rockspecimen failure And the critical value of decrease per-centage is 80 e deformation modulus is less sensitive tothe change of joint connectivity whereas it is more sensitiveto the change of the inclination angle

ree different types (shear split and mixed) of failuremodes were observed e higher values of confiningpressure can restraint the lateral deformation At 30deg jointorientation the predominant failure was splitting failureSimilarly mixed failures were observed at 45deg joint ori-entations because of the concentration of stresses at the tipof the joints On the contrary shear failure was quitecommon due to sufficient length of 60deg joint orientationComparatively the effect of joint connectivity rate wassmaller than that of the joint orientation However therewas a clear variation in the crack propagation when thejoint angle was 60deg e position of joints plays a significantrole for anisotropic failures

A few equations were introduced based on Griffithrsquostheory to evaluate the maximum tensile stress at the tips ofthe crack e results based on the calculation reveal theimpaction of joint characteristics on the stress extremum

Abbreviations

c e angle of principal stress σ1 and Y axisα e central angle of the ellipsem e ratio of the minor axis (b) to major axis (a) of the

ellipseλ e ratio of two principal stresses between σ3 and σ1σ0 e maximum tensile stressσ1 e maximum principal stressσ3 e minimum principal stressσb Tangential stress on the perimeter of a slender ellipse

Data Availability

e data used to support the findings of this study may bereleased upon application to the China ree GorgesUniversity Review Board which can be done by contactingthe author ldquoGuoyong Duan (dgyhhueducn)rdquo

Conflicts of Interest

e authors wish to confirm that there are no knownconflicts of interest associated with this publication andthere has been no significant financial support for this workthat could have influenced its outcome Accordingly thealteration of confining pressure is of most significance onthe variation of the tensile stress while the joint anglecontributes the minimal impact

Acknowledgments

e authors gratefully acknowledge the financial supportfrom the National Natural Science Foundation of China(Grant nos 51439003 and 51279091) the Fundamental

Research Funds for the Central Universities (Hohai Uni-versity) (Grant no 2016B43014) Open Fund of Key Lab-oratory of Geological Hazards on ree Gorges ReservoirArea (China ree Gorges University) (Grant no2015KDZ12) and Key Project of Scientific Research Plan ofHubei Provincial Department of Education (Grant noD20181202)

References

[1] T Liu P Cao and H Lin ldquoEvolution procedure of multiplerock cracks under seepage pressurerdquo Mathematical Problemsin Engineering vol 2013 Article ID 738013 11 pages 2013

[2] E Hoek ldquoStrength of jointed rock massesrdquo Geotechniquevol 33 no 3 pp 187ndash223 1983

[3] E Hoek and Z T Bieniawski ldquoBrittle fracture propagation inrock under compressionrdquo International Journal of Fracturevol 26 no 4 pp 276ndash294 1984

[4] M Basista and D Gross ldquoe sliding crack model of brittledeformation an internal variable approachrdquo InternationalJournal of Solids and Structures vol 35 no 5-6 pp 487ndash5091998

[5] I Vardoulakis J F Labuz E Papamichos and J TronvollldquoContinuum fracture mechanics of uniaxial compression onbrittle materialsrdquo International Journal of Solids and Struc-tures vol 35 no 31-32 pp 4313ndash4335 1998

[6] A Paluszny and S K Matthai ldquoNumerical modeling ofdiscrete multi-crack growth applied to pattern formation ingeological brittle mediardquo International Journal of Solids andStructures vol 46 no 18-19 pp 3383ndash3397 2009

[7] R-H Cao P Cao H Lin C-Z Pu and K Ou ldquoMechanicalbehavior of brittle rock-like specimens with pre-existingfissures under uniaxial loading experimental studies andparticle mechanics approachrdquo Rock Mechanics and RockEngineering vol 49 no 3 pp 763ndash783 2016

[8] X Fan R Chen H Lin H Lai C Zhang and Q ZhaoldquoCracking and failure in rock specimen containing combinedflaw and hole under uniaxial compressionrdquo Advances in CivilEngineering vol 2018 Article ID 9818250 15 pages 2018

[9] D Bing H Guicheng and Z Zhijun ldquoA numerical researchon crack process of gypsum containing single flaw withdifferent angle and length in uniaxial loadingrdquo Shock andVibration vol 2018 Article ID 2968205 16 pages 2018

[10] J Zhang ldquoInvestigation of relation between fracture scale andacoustic emission time-frequency parameters in rocksrdquo Shockand Vibration vol 2018 Article ID 3057628 14 pages 2018

[11] X Meng and W Liu ldquoStudy on failure process and perme-ation evolution of single-cracked rockrdquo Advances in MaterialsScience and Engineering vol 2018 Article ID 79156529 pages 2018

[12] M Sagong D Park J Yoo and J S Lee ldquoExperimental andnumerical analyses of an opening in a jointed rockmass underbiaxial compressionrdquo International Journal of RockMechanicsand Mining Sciences vol 48 no 7 pp 1055ndash1067 2011

[13] W Chen H Konietzky X Tan and T Fruhwirt ldquoPre-failuredamage analysis for brittle rocks under triaxial compressionrdquoComputers and Geotechnics vol 74 pp 45ndash55 2016

[14] S-Q Yang P G Ranjith Y-H Huang et al ldquoExperimentalinvestigation on mechanical damage characteristics ofsandstone under triaxial cyclic loadingrdquo Geophysical JournalInternational vol 201 no 2 pp 662ndash682 2015

[15] N Erarslan ldquoMicrostructural investigation of subcriticalcrack propagation and fracture process zone (FPZ) by the

Advances in Materials Science and Engineering 13

joints with thin steel sheets A release agent was presmearedinside the molds and the steel sheets to extrude the samplessmoothly after the initial set time Due to the presence andextraction of thin sheets the joints were developed duringsolidification en the cement mortar samples were kept ina curing room (at 18degC room temperature 99 relativehumidity and atmospheric pressure for 28 days) No fillerswere employed inside the joint openings (all of the openingwidths were less than 1mm) To get flat surfaces the tips ofthe specimens were trimmed off

21 Joint Geometries Created to Observe the Influence of JointConnectivity Rate e first set of triaxial compression testwas executed to investigate the effect of joint connectivityrate on the mechanical properties of the specimens withpartially cut-through joints Here the angle of inclinationwas kept constant It is defined as the ratio between the jointarea and the joint location plane across the specimen asshown in Figure 2 Meanwhile k stands for joint connec-tivity rate and three different values were considered (030045 and 060) However all the other joint geometrical

Lianziya hazardous rock mass

Yangtze river

Three gorges dam

Xiangxi town

Guojiaba town

Quyuan town

Taipingxi town

FIGURE 1 Lianziya hazardous rock mass in ree Gorges Reservoir area

Table 1 e parameter comparisons between sandstone and cement mortar

Sandstone Cement mortarCohesion (MPa) 162 107Internal friction angle (deg) 35 315Uniaxial compression strength (MPa) 3929 3265Elasticity modulus (GPa) 1025 924Poissonrsquos ratio 040 038Unit weight (kNm3) 220 206

(a)

The joint plane

(b)

Figure 2 Schematic diagram of specimens (a) Cylindrical plastic molds used for sample preparation (b) Joint location and jointconnectivity rate

Advances in Materials Science and Engineering 3

parameters were kept constant e joints were symmetri-cally located at the center of the specimen to minimize theconceivable effect of eccentricity

22 Joint Geometries Created to Observe the Influence of JointOrientation e other set of triaxial compression test wasperformed to investigate the effect of joint orientation on themechanical response of the specimens with partially cut-through joints In this case the joint connectivity rate waskept constante centroid of the joint was located at 50mmdistance from the tip of the specimen ree different valueswere adopted as joint orientations (ie 30 45 and 60deg) ejoint orientations were measured from the horizontal axisExcept the joint orientation the other geometrical param-eters were kept constant Based on the MohrndashCoulombfailure criterion the dip angle of the structural plane withrespect to the intact failure surface is slightly larger than 45degAs a result one can investigate the impact of joint orien-tation by altering the angles Figure 3 shows the geometricalspecifications used for preparing nine different specimens

3 Description of Laboratory Test

In this testing program a computerized triaxial compressivetesting apparatus was used as shown in Figure 4

is device (RMT-150C) was designed and manufac-tured by the Institute of Rock and Soil Mechanics (affiliatedto the Chinese Academy of Sciences) It can simulate theloading and the unloading conditions and perform differenttypes of tests (uniaxial test direct shear test Brazil disk splittest etc)

e PCI-2 AE screening systemwas used to eliminate thespecimens with defects

4 Experimental Methodology

To investigate the mechanical properties of rock mass withdisconnected joints especially under loading and unloadingconditions five different values of confining pressures anddifferent kinds of stress paths were adopted as listed inTable 1

e adopted testing procedures for the cylindrical rockspecimens are as follows

(1) Initially the ldquoforce-confining pressurerdquo mode wasselected and a predefined axial force and confiningpressures were applied on the specimens simulta-neously at the loading rate of 02 kNs and 01MPas respectively e axial load was increasing at therate of 05 kNs until failure occurred at a constantconfining pressure en the maximum triaxialcompression strength was determined

(2) e same type of specimen was chosen(3) e ldquoforce-confining pressurerdquo mode was selected

and a predefined axial force and confining pressureswere applied on the specimens simultaneously at theloading rate of 02 kNs and 01MPas respectively

(4) After setting the confining pressure constant theaxial load was applied at the rate of 05 kNs up to the70 of the maximum triaxial compressive strength

(5) en the confining pressure was unloaded to failureat a rate of 001MPas e predefined loadings forthe specimen with 30deg joint orientation and 03connectivity rates are shown in Table 2

5 Testing Results and Discussion

As it was mentioned in Section 4 the testing programme wasdesigned to analyze the effect of joint orientation andconnectivity rate on the mechanical properties of rockspecimens with partially cut-through joints eir effects onthe compressive strength and deformation characteristicsare discussed below

Brittle failure occurred frequently in the triaxialunloading tests e process of unloading was conducted bydecreasing the confining pressure Hence the percentage ofreduction in the confining pressure was a key factor whichaffected the sample failure During the unloading tests allthe parameters were recognized as equivalent variableswhich were different from the typical parameters

rough the analysis of the deformation modulus of thespecimens with different dip angles and connectivity ratiosthe variation of different types of deformation modulus withunloading is shown in Figure 5 e deformation modulus isrepresented as E50 and Δσ3 is the reduction percentage ofthe confining pressure e results show that the de-formation modulus of specimens with different connectivityratios is similar under the same confining pressure level As aresult the deformation modulus is less sensitive to thechange of joint connectivity As the inclination angle in-creases the average value of the deformation modulus en-largeserefore the deformation modulus is more sensitiveto the change of the inclination angle When the confiningpressure level varies the deformation modulus of the sametype of specimens fluctuates slightly It can be concluded thatthe confining pressures ranging from 0MPa to 10MPa havea little effect on the deformation modulus According to thevariation of the deformation modulus a large decrease in theamplitude arouses at the unloading percentages of 0sim20and 80sim100 Especially after the 80 reduction ofconfining pressure the deformation modulus declinedrapidly However when the unloading loss ranges from20ndash80 the decreasing amplitude is stable relativelyAccelerated variation crops up at the early stage and the latestage of unloading During the triaxial unloading processthe internal strain of the specimen is accumulated by theconfining pressure constraint When the confining pressureis declining accumulated energy release initiates graduallye internal stress and strain are adjusted while maintainingthe residual bearing capacity Accordingly the middle stageof deformation modulus alteration is relatively flat At theend of unloading process the residual confining pressure isinsufficient to restrain the energy inside the specimen andthereupon failure occurs Hence the development ofunloading procedure can be divided into two stages in

4 Advances in Materials Science and Engineering

consonance with the decrease rate of deformation modulusere are 0sim80 and 80sim100 respectively

6 Failure Modes

e failure modes of jointed samples were identified by thelocation and the size of cracks e influence of joint pa-rameters on the crack propagation was discussed As shownin Figure 6 the failure modes were categorized into threegroups

(1) e shear failure occurred frequently under com-pression condition It can be recognized by the shearplane Ignoring stress concentrations on theboundaries of samples the shear plane was relativelyflat is may be induced due to the conversion fromdefects to weak structural planes Based on the as-sumption of sliding mostly occurs along the weak

planes in rockmass the joints are considered as weakstructural planes However rupture did not tran-spired along the crack orientation completely duringthe test In other words the prefabricated cracks didnot thoroughly affect the crack propagation inseveral cases (Figure 7(a)) e condition of a sampleseparated into two parts along the joints accountedfor two third percentage Joint geometries and thedegree of coincidence between the joint and therupture trajectory can be used to characterize thefailure modes (Figure 3)

(2) Generally splitting failure arises at lower confiningpressures e lateral deformation increases muchmore rapidly in the circumstances And the splittingcracks were nearly parallel to themaximum principalstress of the sample Meanwhile the secondarycracks propagated from the crack tips during the

30deg

k = 03

Φ 50

100

(a)

30deg

k = 045

Φ 50

100

(b)30deg

k = 06

Φ 50

100

(c)

45deg

k = 03

Φ 50

100

(d)

45degk = 04

5

Φ 50

100

(e)

45degk = 06

Φ 50

100

(f )

60deg

k = 0

3

Φ 50

100

(g)

60degk = 0

45

Φ 50

100

(h)

60degk = 0

6Φ 50

100

(i)

Figure 3 Geometrical specifications

Advances in Materials Science and Engineering 5

unloading tests It is noteworthy that splitting failureoccurs in the intact rock samples in the procedure ofuniaxial compression Nevertheless similar phe-nomena were observed on the specimens withsmaller joint angles and connectivity rates In otherwords the likelihood of splitting failure is high whenthe influence of joint is insignificant Meanwhile theprobability of splitting failure is also dependent onthe angle of the joint and the vertical axis of thesample Based upon the results samples with 30degjoint orientation were more inclined to splittingfailure

(3) A mixed type of failure (tension and shear) wasdeveloped at higher joint orientation and mediumjoint connectivity rate It is characterized by slidingof joints and development of tension cracks from thecrack tip Based on the rupture morphology theprecondition of mixed failure hinges on middlevalues of joint orientation and connectivity rate asshown in Figures 7(d)ndash7(f) Considering the ten-dency of shear along the joint plane tension cracksdeveloped by the reason of stress concentrations at

the tips of the cracks It is an important characteristicthat reveals the dislocation of sample pieces beforethe ultimate failure e two parallel tension frac-tures were normal to the original crack In mostcases the secondary cracks were connected to thetips of joints where stress concentrations are in-evitable erefore the propagation of cracks can bedetermined by analyzing the geometries of joints andloading conditions

61 Effect of Confining Pressure on the Failure of the SamplesIn the triaxial test the mechanical properties of rockspecimens can be affected by the confining pressure defi-nitely e effect of unloading was remarkable when theconfining pressures were 2 and 4megapascals (based uponthe analysis of 30 diagrams) As a result the specimensexhibited different levels of damages at different confiningpressures During the unloading test parts of the specimenswere shattering when the confining pressure was lower forexample 2 and 4megapascals In this case the fragments

Figure 4 Triaxial compressive testing apparatus

Table 2 Stress state for the triaxial compression and unloading compression

Stress state Minimum principal stress σ3 (MPa) Maximum principal stress σ1 (MPa) Remarks

σ1

σ1σ1 gt σ2 = σ3

σ3σ3

10 61526 Triaxial compression8 55468 Triaxial compression6 47028 Triaxial compression4 37239 Triaxial compression2 29256 Triaxial compression10 43106 Triaxial unloading compression8 38677 Triaxial unloading compression6 32110 Triaxial unloading compression4 25502 Triaxial unloading compression2 20060 Triaxial unloading compression

6 Advances in Materials Science and Engineering

0 20 40 60 80 1000

1

2

3

4

5

6

7

8E 5

0 (G

Pa)

∆σ3 ()

10 MPa8 MPa6 MPa

4 MPa2 MPa

(a)

E 50 (

GPa

)

∆σ3 ()0 20 40 60 80 100

0

1

2

3

4

5

6

7

8

10 MPa8 MPa6 MPa

4 MPa2 MPa

(b)

E 50 (

GPa

)

∆σ3 ()

10 MPa8 MPa6 MPa

4 MPa2 MPa

0 20 40 60 80 1003

4

5

6

7

8

(c)

E 50 (

GPa

)

∆σ3 ()

10 MPa8 MPa6 MPa

4 MPa2 MPa

0 20 40 60 80 1000

4

8

2

16

20

(d)

E 50 (

GPa

)

∆σ3 ()

10 MPa8 MPa6 MPa

4 MPa2 MPa

0 20 40 60 80 1000

2

4

6

8

10

(e)

E 50 (

GPa

)

∆σ3 ()

10 MPa8 MPa6 MPa

4 MPa2 MPa

0 20 40 60 80 1000

2

4

6

8

0

(f )

Figure 5 Continued

Advances in Materials Science and Engineering 7

E 50 (

GPa

)

∆σ3 ()

10 MPa8 MPa6 MPa

4 MPa2 MPa

0 20 40 60 80 1000

2

4

6

8

10

12

14

(g)

E 50 (

GPa

)

∆σ3 ()

10 MPa8 MPa6 MPa

4 MPa2 MPa

0 20 40 60 80 1000

2

4

6

(h)

E 50 (

GPa

)

∆σ3 ()

10 MPa8 MPa6 MPa

4 MPa2 MPa

0 20 40 60 80 1000

2

4

6

8

(i)

Figure 5 Relationship between deformation modulus and unloading percentage (joint orientation defined as β joint connectivity ratedefined as k) (a) β 30deg k 03 (b) β 30deg k 045 (c) β 30deg k 06 (d) β 45deg k 03 (e) β 45deg k 045 (f ) β 45deg k 06 (g) β 60degk 03 (h) β 60deg k 045 (i) β 60deg k 06

ndash

Failure modes

Shear failure

Splitting failure

Mixed failure (tension and shear)

Slide along the joint planeCompletely

Partly

Does not slide along the joint plane

ndash

Figure 6 Failure modes

8 Advances in Materials Science and Engineering

2 MPa 6 MPa4 MPa 8 MPa 10 MPa

(a)

2 MPa 6 MPa4 MPa 8 MPa 10 MPa

(b)

2 MPa 6 MPa4 MPa 8 MPa 10 MPa

(c)

2 MPa 6 MPa4 MPa 8 MPa 10 MPa

(d)

2 MPa 6 MPa4 MPa 8 MPa 10 MPa

(e)

Figure 7 Continued

Advances in Materials Science and Engineering 9

cannot be reconstructed to form intact specimens(Figure 7(d)) e amount of cracks initiated during the testwas also considered as a parameter to characterize the effectof confining pressures Note that the amount of crack isinversely proportional to the confining pressure Here theconfining pressure was designated by σ3 e fractureexhibited a continuous line-type crack at higher confiningpressures (ie 8 and 10megapascals) However the crackshowed a fork-like pattern when the confining pressure was

less than 4MPa At lower confining pressures samples weremore vulnerable for damages during the unloading test ephenomena were explained in terms of the restraint effect ofconfining pressure on the lateral deformation of the spec-imens According to MohrndashCoulomb failure criterion thedeviatoric stress (σ1 minus σ3) will increase when the confiningpressure (σ3) decreases and the failure envelop can be easilyreached Meanwhile internal defects were rapidly propa-gating as the deviatoric stress increased Lastly perforated

2 MPa 6 MPa4 MPa 8 MPa 10 MPa

(f )

2 MPa 6 MPa4 MPa 8 MPa 10 MPa

(g)

2 MPa 6 MPa4 MPa 8 MPa 10 MPa

(h)

2 MPa 6 MPa4 MPa 8 MPa 10 MPa

(i)

Figure 7 Sample failures under different pressures (joint orientation defined as β joint connectivity rate defined as k) (a) β 30deg k 03 (b)β 30deg k 045 (c) β 30deg k 06 (d) β 45deg k 03 (e) β 45deg k 045 (f ) β 45deg k 06 (g) β 60deg k 03 (h) β 60deg k 045 (i)β 60deg k 06

10 Advances in Materials Science and Engineering

cracks emerged On the contrary internal defects weresqueezed by the confining pressure Hence higher values ofconfining pressures can impose a significant restraint againstthe lateral deformations Similarly excavation can beregarded as unloading condition in rock slope engineering

62 Effect of Orientation on the Failure Modes In naturerocks have different orientations of joints which affect theirphysical and mechanical properties In this study the re-lationships between joint orientations and crack propaga-tions were studied Obviously the failure modes for differentjoint orientations were divergent As shown in Figure 6splitting failure was the predominant failure type forspecimens with 30deg joint orientations (Figures 7(a)ndash7(c))Hence joints with mild slopes have relatively small impacton the properties of rocks On the contrary mixed types offailures have been noticed on the specimens with 45deg jointsDuring testing the specimens were fragmented into piecesIn fact it was difficult to trace the failure paths However at60deg joint orientation shear failure was the predominant oneA flat surface was formed as a result of flaw cuttings Henceat 60deg joint orientation dislocations of pieces occurred dueto a noticeable sliding

63 Effect of Joint Connectivity Rate on the Failure ModesFor different joint angles the effect of connectivity rate (k) isdistinct It had a considerable effect when the joint angleswere 45deg or 60deg However at 30deg joint angle the effect of jointconnectivity was relatively small e path of crack prop-agation was twisted when the value of kwas 03 (Figures 7(a)7(d) and 7(g)) is is because of the restraint action ofuncracked section of the specimens against sliding erupture face was rough since it is a result of multiple factorsOn the one hand joint and rock bridge alter or replace eachother in accordance with the connectivity rate On the otherhand the nonlinear variation of tension stress at the tip offlaws exists As is shown in Figure 6 the crack path wasbecoming edge shaped when the value of k increasedMeanwhile the failure modes gradually changed by thejoints Higher values of k yielded the straight plane ofruptures Nonetheless the concentrations of stresses werenoticeable as shown in the development of secondary cracks(Figures 7(e) and 7(f )) e cracks were extended between

the tips of the joints and the edges of the specimens when thejoint angle and the connectivity rate were 45deg and 045respectively For small values of k the crack pattern becamepolygonal However the crack pattern was transforminginto an arc shape when the value of k increased Besides thesecondary cracks became normal to the edge of the speci-mens during shearing At 60deg joint orientation there was noremarkable effect on the failure modes due to the variationsof joint connectivity rate (k) All surfaces of failures werenearly flat Nonetheless there was a slight discrepancy be-tween them Similar failure modes were developed whenjoint orientation and connectivity rate were 45deg and 03respectively e significant effect of joint connectivity ratewas obvious Hence shear failures can easily develop andtheir probability of occurrence depends on the values of kShear failures were developed at higher values of k(Figures 7(h) and 7(i))

Table 3 summarizes different types of failure modes forspecimens with diverse joint orientations and connectivityrates e failure mode was governed by the joint param-eters Fundamentally the failure modes depend on the jointorientations initially For instance splitting failure is un-likely to occur when the joint angle is above 45deg Meanwhileconnectivity rate determines the extent of shear failuree kvalues control the transformation between mixed andcomplete shear failures At 60deg joint orientation both typesof failures were observed e critical value for k was 03 inthis case

7 Discussions of Crack Force Analysis Based onGriffithrsquos Theory

MohrndashCoulomb and Griffith criteria are conventionallyused to interpret the failure of rock specimens It is no-ticeable that the MohrndashCoulomb criterion cannot clarify theinfluence of inside flaws on rock sample failure Howevermaximum tensile stress in the perimeter of Griffithrsquos crack(Figure 8) can be analyzed according to Griffithrsquos theory forthe sake of exploring the effectiveness of the internal defects

Han et al [26] deduced equation (2) a function of α andc considering primary stress of equation (1) and Inglisrsquosformula

σy σ1 + σ3

2+σ1 minus σ3

2cos 2c

σx σ1 + σ3

2minusσ1 minus σ3

2cos 2c

τxy minusσ1 minus σ3

2sin 2c

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(1)

σb σ1

m2 + α2m 1 + α21113872 1113873(1 + λ) +(1 + m) mminus α21113872 1113873(1minus α)cos 2c + α(1 + m)

2(1minus λ)sin 2c1113966 1113967 (2)

Advances in Materials Science and Engineering 11

As shown in the specimen schematic diagram α and thecrack width can be valued as 0degand 1mm respectively sincethe crack is long and thin Hence the following equation isattained from equation (2)

σb σ1m

(1 + λ) +(1 + m)(1minus λ)cos 2c1113864 1113865 (3)

e stress extremum hinges on σ1 λ and m at the tips ofthe crack in the condition of c minus(π2) e equation isobtained as follows

σ0 σ1m

2λminusm(1minus λ) (4)

According to equation (4) the maximum tensile stress atthe tips of flaw was calculated Meanwhile the influences onthe maximum tensile stress were analyzed considering jointinclined angles joint connectivity and confining pressures(Figures 9 and 10) Obviously the maximum tensile stressincreases promptly in accordance with the confining pres-sures Simultaneously higher joint connectivity rates lead toa large stress extremum And the confining pressure enlargesthis trend As it is shown in Figure 10 the escalation of jointorientation boosts the maximum tensile stress at the tips offlaw erefore the alteration of confining pressure is ofmost significant on the variation of the tensile stress whilethe joint angle contributes the minimal impact

With confining pressure (ie λ) declining gradually inthe unloading stage the longitudinal crack may arise due toextremum of tensile stress on the tips and when it satisfiesthe fracture strength the rupture of the longitudinal crackappears along the plane of the crack and may propagatecontinuously to the specimenrsquos failure Considering the axialstress stops expansion in the unloading stage the specimencannot be broken by some slanting cracks or crossed cracks

directly during the compression erefore the splittingfailure coupled with shear failure emerges finally

8 Conclusions

In the preceding pages the crack propagation was studied onthe specimens with partially cut-through joints e primeobjective of this study was to examine the effect of jointorientations and joint connectivity rates on the mechanicalcharacteristics of specimens Even though nonlinear me-chanical characteristics were observed the macroscopicbehavior was studied based upon the different values of jointgeometries Furthermore nine types of specimens were usedto investigate the failure mechanisms at different confiningpressures e confining pressure joint orientation andconnectivity rate were considered as key parameters toanalyze different failure modes Finally Griffithrsquos theory wasused to analyze the maximum tensile stress Nevertheless afurther study is important to extend the application of thisstudy in real engineering problemsemain findings of thispaper are summarized below

A nonlinear correlation was observed among the elas-ticity modulus joint orientation and joint connectivity rateMeanwhile the confining pressure was arithmetically

Y

σy

σx

Xα

γ

σx

σy

σ3σ1

τxy τxyτyx

τyx

Figure 8 Stress state of the crack based on Griffithrsquos theory [26]

0

100

200

300

400

500

600

2 4 6 8 10

σ 0 (M

Pa)

σ3 (MPa)

β = 30deg k = 03β = 30deg k = 045β = 30deg k = 06

Figure 9 e stress extremum with 30deg joint angle under differentconfining pressures

0

200

400

600

800

1000

30 45 60

σ 0 (M

Pa)

β (deg)k = 03k = 045k = 06

Figure 10 e stress extremum under different joint orientationsand connectivity rates

Table 3 Failure modes for different types of samples

Connectivity rateOrientation 03 045 0630 Splitting Splitting Splitting45 Mixed Mixed Mixed60 Mixed Shear Shear

12 Advances in Materials Science and Engineering

increasing During the unloading stage the reduction ofdeformation modulus can be used to predict the rockspecimen failure And the critical value of decrease per-centage is 80 e deformation modulus is less sensitive tothe change of joint connectivity whereas it is more sensitiveto the change of the inclination angle

ree different types (shear split and mixed) of failuremodes were observed e higher values of confiningpressure can restraint the lateral deformation At 30deg jointorientation the predominant failure was splitting failureSimilarly mixed failures were observed at 45deg joint ori-entations because of the concentration of stresses at the tipof the joints On the contrary shear failure was quitecommon due to sufficient length of 60deg joint orientationComparatively the effect of joint connectivity rate wassmaller than that of the joint orientation However therewas a clear variation in the crack propagation when thejoint angle was 60deg e position of joints plays a significantrole for anisotropic failures

A few equations were introduced based on Griffithrsquostheory to evaluate the maximum tensile stress at the tips ofthe crack e results based on the calculation reveal theimpaction of joint characteristics on the stress extremum

Abbreviations

c e angle of principal stress σ1 and Y axisα e central angle of the ellipsem e ratio of the minor axis (b) to major axis (a) of the

ellipseλ e ratio of two principal stresses between σ3 and σ1σ0 e maximum tensile stressσ1 e maximum principal stressσ3 e minimum principal stressσb Tangential stress on the perimeter of a slender ellipse

Data Availability

e data used to support the findings of this study may bereleased upon application to the China ree GorgesUniversity Review Board which can be done by contactingthe author ldquoGuoyong Duan (dgyhhueducn)rdquo

Conflicts of Interest

e authors wish to confirm that there are no knownconflicts of interest associated with this publication andthere has been no significant financial support for this workthat could have influenced its outcome Accordingly thealteration of confining pressure is of most significance onthe variation of the tensile stress while the joint anglecontributes the minimal impact

Acknowledgments

e authors gratefully acknowledge the financial supportfrom the National Natural Science Foundation of China(Grant nos 51439003 and 51279091) the Fundamental

Research Funds for the Central Universities (Hohai Uni-versity) (Grant no 2016B43014) Open Fund of Key Lab-oratory of Geological Hazards on ree Gorges ReservoirArea (China ree Gorges University) (Grant no2015KDZ12) and Key Project of Scientific Research Plan ofHubei Provincial Department of Education (Grant noD20181202)

References

[1] T Liu P Cao and H Lin ldquoEvolution procedure of multiplerock cracks under seepage pressurerdquo Mathematical Problemsin Engineering vol 2013 Article ID 738013 11 pages 2013

[2] E Hoek ldquoStrength of jointed rock massesrdquo Geotechniquevol 33 no 3 pp 187ndash223 1983

[3] E Hoek and Z T Bieniawski ldquoBrittle fracture propagation inrock under compressionrdquo International Journal of Fracturevol 26 no 4 pp 276ndash294 1984

[4] M Basista and D Gross ldquoe sliding crack model of brittledeformation an internal variable approachrdquo InternationalJournal of Solids and Structures vol 35 no 5-6 pp 487ndash5091998

[5] I Vardoulakis J F Labuz E Papamichos and J TronvollldquoContinuum fracture mechanics of uniaxial compression onbrittle materialsrdquo International Journal of Solids and Struc-tures vol 35 no 31-32 pp 4313ndash4335 1998

[6] A Paluszny and S K Matthai ldquoNumerical modeling ofdiscrete multi-crack growth applied to pattern formation ingeological brittle mediardquo International Journal of Solids andStructures vol 46 no 18-19 pp 3383ndash3397 2009

[7] R-H Cao P Cao H Lin C-Z Pu and K Ou ldquoMechanicalbehavior of brittle rock-like specimens with pre-existingfissures under uniaxial loading experimental studies andparticle mechanics approachrdquo Rock Mechanics and RockEngineering vol 49 no 3 pp 763ndash783 2016

[8] X Fan R Chen H Lin H Lai C Zhang and Q ZhaoldquoCracking and failure in rock specimen containing combinedflaw and hole under uniaxial compressionrdquo Advances in CivilEngineering vol 2018 Article ID 9818250 15 pages 2018

[9] D Bing H Guicheng and Z Zhijun ldquoA numerical researchon crack process of gypsum containing single flaw withdifferent angle and length in uniaxial loadingrdquo Shock andVibration vol 2018 Article ID 2968205 16 pages 2018

[10] J Zhang ldquoInvestigation of relation between fracture scale andacoustic emission time-frequency parameters in rocksrdquo Shockand Vibration vol 2018 Article ID 3057628 14 pages 2018

[11] X Meng and W Liu ldquoStudy on failure process and perme-ation evolution of single-cracked rockrdquo Advances in MaterialsScience and Engineering vol 2018 Article ID 79156529 pages 2018

[12] M Sagong D Park J Yoo and J S Lee ldquoExperimental andnumerical analyses of an opening in a jointed rockmass underbiaxial compressionrdquo International Journal of RockMechanicsand Mining Sciences vol 48 no 7 pp 1055ndash1067 2011

[13] W Chen H Konietzky X Tan and T Fruhwirt ldquoPre-failuredamage analysis for brittle rocks under triaxial compressionrdquoComputers and Geotechnics vol 74 pp 45ndash55 2016

[14] S-Q Yang P G Ranjith Y-H Huang et al ldquoExperimentalinvestigation on mechanical damage characteristics ofsandstone under triaxial cyclic loadingrdquo Geophysical JournalInternational vol 201 no 2 pp 662ndash682 2015

[15] N Erarslan ldquoMicrostructural investigation of subcriticalcrack propagation and fracture process zone (FPZ) by the

Advances in Materials Science and Engineering 13

parameters were kept constant e joints were symmetri-cally located at the center of the specimen to minimize theconceivable effect of eccentricity

22 Joint Geometries Created to Observe the Influence of JointOrientation e other set of triaxial compression test wasperformed to investigate the effect of joint orientation on themechanical response of the specimens with partially cut-through joints In this case the joint connectivity rate waskept constante centroid of the joint was located at 50mmdistance from the tip of the specimen ree different valueswere adopted as joint orientations (ie 30 45 and 60deg) ejoint orientations were measured from the horizontal axisExcept the joint orientation the other geometrical param-eters were kept constant Based on the MohrndashCoulombfailure criterion the dip angle of the structural plane withrespect to the intact failure surface is slightly larger than 45degAs a result one can investigate the impact of joint orien-tation by altering the angles Figure 3 shows the geometricalspecifications used for preparing nine different specimens

3 Description of Laboratory Test

In this testing program a computerized triaxial compressivetesting apparatus was used as shown in Figure 4

is device (RMT-150C) was designed and manufac-tured by the Institute of Rock and Soil Mechanics (affiliatedto the Chinese Academy of Sciences) It can simulate theloading and the unloading conditions and perform differenttypes of tests (uniaxial test direct shear test Brazil disk splittest etc)

e PCI-2 AE screening systemwas used to eliminate thespecimens with defects

4 Experimental Methodology

To investigate the mechanical properties of rock mass withdisconnected joints especially under loading and unloadingconditions five different values of confining pressures anddifferent kinds of stress paths were adopted as listed inTable 1

e adopted testing procedures for the cylindrical rockspecimens are as follows

(1) Initially the ldquoforce-confining pressurerdquo mode wasselected and a predefined axial force and confiningpressures were applied on the specimens simulta-neously at the loading rate of 02 kNs and 01MPas respectively e axial load was increasing at therate of 05 kNs until failure occurred at a constantconfining pressure en the maximum triaxialcompression strength was determined

(2) e same type of specimen was chosen(3) e ldquoforce-confining pressurerdquo mode was selected

and a predefined axial force and confining pressureswere applied on the specimens simultaneously at theloading rate of 02 kNs and 01MPas respectively

(4) After setting the confining pressure constant theaxial load was applied at the rate of 05 kNs up to the70 of the maximum triaxial compressive strength

(5) en the confining pressure was unloaded to failureat a rate of 001MPas e predefined loadings forthe specimen with 30deg joint orientation and 03connectivity rates are shown in Table 2

5 Testing Results and Discussion

As it was mentioned in Section 4 the testing programme wasdesigned to analyze the effect of joint orientation andconnectivity rate on the mechanical properties of rockspecimens with partially cut-through joints eir effects onthe compressive strength and deformation characteristicsare discussed below

Brittle failure occurred frequently in the triaxialunloading tests e process of unloading was conducted bydecreasing the confining pressure Hence the percentage ofreduction in the confining pressure was a key factor whichaffected the sample failure During the unloading tests allthe parameters were recognized as equivalent variableswhich were different from the typical parameters

rough the analysis of the deformation modulus of thespecimens with different dip angles and connectivity ratiosthe variation of different types of deformation modulus withunloading is shown in Figure 5 e deformation modulus isrepresented as E50 and Δσ3 is the reduction percentage ofthe confining pressure e results show that the de-formation modulus of specimens with different connectivityratios is similar under the same confining pressure level As aresult the deformation modulus is less sensitive to thechange of joint connectivity As the inclination angle in-creases the average value of the deformation modulus en-largeserefore the deformation modulus is more sensitiveto the change of the inclination angle When the confiningpressure level varies the deformation modulus of the sametype of specimens fluctuates slightly It can be concluded thatthe confining pressures ranging from 0MPa to 10MPa havea little effect on the deformation modulus According to thevariation of the deformation modulus a large decrease in theamplitude arouses at the unloading percentages of 0sim20and 80sim100 Especially after the 80 reduction ofconfining pressure the deformation modulus declinedrapidly However when the unloading loss ranges from20ndash80 the decreasing amplitude is stable relativelyAccelerated variation crops up at the early stage and the latestage of unloading During the triaxial unloading processthe internal strain of the specimen is accumulated by theconfining pressure constraint When the confining pressureis declining accumulated energy release initiates graduallye internal stress and strain are adjusted while maintainingthe residual bearing capacity Accordingly the middle stageof deformation modulus alteration is relatively flat At theend of unloading process the residual confining pressure isinsufficient to restrain the energy inside the specimen andthereupon failure occurs Hence the development ofunloading procedure can be divided into two stages in

4 Advances in Materials Science and Engineering

consonance with the decrease rate of deformation modulusere are 0sim80 and 80sim100 respectively

6 Failure Modes

e failure modes of jointed samples were identified by thelocation and the size of cracks e influence of joint pa-rameters on the crack propagation was discussed As shownin Figure 6 the failure modes were categorized into threegroups

(1) e shear failure occurred frequently under com-pression condition It can be recognized by the shearplane Ignoring stress concentrations on theboundaries of samples the shear plane was relativelyflat is may be induced due to the conversion fromdefects to weak structural planes Based on the as-sumption of sliding mostly occurs along the weak

planes in rockmass the joints are considered as weakstructural planes However rupture did not tran-spired along the crack orientation completely duringthe test In other words the prefabricated cracks didnot thoroughly affect the crack propagation inseveral cases (Figure 7(a)) e condition of a sampleseparated into two parts along the joints accountedfor two third percentage Joint geometries and thedegree of coincidence between the joint and therupture trajectory can be used to characterize thefailure modes (Figure 3)

(2) Generally splitting failure arises at lower confiningpressures e lateral deformation increases muchmore rapidly in the circumstances And the splittingcracks were nearly parallel to themaximum principalstress of the sample Meanwhile the secondarycracks propagated from the crack tips during the

30deg

k = 03

Φ 50

100

(a)

30deg

k = 045

Φ 50

100

(b)30deg

k = 06

Φ 50

100

(c)

45deg

k = 03

Φ 50

100

(d)

45degk = 04

5

Φ 50

100

(e)

45degk = 06

Φ 50

100

(f )

60deg

k = 0

3

Φ 50

100

(g)

60degk = 0

45

Φ 50

100

(h)

60degk = 0

6Φ 50

100

(i)

Figure 3 Geometrical specifications

Advances in Materials Science and Engineering 5

unloading tests It is noteworthy that splitting failureoccurs in the intact rock samples in the procedure ofuniaxial compression Nevertheless similar phe-nomena were observed on the specimens withsmaller joint angles and connectivity rates In otherwords the likelihood of splitting failure is high whenthe influence of joint is insignificant Meanwhile theprobability of splitting failure is also dependent onthe angle of the joint and the vertical axis of thesample Based upon the results samples with 30degjoint orientation were more inclined to splittingfailure

(3) A mixed type of failure (tension and shear) wasdeveloped at higher joint orientation and mediumjoint connectivity rate It is characterized by slidingof joints and development of tension cracks from thecrack tip Based on the rupture morphology theprecondition of mixed failure hinges on middlevalues of joint orientation and connectivity rate asshown in Figures 7(d)ndash7(f) Considering the ten-dency of shear along the joint plane tension cracksdeveloped by the reason of stress concentrations at

the tips of the cracks It is an important characteristicthat reveals the dislocation of sample pieces beforethe ultimate failure e two parallel tension frac-tures were normal to the original crack In mostcases the secondary cracks were connected to thetips of joints where stress concentrations are in-evitable erefore the propagation of cracks can bedetermined by analyzing the geometries of joints andloading conditions

61 Effect of Confining Pressure on the Failure of the SamplesIn the triaxial test the mechanical properties of rockspecimens can be affected by the confining pressure defi-nitely e effect of unloading was remarkable when theconfining pressures were 2 and 4megapascals (based uponthe analysis of 30 diagrams) As a result the specimensexhibited different levels of damages at different confiningpressures During the unloading test parts of the specimenswere shattering when the confining pressure was lower forexample 2 and 4megapascals In this case the fragments

Figure 4 Triaxial compressive testing apparatus

Table 2 Stress state for the triaxial compression and unloading compression

Stress state Minimum principal stress σ3 (MPa) Maximum principal stress σ1 (MPa) Remarks

σ1

σ1σ1 gt σ2 = σ3

σ3σ3

10 61526 Triaxial compression8 55468 Triaxial compression6 47028 Triaxial compression4 37239 Triaxial compression2 29256 Triaxial compression10 43106 Triaxial unloading compression8 38677 Triaxial unloading compression6 32110 Triaxial unloading compression4 25502 Triaxial unloading compression2 20060 Triaxial unloading compression

6 Advances in Materials Science and Engineering

0 20 40 60 80 1000

1

2

3

4

5

6

7

8E 5

0 (G

Pa)

∆σ3 ()

10 MPa8 MPa6 MPa

4 MPa2 MPa

(a)

E 50 (

GPa

)

∆σ3 ()0 20 40 60 80 100

0

1

2

3

4

5

6

7

8

10 MPa8 MPa6 MPa

4 MPa2 MPa

(b)

E 50 (

GPa

)

∆σ3 ()

10 MPa8 MPa6 MPa

4 MPa2 MPa

0 20 40 60 80 1003

4

5

6

7

8

(c)

E 50 (

GPa

)

∆σ3 ()

10 MPa8 MPa6 MPa

4 MPa2 MPa

0 20 40 60 80 1000

4

8

2

16

20

(d)

E 50 (

GPa

)

∆σ3 ()

10 MPa8 MPa6 MPa

4 MPa2 MPa

0 20 40 60 80 1000

2

4

6

8

10

(e)

E 50 (

GPa

)

∆σ3 ()

10 MPa8 MPa6 MPa

4 MPa2 MPa

0 20 40 60 80 1000

2

4

6

8

0

(f )

Figure 5 Continued

Advances in Materials Science and Engineering 7

E 50 (

GPa

)

∆σ3 ()

10 MPa8 MPa6 MPa

4 MPa2 MPa

0 20 40 60 80 1000

2

4

6

8

10

12

14

(g)

E 50 (

GPa

)

∆σ3 ()

10 MPa8 MPa6 MPa

4 MPa2 MPa

0 20 40 60 80 1000

2

4

6

(h)

E 50 (

GPa

)

∆σ3 ()

10 MPa8 MPa6 MPa

4 MPa2 MPa

0 20 40 60 80 1000

2

4

6

8

(i)

Figure 5 Relationship between deformation modulus and unloading percentage (joint orientation defined as β joint connectivity ratedefined as k) (a) β 30deg k 03 (b) β 30deg k 045 (c) β 30deg k 06 (d) β 45deg k 03 (e) β 45deg k 045 (f ) β 45deg k 06 (g) β 60degk 03 (h) β 60deg k 045 (i) β 60deg k 06

ndash

Failure modes

Shear failure

Splitting failure

Mixed failure (tension and shear)

Slide along the joint planeCompletely

Partly

Does not slide along the joint plane

ndash

Figure 6 Failure modes

8 Advances in Materials Science and Engineering

2 MPa 6 MPa4 MPa 8 MPa 10 MPa

(a)

2 MPa 6 MPa4 MPa 8 MPa 10 MPa

(b)

2 MPa 6 MPa4 MPa 8 MPa 10 MPa

(c)

2 MPa 6 MPa4 MPa 8 MPa 10 MPa

(d)

2 MPa 6 MPa4 MPa 8 MPa 10 MPa

(e)

Figure 7 Continued

Advances in Materials Science and Engineering 9

cannot be reconstructed to form intact specimens(Figure 7(d)) e amount of cracks initiated during the testwas also considered as a parameter to characterize the effectof confining pressures Note that the amount of crack isinversely proportional to the confining pressure Here theconfining pressure was designated by σ3 e fractureexhibited a continuous line-type crack at higher confiningpressures (ie 8 and 10megapascals) However the crackshowed a fork-like pattern when the confining pressure was

less than 4MPa At lower confining pressures samples weremore vulnerable for damages during the unloading test ephenomena were explained in terms of the restraint effect ofconfining pressure on the lateral deformation of the spec-imens According to MohrndashCoulomb failure criterion thedeviatoric stress (σ1 minus σ3) will increase when the confiningpressure (σ3) decreases and the failure envelop can be easilyreached Meanwhile internal defects were rapidly propa-gating as the deviatoric stress increased Lastly perforated

2 MPa 6 MPa4 MPa 8 MPa 10 MPa

(f )

2 MPa 6 MPa4 MPa 8 MPa 10 MPa

(g)

2 MPa 6 MPa4 MPa 8 MPa 10 MPa

(h)

2 MPa 6 MPa4 MPa 8 MPa 10 MPa

(i)

Figure 7 Sample failures under different pressures (joint orientation defined as β joint connectivity rate defined as k) (a) β 30deg k 03 (b)β 30deg k 045 (c) β 30deg k 06 (d) β 45deg k 03 (e) β 45deg k 045 (f ) β 45deg k 06 (g) β 60deg k 03 (h) β 60deg k 045 (i)β 60deg k 06

10 Advances in Materials Science and Engineering

cracks emerged On the contrary internal defects weresqueezed by the confining pressure Hence higher values ofconfining pressures can impose a significant restraint againstthe lateral deformations Similarly excavation can beregarded as unloading condition in rock slope engineering

62 Effect of Orientation on the Failure Modes In naturerocks have different orientations of joints which affect theirphysical and mechanical properties In this study the re-lationships between joint orientations and crack propaga-tions were studied Obviously the failure modes for differentjoint orientations were divergent As shown in Figure 6splitting failure was the predominant failure type forspecimens with 30deg joint orientations (Figures 7(a)ndash7(c))Hence joints with mild slopes have relatively small impacton the properties of rocks On the contrary mixed types offailures have been noticed on the specimens with 45deg jointsDuring testing the specimens were fragmented into piecesIn fact it was difficult to trace the failure paths However at60deg joint orientation shear failure was the predominant oneA flat surface was formed as a result of flaw cuttings Henceat 60deg joint orientation dislocations of pieces occurred dueto a noticeable sliding

63 Effect of Joint Connectivity Rate on the Failure ModesFor different joint angles the effect of connectivity rate (k) isdistinct It had a considerable effect when the joint angleswere 45deg or 60deg However at 30deg joint angle the effect of jointconnectivity was relatively small e path of crack prop-agation was twisted when the value of kwas 03 (Figures 7(a)7(d) and 7(g)) is is because of the restraint action ofuncracked section of the specimens against sliding erupture face was rough since it is a result of multiple factorsOn the one hand joint and rock bridge alter or replace eachother in accordance with the connectivity rate On the otherhand the nonlinear variation of tension stress at the tip offlaws exists As is shown in Figure 6 the crack path wasbecoming edge shaped when the value of k increasedMeanwhile the failure modes gradually changed by thejoints Higher values of k yielded the straight plane ofruptures Nonetheless the concentrations of stresses werenoticeable as shown in the development of secondary cracks(Figures 7(e) and 7(f )) e cracks were extended between

the tips of the joints and the edges of the specimens when thejoint angle and the connectivity rate were 45deg and 045respectively For small values of k the crack pattern becamepolygonal However the crack pattern was transforminginto an arc shape when the value of k increased Besides thesecondary cracks became normal to the edge of the speci-mens during shearing At 60deg joint orientation there was noremarkable effect on the failure modes due to the variationsof joint connectivity rate (k) All surfaces of failures werenearly flat Nonetheless there was a slight discrepancy be-tween them Similar failure modes were developed whenjoint orientation and connectivity rate were 45deg and 03respectively e significant effect of joint connectivity ratewas obvious Hence shear failures can easily develop andtheir probability of occurrence depends on the values of kShear failures were developed at higher values of k(Figures 7(h) and 7(i))

Table 3 summarizes different types of failure modes forspecimens with diverse joint orientations and connectivityrates e failure mode was governed by the joint param-eters Fundamentally the failure modes depend on the jointorientations initially For instance splitting failure is un-likely to occur when the joint angle is above 45deg Meanwhileconnectivity rate determines the extent of shear failuree kvalues control the transformation between mixed andcomplete shear failures At 60deg joint orientation both typesof failures were observed e critical value for k was 03 inthis case

7 Discussions of Crack Force Analysis Based onGriffithrsquos Theory

MohrndashCoulomb and Griffith criteria are conventionallyused to interpret the failure of rock specimens It is no-ticeable that the MohrndashCoulomb criterion cannot clarify theinfluence of inside flaws on rock sample failure Howevermaximum tensile stress in the perimeter of Griffithrsquos crack(Figure 8) can be analyzed according to Griffithrsquos theory forthe sake of exploring the effectiveness of the internal defects

Han et al [26] deduced equation (2) a function of α andc considering primary stress of equation (1) and Inglisrsquosformula

σy σ1 + σ3

2+σ1 minus σ3

2cos 2c

σx σ1 + σ3

2minusσ1 minus σ3

2cos 2c

τxy minusσ1 minus σ3

2sin 2c

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(1)

σb σ1

m2 + α2m 1 + α21113872 1113873(1 + λ) +(1 + m) mminus α21113872 1113873(1minus α)cos 2c + α(1 + m)

2(1minus λ)sin 2c1113966 1113967 (2)

Advances in Materials Science and Engineering 11

As shown in the specimen schematic diagram α and thecrack width can be valued as 0degand 1mm respectively sincethe crack is long and thin Hence the following equation isattained from equation (2)

σb σ1m

(1 + λ) +(1 + m)(1minus λ)cos 2c1113864 1113865 (3)

e stress extremum hinges on σ1 λ and m at the tips ofthe crack in the condition of c minus(π2) e equation isobtained as follows

σ0 σ1m

2λminusm(1minus λ) (4)

According to equation (4) the maximum tensile stress atthe tips of flaw was calculated Meanwhile the influences onthe maximum tensile stress were analyzed considering jointinclined angles joint connectivity and confining pressures(Figures 9 and 10) Obviously the maximum tensile stressincreases promptly in accordance with the confining pres-sures Simultaneously higher joint connectivity rates lead toa large stress extremum And the confining pressure enlargesthis trend As it is shown in Figure 10 the escalation of jointorientation boosts the maximum tensile stress at the tips offlaw erefore the alteration of confining pressure is ofmost significant on the variation of the tensile stress whilethe joint angle contributes the minimal impact

With confining pressure (ie λ) declining gradually inthe unloading stage the longitudinal crack may arise due toextremum of tensile stress on the tips and when it satisfiesthe fracture strength the rupture of the longitudinal crackappears along the plane of the crack and may propagatecontinuously to the specimenrsquos failure Considering the axialstress stops expansion in the unloading stage the specimencannot be broken by some slanting cracks or crossed cracks

directly during the compression erefore the splittingfailure coupled with shear failure emerges finally

8 Conclusions

In the preceding pages the crack propagation was studied onthe specimens with partially cut-through joints e primeobjective of this study was to examine the effect of jointorientations and joint connectivity rates on the mechanicalcharacteristics of specimens Even though nonlinear me-chanical characteristics were observed the macroscopicbehavior was studied based upon the different values of jointgeometries Furthermore nine types of specimens were usedto investigate the failure mechanisms at different confiningpressures e confining pressure joint orientation andconnectivity rate were considered as key parameters toanalyze different failure modes Finally Griffithrsquos theory wasused to analyze the maximum tensile stress Nevertheless afurther study is important to extend the application of thisstudy in real engineering problemsemain findings of thispaper are summarized below

A nonlinear correlation was observed among the elas-ticity modulus joint orientation and joint connectivity rateMeanwhile the confining pressure was arithmetically

Y

σy

σx

Xα

γ

σx

σy

σ3σ1

τxy τxyτyx

τyx

Figure 8 Stress state of the crack based on Griffithrsquos theory [26]

0

100

200

300

400

500

600

2 4 6 8 10

σ 0 (M

Pa)

σ3 (MPa)

β = 30deg k = 03β = 30deg k = 045β = 30deg k = 06

Figure 9 e stress extremum with 30deg joint angle under differentconfining pressures

0

200

400

600

800

1000

30 45 60

σ 0 (M

Pa)

β (deg)k = 03k = 045k = 06

Figure 10 e stress extremum under different joint orientationsand connectivity rates

Table 3 Failure modes for different types of samples

Connectivity rateOrientation 03 045 0630 Splitting Splitting Splitting45 Mixed Mixed Mixed60 Mixed Shear Shear

12 Advances in Materials Science and Engineering

increasing During the unloading stage the reduction ofdeformation modulus can be used to predict the rockspecimen failure And the critical value of decrease per-centage is 80 e deformation modulus is less sensitive tothe change of joint connectivity whereas it is more sensitiveto the change of the inclination angle

ree different types (shear split and mixed) of failuremodes were observed e higher values of confiningpressure can restraint the lateral deformation At 30deg jointorientation the predominant failure was splitting failureSimilarly mixed failures were observed at 45deg joint ori-entations because of the concentration of stresses at the tipof the joints On the contrary shear failure was quitecommon due to sufficient length of 60deg joint orientationComparatively the effect of joint connectivity rate wassmaller than that of the joint orientation However therewas a clear variation in the crack propagation when thejoint angle was 60deg e position of joints plays a significantrole for anisotropic failures

A few equations were introduced based on Griffithrsquostheory to evaluate the maximum tensile stress at the tips ofthe crack e results based on the calculation reveal theimpaction of joint characteristics on the stress extremum

Abbreviations

c e angle of principal stress σ1 and Y axisα e central angle of the ellipsem e ratio of the minor axis (b) to major axis (a) of the

ellipseλ e ratio of two principal stresses between σ3 and σ1σ0 e maximum tensile stressσ1 e maximum principal stressσ3 e minimum principal stressσb Tangential stress on the perimeter of a slender ellipse

Data Availability

e data used to support the findings of this study may bereleased upon application to the China ree GorgesUniversity Review Board which can be done by contactingthe author ldquoGuoyong Duan (dgyhhueducn)rdquo

Conflicts of Interest

e authors wish to confirm that there are no knownconflicts of interest associated with this publication andthere has been no significant financial support for this workthat could have influenced its outcome Accordingly thealteration of confining pressure is of most significance onthe variation of the tensile stress while the joint anglecontributes the minimal impact

Acknowledgments

e authors gratefully acknowledge the financial supportfrom the National Natural Science Foundation of China(Grant nos 51439003 and 51279091) the Fundamental

Research Funds for the Central Universities (Hohai Uni-versity) (Grant no 2016B43014) Open Fund of Key Lab-oratory of Geological Hazards on ree Gorges ReservoirArea (China ree Gorges University) (Grant no2015KDZ12) and Key Project of Scientific Research Plan ofHubei Provincial Department of Education (Grant noD20181202)

References

[1] T Liu P Cao and H Lin ldquoEvolution procedure of multiplerock cracks under seepage pressurerdquo Mathematical Problemsin Engineering vol 2013 Article ID 738013 11 pages 2013

[2] E Hoek ldquoStrength of jointed rock massesrdquo Geotechniquevol 33 no 3 pp 187ndash223 1983

[3] E Hoek and Z T Bieniawski ldquoBrittle fracture propagation inrock under compressionrdquo International Journal of Fracturevol 26 no 4 pp 276ndash294 1984

[4] M Basista and D Gross ldquoe sliding crack model of brittledeformation an internal variable approachrdquo InternationalJournal of Solids and Structures vol 35 no 5-6 pp 487ndash5091998

[5] I Vardoulakis J F Labuz E Papamichos and J TronvollldquoContinuum fracture mechanics of uniaxial compression onbrittle materialsrdquo International Journal of Solids and Struc-tures vol 35 no 31-32 pp 4313ndash4335 1998

[6] A Paluszny and S K Matthai ldquoNumerical modeling ofdiscrete multi-crack growth applied to pattern formation ingeological brittle mediardquo International Journal of Solids andStructures vol 46 no 18-19 pp 3383ndash3397 2009

[7] R-H Cao P Cao H Lin C-Z Pu and K Ou ldquoMechanicalbehavior of brittle rock-like specimens with pre-existingfissures under uniaxial loading experimental studies andparticle mechanics approachrdquo Rock Mechanics and RockEngineering vol 49 no 3 pp 763ndash783 2016

[8] X Fan R Chen H Lin H Lai C Zhang and Q ZhaoldquoCracking and failure in rock specimen containing combinedflaw and hole under uniaxial compressionrdquo Advances in CivilEngineering vol 2018 Article ID 9818250 15 pages 2018

[9] D Bing H Guicheng and Z Zhijun ldquoA numerical researchon crack process of gypsum containing single flaw withdifferent angle and length in uniaxial loadingrdquo Shock andVibration vol 2018 Article ID 2968205 16 pages 2018

[10] J Zhang ldquoInvestigation of relation between fracture scale andacoustic emission time-frequency parameters in rocksrdquo Shockand Vibration vol 2018 Article ID 3057628 14 pages 2018

[11] X Meng and W Liu ldquoStudy on failure process and perme-ation evolution of single-cracked rockrdquo Advances in MaterialsScience and Engineering vol 2018 Article ID 79156529 pages 2018

[12] M Sagong D Park J Yoo and J S Lee ldquoExperimental andnumerical analyses of an opening in a jointed rockmass underbiaxial compressionrdquo International Journal of RockMechanicsand Mining Sciences vol 48 no 7 pp 1055ndash1067 2011

[13] W Chen H Konietzky X Tan and T Fruhwirt ldquoPre-failuredamage analysis for brittle rocks under triaxial compressionrdquoComputers and Geotechnics vol 74 pp 45ndash55 2016

[14] S-Q Yang P G Ranjith Y-H Huang et al ldquoExperimentalinvestigation on mechanical damage characteristics ofsandstone under triaxial cyclic loadingrdquo Geophysical JournalInternational vol 201 no 2 pp 662ndash682 2015

[15] N Erarslan ldquoMicrostructural investigation of subcriticalcrack propagation and fracture process zone (FPZ) by the

Advances in Materials Science and Engineering 13

consonance with the decrease rate of deformation modulusere are 0sim80 and 80sim100 respectively

6 Failure Modes

e failure modes of jointed samples were identified by thelocation and the size of cracks e influence of joint pa-rameters on the crack propagation was discussed As shownin Figure 6 the failure modes were categorized into threegroups

(1) e shear failure occurred frequently under com-pression condition It can be recognized by the shearplane Ignoring stress concentrations on theboundaries of samples the shear plane was relativelyflat is may be induced due to the conversion fromdefects to weak structural planes Based on the as-sumption of sliding mostly occurs along the weak

planes in rockmass the joints are considered as weakstructural planes However rupture did not tran-spired along the crack orientation completely duringthe test In other words the prefabricated cracks didnot thoroughly affect the crack propagation inseveral cases (Figure 7(a)) e condition of a sampleseparated into two parts along the joints accountedfor two third percentage Joint geometries and thedegree of coincidence between the joint and therupture trajectory can be used to characterize thefailure modes (Figure 3)

(2) Generally splitting failure arises at lower confiningpressures e lateral deformation increases muchmore rapidly in the circumstances And the splittingcracks were nearly parallel to themaximum principalstress of the sample Meanwhile the secondarycracks propagated from the crack tips during the

30deg

k = 03

Φ 50

100

(a)

30deg

k = 045

Φ 50

100

(b)30deg

k = 06

Φ 50

100

(c)

45deg

k = 03

Φ 50

100

(d)

45degk = 04

5

Φ 50

100

(e)

45degk = 06

Φ 50

100

(f )

60deg

k = 0

3

Φ 50

100

(g)

60degk = 0

45

Φ 50

100

(h)

60degk = 0

6Φ 50

100

(i)

Figure 3 Geometrical specifications

Advances in Materials Science and Engineering 5

unloading tests It is noteworthy that splitting failureoccurs in the intact rock samples in the procedure ofuniaxial compression Nevertheless similar phe-nomena were observed on the specimens withsmaller joint angles and connectivity rates In otherwords the likelihood of splitting failure is high whenthe influence of joint is insignificant Meanwhile theprobability of splitting failure is also dependent onthe angle of the joint and the vertical axis of thesample Based upon the results samples with 30degjoint orientation were more inclined to splittingfailure

(3) A mixed type of failure (tension and shear) wasdeveloped at higher joint orientation and mediumjoint connectivity rate It is characterized by slidingof joints and development of tension cracks from thecrack tip Based on the rupture morphology theprecondition of mixed failure hinges on middlevalues of joint orientation and connectivity rate asshown in Figures 7(d)ndash7(f) Considering the ten-dency of shear along the joint plane tension cracksdeveloped by the reason of stress concentrations at

the tips of the cracks It is an important characteristicthat reveals the dislocation of sample pieces beforethe ultimate failure e two parallel tension frac-tures were normal to the original crack In mostcases the secondary cracks were connected to thetips of joints where stress concentrations are in-evitable erefore the propagation of cracks can bedetermined by analyzing the geometries of joints andloading conditions

61 Effect of Confining Pressure on the Failure of the SamplesIn the triaxial test the mechanical properties of rockspecimens can be affected by the confining pressure defi-nitely e effect of unloading was remarkable when theconfining pressures were 2 and 4megapascals (based uponthe analysis of 30 diagrams) As a result the specimensexhibited different levels of damages at different confiningpressures During the unloading test parts of the specimenswere shattering when the confining pressure was lower forexample 2 and 4megapascals In this case the fragments

Figure 4 Triaxial compressive testing apparatus

Table 2 Stress state for the triaxial compression and unloading compression

Stress state Minimum principal stress σ3 (MPa) Maximum principal stress σ1 (MPa) Remarks

σ1

σ1σ1 gt σ2 = σ3

σ3σ3

10 61526 Triaxial compression8 55468 Triaxial compression6 47028 Triaxial compression4 37239 Triaxial compression2 29256 Triaxial compression10 43106 Triaxial unloading compression8 38677 Triaxial unloading compression6 32110 Triaxial unloading compression4 25502 Triaxial unloading compression2 20060 Triaxial unloading compression

6 Advances in Materials Science and Engineering

0 20 40 60 80 1000

1

2

3

4

5

6

7

8E 5

0 (G

Pa)

∆σ3 ()

10 MPa8 MPa6 MPa

4 MPa2 MPa

(a)

E 50 (

GPa

)

∆σ3 ()0 20 40 60 80 100

0

1

2

3

4

5

6

7

8

10 MPa8 MPa6 MPa

4 MPa2 MPa

(b)

E 50 (

GPa

)

∆σ3 ()

10 MPa8 MPa6 MPa

4 MPa2 MPa

0 20 40 60 80 1003

4

5

6

7

8

(c)

E 50 (

GPa

)

∆σ3 ()

10 MPa8 MPa6 MPa

4 MPa2 MPa

0 20 40 60 80 1000

4

8

2

16

20

(d)

E 50 (

GPa

)

∆σ3 ()

10 MPa8 MPa6 MPa

4 MPa2 MPa

0 20 40 60 80 1000

2

4

6

8

10

(e)

E 50 (

GPa

)

∆σ3 ()

10 MPa8 MPa6 MPa

4 MPa2 MPa

0 20 40 60 80 1000

2

4

6

8

0

(f )

Figure 5 Continued

Advances in Materials Science and Engineering 7

E 50 (

GPa

)

∆σ3 ()

10 MPa8 MPa6 MPa

4 MPa2 MPa

0 20 40 60 80 1000

2

4

6

8

10

12

14

(g)

E 50 (

GPa

)

∆σ3 ()

10 MPa8 MPa6 MPa

4 MPa2 MPa

0 20 40 60 80 1000

2

4

6

(h)

E 50 (

GPa

)

∆σ3 ()

10 MPa8 MPa6 MPa

4 MPa2 MPa

0 20 40 60 80 1000

2

4

6

8

(i)

Figure 5 Relationship between deformation modulus and unloading percentage (joint orientation defined as β joint connectivity ratedefined as k) (a) β 30deg k 03 (b) β 30deg k 045 (c) β 30deg k 06 (d) β 45deg k 03 (e) β 45deg k 045 (f ) β 45deg k 06 (g) β 60degk 03 (h) β 60deg k 045 (i) β 60deg k 06

ndash

Failure modes

Shear failure

Splitting failure

Mixed failure (tension and shear)

Slide along the joint planeCompletely

Partly

Does not slide along the joint plane

ndash

Figure 6 Failure modes

8 Advances in Materials Science and Engineering

2 MPa 6 MPa4 MPa 8 MPa 10 MPa

(a)

2 MPa 6 MPa4 MPa 8 MPa 10 MPa

(b)

2 MPa 6 MPa4 MPa 8 MPa 10 MPa

(c)

2 MPa 6 MPa4 MPa 8 MPa 10 MPa

(d)

2 MPa 6 MPa4 MPa 8 MPa 10 MPa

(e)

Figure 7 Continued

Advances in Materials Science and Engineering 9

cannot be reconstructed to form intact specimens(Figure 7(d)) e amount of cracks initiated during the testwas also considered as a parameter to characterize the effectof confining pressures Note that the amount of crack isinversely proportional to the confining pressure Here theconfining pressure was designated by σ3 e fractureexhibited a continuous line-type crack at higher confiningpressures (ie 8 and 10megapascals) However the crackshowed a fork-like pattern when the confining pressure was

less than 4MPa At lower confining pressures samples weremore vulnerable for damages during the unloading test ephenomena were explained in terms of the restraint effect ofconfining pressure on the lateral deformation of the spec-imens According to MohrndashCoulomb failure criterion thedeviatoric stress (σ1 minus σ3) will increase when the confiningpressure (σ3) decreases and the failure envelop can be easilyreached Meanwhile internal defects were rapidly propa-gating as the deviatoric stress increased Lastly perforated

2 MPa 6 MPa4 MPa 8 MPa 10 MPa

(f )

2 MPa 6 MPa4 MPa 8 MPa 10 MPa

(g)

2 MPa 6 MPa4 MPa 8 MPa 10 MPa

(h)

2 MPa 6 MPa4 MPa 8 MPa 10 MPa

(i)

Figure 7 Sample failures under different pressures (joint orientation defined as β joint connectivity rate defined as k) (a) β 30deg k 03 (b)β 30deg k 045 (c) β 30deg k 06 (d) β 45deg k 03 (e) β 45deg k 045 (f ) β 45deg k 06 (g) β 60deg k 03 (h) β 60deg k 045 (i)β 60deg k 06

10 Advances in Materials Science and Engineering

cracks emerged On the contrary internal defects weresqueezed by the confining pressure Hence higher values ofconfining pressures can impose a significant restraint againstthe lateral deformations Similarly excavation can beregarded as unloading condition in rock slope engineering

62 Effect of Orientation on the Failure Modes In naturerocks have different orientations of joints which affect theirphysical and mechanical properties In this study the re-lationships between joint orientations and crack propaga-tions were studied Obviously the failure modes for differentjoint orientations were divergent As shown in Figure 6splitting failure was the predominant failure type forspecimens with 30deg joint orientations (Figures 7(a)ndash7(c))Hence joints with mild slopes have relatively small impacton the properties of rocks On the contrary mixed types offailures have been noticed on the specimens with 45deg jointsDuring testing the specimens were fragmented into piecesIn fact it was difficult to trace the failure paths However at60deg joint orientation shear failure was the predominant oneA flat surface was formed as a result of flaw cuttings Henceat 60deg joint orientation dislocations of pieces occurred dueto a noticeable sliding

63 Effect of Joint Connectivity Rate on the Failure ModesFor different joint angles the effect of connectivity rate (k) isdistinct It had a considerable effect when the joint angleswere 45deg or 60deg However at 30deg joint angle the effect of jointconnectivity was relatively small e path of crack prop-agation was twisted when the value of kwas 03 (Figures 7(a)7(d) and 7(g)) is is because of the restraint action ofuncracked section of the specimens against sliding erupture face was rough since it is a result of multiple factorsOn the one hand joint and rock bridge alter or replace eachother in accordance with the connectivity rate On the otherhand the nonlinear variation of tension stress at the tip offlaws exists As is shown in Figure 6 the crack path wasbecoming edge shaped when the value of k increasedMeanwhile the failure modes gradually changed by thejoints Higher values of k yielded the straight plane ofruptures Nonetheless the concentrations of stresses werenoticeable as shown in the development of secondary cracks(Figures 7(e) and 7(f )) e cracks were extended between

the tips of the joints and the edges of the specimens when thejoint angle and the connectivity rate were 45deg and 045respectively For small values of k the crack pattern becamepolygonal However the crack pattern was transforminginto an arc shape when the value of k increased Besides thesecondary cracks became normal to the edge of the speci-mens during shearing At 60deg joint orientation there was noremarkable effect on the failure modes due to the variationsof joint connectivity rate (k) All surfaces of failures werenearly flat Nonetheless there was a slight discrepancy be-tween them Similar failure modes were developed whenjoint orientation and connectivity rate were 45deg and 03respectively e significant effect of joint connectivity ratewas obvious Hence shear failures can easily develop andtheir probability of occurrence depends on the values of kShear failures were developed at higher values of k(Figures 7(h) and 7(i))

Table 3 summarizes different types of failure modes forspecimens with diverse joint orientations and connectivityrates e failure mode was governed by the joint param-eters Fundamentally the failure modes depend on the jointorientations initially For instance splitting failure is un-likely to occur when the joint angle is above 45deg Meanwhileconnectivity rate determines the extent of shear failuree kvalues control the transformation between mixed andcomplete shear failures At 60deg joint orientation both typesof failures were observed e critical value for k was 03 inthis case

7 Discussions of Crack Force Analysis Based onGriffithrsquos Theory

MohrndashCoulomb and Griffith criteria are conventionallyused to interpret the failure of rock specimens It is no-ticeable that the MohrndashCoulomb criterion cannot clarify theinfluence of inside flaws on rock sample failure Howevermaximum tensile stress in the perimeter of Griffithrsquos crack(Figure 8) can be analyzed according to Griffithrsquos theory forthe sake of exploring the effectiveness of the internal defects

Han et al [26] deduced equation (2) a function of α andc considering primary stress of equation (1) and Inglisrsquosformula

σy σ1 + σ3

2+σ1 minus σ3

2cos 2c

σx σ1 + σ3

2minusσ1 minus σ3

2cos 2c

τxy minusσ1 minus σ3

2sin 2c

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(1)

σb σ1

m2 + α2m 1 + α21113872 1113873(1 + λ) +(1 + m) mminus α21113872 1113873(1minus α)cos 2c + α(1 + m)

2(1minus λ)sin 2c1113966 1113967 (2)

Advances in Materials Science and Engineering 11

As shown in the specimen schematic diagram α and thecrack width can be valued as 0degand 1mm respectively sincethe crack is long and thin Hence the following equation isattained from equation (2)

σb σ1m

(1 + λ) +(1 + m)(1minus λ)cos 2c1113864 1113865 (3)

e stress extremum hinges on σ1 λ and m at the tips ofthe crack in the condition of c minus(π2) e equation isobtained as follows

σ0 σ1m

2λminusm(1minus λ) (4)

According to equation (4) the maximum tensile stress atthe tips of flaw was calculated Meanwhile the influences onthe maximum tensile stress were analyzed considering jointinclined angles joint connectivity and confining pressures(Figures 9 and 10) Obviously the maximum tensile stressincreases promptly in accordance with the confining pres-sures Simultaneously higher joint connectivity rates lead toa large stress extremum And the confining pressure enlargesthis trend As it is shown in Figure 10 the escalation of jointorientation boosts the maximum tensile stress at the tips offlaw erefore the alteration of confining pressure is ofmost significant on the variation of the tensile stress whilethe joint angle contributes the minimal impact

With confining pressure (ie λ) declining gradually inthe unloading stage the longitudinal crack may arise due toextremum of tensile stress on the tips and when it satisfiesthe fracture strength the rupture of the longitudinal crackappears along the plane of the crack and may propagatecontinuously to the specimenrsquos failure Considering the axialstress stops expansion in the unloading stage the specimencannot be broken by some slanting cracks or crossed cracks

directly during the compression erefore the splittingfailure coupled with shear failure emerges finally

8 Conclusions

In the preceding pages the crack propagation was studied onthe specimens with partially cut-through joints e primeobjective of this study was to examine the effect of jointorientations and joint connectivity rates on the mechanicalcharacteristics of specimens Even though nonlinear me-chanical characteristics were observed the macroscopicbehavior was studied based upon the different values of jointgeometries Furthermore nine types of specimens were usedto investigate the failure mechanisms at different confiningpressures e confining pressure joint orientation andconnectivity rate were considered as key parameters toanalyze different failure modes Finally Griffithrsquos theory wasused to analyze the maximum tensile stress Nevertheless afurther study is important to extend the application of thisstudy in real engineering problemsemain findings of thispaper are summarized below

A nonlinear correlation was observed among the elas-ticity modulus joint orientation and joint connectivity rateMeanwhile the confining pressure was arithmetically

Y

σy

σx

Xα

γ

σx

σy

σ3σ1

τxy τxyτyx

τyx

Figure 8 Stress state of the crack based on Griffithrsquos theory [26]

0

100

200

300

400

500

600

2 4 6 8 10

σ 0 (M

Pa)

σ3 (MPa)

β = 30deg k = 03β = 30deg k = 045β = 30deg k = 06

Figure 9 e stress extremum with 30deg joint angle under differentconfining pressures

0

200

400

600

800

1000

30 45 60

σ 0 (M

Pa)

β (deg)k = 03k = 045k = 06

Figure 10 e stress extremum under different joint orientationsand connectivity rates

Table 3 Failure modes for different types of samples

Connectivity rateOrientation 03 045 0630 Splitting Splitting Splitting45 Mixed Mixed Mixed60 Mixed Shear Shear

12 Advances in Materials Science and Engineering

increasing During the unloading stage the reduction ofdeformation modulus can be used to predict the rockspecimen failure And the critical value of decrease per-centage is 80 e deformation modulus is less sensitive tothe change of joint connectivity whereas it is more sensitiveto the change of the inclination angle

ree different types (shear split and mixed) of failuremodes were observed e higher values of confiningpressure can restraint the lateral deformation At 30deg jointorientation the predominant failure was splitting failureSimilarly mixed failures were observed at 45deg joint ori-entations because of the concentration of stresses at the tipof the joints On the contrary shear failure was quitecommon due to sufficient length of 60deg joint orientationComparatively the effect of joint connectivity rate wassmaller than that of the joint orientation However therewas a clear variation in the crack propagation when thejoint angle was 60deg e position of joints plays a significantrole for anisotropic failures

A few equations were introduced based on Griffithrsquostheory to evaluate the maximum tensile stress at the tips ofthe crack e results based on the calculation reveal theimpaction of joint characteristics on the stress extremum

Abbreviations

c e angle of principal stress σ1 and Y axisα e central angle of the ellipsem e ratio of the minor axis (b) to major axis (a) of the

ellipseλ e ratio of two principal stresses between σ3 and σ1σ0 e maximum tensile stressσ1 e maximum principal stressσ3 e minimum principal stressσb Tangential stress on the perimeter of a slender ellipse

Data Availability

e data used to support the findings of this study may bereleased upon application to the China ree GorgesUniversity Review Board which can be done by contactingthe author ldquoGuoyong Duan (dgyhhueducn)rdquo

Conflicts of Interest

e authors wish to confirm that there are no knownconflicts of interest associated with this publication andthere has been no significant financial support for this workthat could have influenced its outcome Accordingly thealteration of confining pressure is of most significance onthe variation of the tensile stress while the joint anglecontributes the minimal impact

Acknowledgments

e authors gratefully acknowledge the financial supportfrom the National Natural Science Foundation of China(Grant nos 51439003 and 51279091) the Fundamental

Research Funds for the Central Universities (Hohai Uni-versity) (Grant no 2016B43014) Open Fund of Key Lab-oratory of Geological Hazards on ree Gorges ReservoirArea (China ree Gorges University) (Grant no2015KDZ12) and Key Project of Scientific Research Plan ofHubei Provincial Department of Education (Grant noD20181202)

References

[1] T Liu P Cao and H Lin ldquoEvolution procedure of multiplerock cracks under seepage pressurerdquo Mathematical Problemsin Engineering vol 2013 Article ID 738013 11 pages 2013

[2] E Hoek ldquoStrength of jointed rock massesrdquo Geotechniquevol 33 no 3 pp 187ndash223 1983

[3] E Hoek and Z T Bieniawski ldquoBrittle fracture propagation inrock under compressionrdquo International Journal of Fracturevol 26 no 4 pp 276ndash294 1984

[4] M Basista and D Gross ldquoe sliding crack model of brittledeformation an internal variable approachrdquo InternationalJournal of Solids and Structures vol 35 no 5-6 pp 487ndash5091998

[5] I Vardoulakis J F Labuz E Papamichos and J TronvollldquoContinuum fracture mechanics of uniaxial compression onbrittle materialsrdquo International Journal of Solids and Struc-tures vol 35 no 31-32 pp 4313ndash4335 1998

[6] A Paluszny and S K Matthai ldquoNumerical modeling ofdiscrete multi-crack growth applied to pattern formation ingeological brittle mediardquo International Journal of Solids andStructures vol 46 no 18-19 pp 3383ndash3397 2009

[7] R-H Cao P Cao H Lin C-Z Pu and K Ou ldquoMechanicalbehavior of brittle rock-like specimens with pre-existingfissures under uniaxial loading experimental studies andparticle mechanics approachrdquo Rock Mechanics and RockEngineering vol 49 no 3 pp 763ndash783 2016

[8] X Fan R Chen H Lin H Lai C Zhang and Q ZhaoldquoCracking and failure in rock specimen containing combinedflaw and hole under uniaxial compressionrdquo Advances in CivilEngineering vol 2018 Article ID 9818250 15 pages 2018

[9] D Bing H Guicheng and Z Zhijun ldquoA numerical researchon crack process of gypsum containing single flaw withdifferent angle and length in uniaxial loadingrdquo Shock andVibration vol 2018 Article ID 2968205 16 pages 2018

[10] J Zhang ldquoInvestigation of relation between fracture scale andacoustic emission time-frequency parameters in rocksrdquo Shockand Vibration vol 2018 Article ID 3057628 14 pages 2018

[11] X Meng and W Liu ldquoStudy on failure process and perme-ation evolution of single-cracked rockrdquo Advances in MaterialsScience and Engineering vol 2018 Article ID 79156529 pages 2018

[12] M Sagong D Park J Yoo and J S Lee ldquoExperimental andnumerical analyses of an opening in a jointed rockmass underbiaxial compressionrdquo International Journal of RockMechanicsand Mining Sciences vol 48 no 7 pp 1055ndash1067 2011

[13] W Chen H Konietzky X Tan and T Fruhwirt ldquoPre-failuredamage analysis for brittle rocks under triaxial compressionrdquoComputers and Geotechnics vol 74 pp 45ndash55 2016

[14] S-Q Yang P G Ranjith Y-H Huang et al ldquoExperimentalinvestigation on mechanical damage characteristics ofsandstone under triaxial cyclic loadingrdquo Geophysical JournalInternational vol 201 no 2 pp 662ndash682 2015

[15] N Erarslan ldquoMicrostructural investigation of subcriticalcrack propagation and fracture process zone (FPZ) by the

Advances in Materials Science and Engineering 13

unloading tests It is noteworthy that splitting failureoccurs in the intact rock samples in the procedure ofuniaxial compression Nevertheless similar phe-nomena were observed on the specimens withsmaller joint angles and connectivity rates In otherwords the likelihood of splitting failure is high whenthe influence of joint is insignificant Meanwhile theprobability of splitting failure is also dependent onthe angle of the joint and the vertical axis of thesample Based upon the results samples with 30degjoint orientation were more inclined to splittingfailure

(3) A mixed type of failure (tension and shear) wasdeveloped at higher joint orientation and mediumjoint connectivity rate It is characterized by slidingof joints and development of tension cracks from thecrack tip Based on the rupture morphology theprecondition of mixed failure hinges on middlevalues of joint orientation and connectivity rate asshown in Figures 7(d)ndash7(f) Considering the ten-dency of shear along the joint plane tension cracksdeveloped by the reason of stress concentrations at

the tips of the cracks It is an important characteristicthat reveals the dislocation of sample pieces beforethe ultimate failure e two parallel tension frac-tures were normal to the original crack In mostcases the secondary cracks were connected to thetips of joints where stress concentrations are in-evitable erefore the propagation of cracks can bedetermined by analyzing the geometries of joints andloading conditions

61 Effect of Confining Pressure on the Failure of the SamplesIn the triaxial test the mechanical properties of rockspecimens can be affected by the confining pressure defi-nitely e effect of unloading was remarkable when theconfining pressures were 2 and 4megapascals (based uponthe analysis of 30 diagrams) As a result the specimensexhibited different levels of damages at different confiningpressures During the unloading test parts of the specimenswere shattering when the confining pressure was lower forexample 2 and 4megapascals In this case the fragments

Figure 4 Triaxial compressive testing apparatus

Table 2 Stress state for the triaxial compression and unloading compression

Stress state Minimum principal stress σ3 (MPa) Maximum principal stress σ1 (MPa) Remarks

σ1

σ1σ1 gt σ2 = σ3

σ3σ3

10 61526 Triaxial compression8 55468 Triaxial compression6 47028 Triaxial compression4 37239 Triaxial compression2 29256 Triaxial compression10 43106 Triaxial unloading compression8 38677 Triaxial unloading compression6 32110 Triaxial unloading compression4 25502 Triaxial unloading compression2 20060 Triaxial unloading compression

6 Advances in Materials Science and Engineering

0 20 40 60 80 1000

1

2

3

4

5

6

7

8E 5

0 (G

Pa)

∆σ3 ()

10 MPa8 MPa6 MPa

4 MPa2 MPa

(a)

E 50 (

GPa

)

∆σ3 ()0 20 40 60 80 100

0

1

2

3

4

5

6

7

8

10 MPa8 MPa6 MPa

4 MPa2 MPa

(b)

E 50 (

GPa

)

∆σ3 ()

10 MPa8 MPa6 MPa

4 MPa2 MPa

0 20 40 60 80 1003

4

5

6

7

8

(c)

E 50 (

GPa

)

∆σ3 ()

10 MPa8 MPa6 MPa

4 MPa2 MPa

0 20 40 60 80 1000

4

8

2

16

20

(d)

E 50 (

GPa

)

∆σ3 ()

10 MPa8 MPa6 MPa

4 MPa2 MPa

0 20 40 60 80 1000

2

4

6

8

10

(e)

E 50 (

GPa

)

∆σ3 ()

10 MPa8 MPa6 MPa

4 MPa2 MPa

0 20 40 60 80 1000

2

4

6

8

0

(f )

Figure 5 Continued

Advances in Materials Science and Engineering 7

E 50 (

GPa

)

∆σ3 ()

10 MPa8 MPa6 MPa

4 MPa2 MPa

0 20 40 60 80 1000

2

4

6

8

10

12

14

(g)

E 50 (

GPa

)

∆σ3 ()

10 MPa8 MPa6 MPa

4 MPa2 MPa

0 20 40 60 80 1000

2

4

6

(h)

E 50 (

GPa

)

∆σ3 ()

10 MPa8 MPa6 MPa

4 MPa2 MPa

0 20 40 60 80 1000

2

4

6

8

(i)

Figure 5 Relationship between deformation modulus and unloading percentage (joint orientation defined as β joint connectivity ratedefined as k) (a) β 30deg k 03 (b) β 30deg k 045 (c) β 30deg k 06 (d) β 45deg k 03 (e) β 45deg k 045 (f ) β 45deg k 06 (g) β 60degk 03 (h) β 60deg k 045 (i) β 60deg k 06

ndash

Failure modes

Shear failure

Splitting failure

Mixed failure (tension and shear)

Slide along the joint planeCompletely

Partly

Does not slide along the joint plane

ndash

Figure 6 Failure modes

8 Advances in Materials Science and Engineering

2 MPa 6 MPa4 MPa 8 MPa 10 MPa

(a)

2 MPa 6 MPa4 MPa 8 MPa 10 MPa

(b)

2 MPa 6 MPa4 MPa 8 MPa 10 MPa

(c)

2 MPa 6 MPa4 MPa 8 MPa 10 MPa

(d)

2 MPa 6 MPa4 MPa 8 MPa 10 MPa

(e)

Figure 7 Continued

Advances in Materials Science and Engineering 9

cannot be reconstructed to form intact specimens(Figure 7(d)) e amount of cracks initiated during the testwas also considered as a parameter to characterize the effectof confining pressures Note that the amount of crack isinversely proportional to the confining pressure Here theconfining pressure was designated by σ3 e fractureexhibited a continuous line-type crack at higher confiningpressures (ie 8 and 10megapascals) However the crackshowed a fork-like pattern when the confining pressure was

less than 4MPa At lower confining pressures samples weremore vulnerable for damages during the unloading test ephenomena were explained in terms of the restraint effect ofconfining pressure on the lateral deformation of the spec-imens According to MohrndashCoulomb failure criterion thedeviatoric stress (σ1 minus σ3) will increase when the confiningpressure (σ3) decreases and the failure envelop can be easilyreached Meanwhile internal defects were rapidly propa-gating as the deviatoric stress increased Lastly perforated

2 MPa 6 MPa4 MPa 8 MPa 10 MPa

(f )

2 MPa 6 MPa4 MPa 8 MPa 10 MPa

(g)

2 MPa 6 MPa4 MPa 8 MPa 10 MPa

(h)

2 MPa 6 MPa4 MPa 8 MPa 10 MPa

(i)

Figure 7 Sample failures under different pressures (joint orientation defined as β joint connectivity rate defined as k) (a) β 30deg k 03 (b)β 30deg k 045 (c) β 30deg k 06 (d) β 45deg k 03 (e) β 45deg k 045 (f ) β 45deg k 06 (g) β 60deg k 03 (h) β 60deg k 045 (i)β 60deg k 06

10 Advances in Materials Science and Engineering

cracks emerged On the contrary internal defects weresqueezed by the confining pressure Hence higher values ofconfining pressures can impose a significant restraint againstthe lateral deformations Similarly excavation can beregarded as unloading condition in rock slope engineering

62 Effect of Orientation on the Failure Modes In naturerocks have different orientations of joints which affect theirphysical and mechanical properties In this study the re-lationships between joint orientations and crack propaga-tions were studied Obviously the failure modes for differentjoint orientations were divergent As shown in Figure 6splitting failure was the predominant failure type forspecimens with 30deg joint orientations (Figures 7(a)ndash7(c))Hence joints with mild slopes have relatively small impacton the properties of rocks On the contrary mixed types offailures have been noticed on the specimens with 45deg jointsDuring testing the specimens were fragmented into piecesIn fact it was difficult to trace the failure paths However at60deg joint orientation shear failure was the predominant oneA flat surface was formed as a result of flaw cuttings Henceat 60deg joint orientation dislocations of pieces occurred dueto a noticeable sliding

63 Effect of Joint Connectivity Rate on the Failure ModesFor different joint angles the effect of connectivity rate (k) isdistinct It had a considerable effect when the joint angleswere 45deg or 60deg However at 30deg joint angle the effect of jointconnectivity was relatively small e path of crack prop-agation was twisted when the value of kwas 03 (Figures 7(a)7(d) and 7(g)) is is because of the restraint action ofuncracked section of the specimens against sliding erupture face was rough since it is a result of multiple factorsOn the one hand joint and rock bridge alter or replace eachother in accordance with the connectivity rate On the otherhand the nonlinear variation of tension stress at the tip offlaws exists As is shown in Figure 6 the crack path wasbecoming edge shaped when the value of k increasedMeanwhile the failure modes gradually changed by thejoints Higher values of k yielded the straight plane ofruptures Nonetheless the concentrations of stresses werenoticeable as shown in the development of secondary cracks(Figures 7(e) and 7(f )) e cracks were extended between

the tips of the joints and the edges of the specimens when thejoint angle and the connectivity rate were 45deg and 045respectively For small values of k the crack pattern becamepolygonal However the crack pattern was transforminginto an arc shape when the value of k increased Besides thesecondary cracks became normal to the edge of the speci-mens during shearing At 60deg joint orientation there was noremarkable effect on the failure modes due to the variationsof joint connectivity rate (k) All surfaces of failures werenearly flat Nonetheless there was a slight discrepancy be-tween them Similar failure modes were developed whenjoint orientation and connectivity rate were 45deg and 03respectively e significant effect of joint connectivity ratewas obvious Hence shear failures can easily develop andtheir probability of occurrence depends on the values of kShear failures were developed at higher values of k(Figures 7(h) and 7(i))

Table 3 summarizes different types of failure modes forspecimens with diverse joint orientations and connectivityrates e failure mode was governed by the joint param-eters Fundamentally the failure modes depend on the jointorientations initially For instance splitting failure is un-likely to occur when the joint angle is above 45deg Meanwhileconnectivity rate determines the extent of shear failuree kvalues control the transformation between mixed andcomplete shear failures At 60deg joint orientation both typesof failures were observed e critical value for k was 03 inthis case

7 Discussions of Crack Force Analysis Based onGriffithrsquos Theory

MohrndashCoulomb and Griffith criteria are conventionallyused to interpret the failure of rock specimens It is no-ticeable that the MohrndashCoulomb criterion cannot clarify theinfluence of inside flaws on rock sample failure Howevermaximum tensile stress in the perimeter of Griffithrsquos crack(Figure 8) can be analyzed according to Griffithrsquos theory forthe sake of exploring the effectiveness of the internal defects

Han et al [26] deduced equation (2) a function of α andc considering primary stress of equation (1) and Inglisrsquosformula

σy σ1 + σ3

2+σ1 minus σ3

2cos 2c

σx σ1 + σ3

2minusσ1 minus σ3

2cos 2c

τxy minusσ1 minus σ3

2sin 2c

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(1)

σb σ1

m2 + α2m 1 + α21113872 1113873(1 + λ) +(1 + m) mminus α21113872 1113873(1minus α)cos 2c + α(1 + m)

2(1minus λ)sin 2c1113966 1113967 (2)

Advances in Materials Science and Engineering 11

As shown in the specimen schematic diagram α and thecrack width can be valued as 0degand 1mm respectively sincethe crack is long and thin Hence the following equation isattained from equation (2)

σb σ1m

(1 + λ) +(1 + m)(1minus λ)cos 2c1113864 1113865 (3)

e stress extremum hinges on σ1 λ and m at the tips ofthe crack in the condition of c minus(π2) e equation isobtained as follows

σ0 σ1m

2λminusm(1minus λ) (4)

According to equation (4) the maximum tensile stress atthe tips of flaw was calculated Meanwhile the influences onthe maximum tensile stress were analyzed considering jointinclined angles joint connectivity and confining pressures(Figures 9 and 10) Obviously the maximum tensile stressincreases promptly in accordance with the confining pres-sures Simultaneously higher joint connectivity rates lead toa large stress extremum And the confining pressure enlargesthis trend As it is shown in Figure 10 the escalation of jointorientation boosts the maximum tensile stress at the tips offlaw erefore the alteration of confining pressure is ofmost significant on the variation of the tensile stress whilethe joint angle contributes the minimal impact

With confining pressure (ie λ) declining gradually inthe unloading stage the longitudinal crack may arise due toextremum of tensile stress on the tips and when it satisfiesthe fracture strength the rupture of the longitudinal crackappears along the plane of the crack and may propagatecontinuously to the specimenrsquos failure Considering the axialstress stops expansion in the unloading stage the specimencannot be broken by some slanting cracks or crossed cracks

directly during the compression erefore the splittingfailure coupled with shear failure emerges finally

8 Conclusions

In the preceding pages the crack propagation was studied onthe specimens with partially cut-through joints e primeobjective of this study was to examine the effect of jointorientations and joint connectivity rates on the mechanicalcharacteristics of specimens Even though nonlinear me-chanical characteristics were observed the macroscopicbehavior was studied based upon the different values of jointgeometries Furthermore nine types of specimens were usedto investigate the failure mechanisms at different confiningpressures e confining pressure joint orientation andconnectivity rate were considered as key parameters toanalyze different failure modes Finally Griffithrsquos theory wasused to analyze the maximum tensile stress Nevertheless afurther study is important to extend the application of thisstudy in real engineering problemsemain findings of thispaper are summarized below

A nonlinear correlation was observed among the elas-ticity modulus joint orientation and joint connectivity rateMeanwhile the confining pressure was arithmetically

Y

σy

σx

Xα

γ

σx

σy

σ3σ1

τxy τxyτyx

τyx

Figure 8 Stress state of the crack based on Griffithrsquos theory [26]

0

100

200

300

400

500

600

2 4 6 8 10

σ 0 (M

Pa)

σ3 (MPa)

β = 30deg k = 03β = 30deg k = 045β = 30deg k = 06

Figure 9 e stress extremum with 30deg joint angle under differentconfining pressures

0

200

400

600

800

1000

30 45 60

σ 0 (M

Pa)

β (deg)k = 03k = 045k = 06

Figure 10 e stress extremum under different joint orientationsand connectivity rates

Table 3 Failure modes for different types of samples

Connectivity rateOrientation 03 045 0630 Splitting Splitting Splitting45 Mixed Mixed Mixed60 Mixed Shear Shear

12 Advances in Materials Science and Engineering

increasing During the unloading stage the reduction ofdeformation modulus can be used to predict the rockspecimen failure And the critical value of decrease per-centage is 80 e deformation modulus is less sensitive tothe change of joint connectivity whereas it is more sensitiveto the change of the inclination angle

ree different types (shear split and mixed) of failuremodes were observed e higher values of confiningpressure can restraint the lateral deformation At 30deg jointorientation the predominant failure was splitting failureSimilarly mixed failures were observed at 45deg joint ori-entations because of the concentration of stresses at the tipof the joints On the contrary shear failure was quitecommon due to sufficient length of 60deg joint orientationComparatively the effect of joint connectivity rate wassmaller than that of the joint orientation However therewas a clear variation in the crack propagation when thejoint angle was 60deg e position of joints plays a significantrole for anisotropic failures

A few equations were introduced based on Griffithrsquostheory to evaluate the maximum tensile stress at the tips ofthe crack e results based on the calculation reveal theimpaction of joint characteristics on the stress extremum

Abbreviations

c e angle of principal stress σ1 and Y axisα e central angle of the ellipsem e ratio of the minor axis (b) to major axis (a) of the

ellipseλ e ratio of two principal stresses between σ3 and σ1σ0 e maximum tensile stressσ1 e maximum principal stressσ3 e minimum principal stressσb Tangential stress on the perimeter of a slender ellipse

Data Availability

e data used to support the findings of this study may bereleased upon application to the China ree GorgesUniversity Review Board which can be done by contactingthe author ldquoGuoyong Duan (dgyhhueducn)rdquo

Conflicts of Interest

e authors wish to confirm that there are no knownconflicts of interest associated with this publication andthere has been no significant financial support for this workthat could have influenced its outcome Accordingly thealteration of confining pressure is of most significance onthe variation of the tensile stress while the joint anglecontributes the minimal impact

Acknowledgments

e authors gratefully acknowledge the financial supportfrom the National Natural Science Foundation of China(Grant nos 51439003 and 51279091) the Fundamental

Research Funds for the Central Universities (Hohai Uni-versity) (Grant no 2016B43014) Open Fund of Key Lab-oratory of Geological Hazards on ree Gorges ReservoirArea (China ree Gorges University) (Grant no2015KDZ12) and Key Project of Scientific Research Plan ofHubei Provincial Department of Education (Grant noD20181202)

References

[1] T Liu P Cao and H Lin ldquoEvolution procedure of multiplerock cracks under seepage pressurerdquo Mathematical Problemsin Engineering vol 2013 Article ID 738013 11 pages 2013

[2] E Hoek ldquoStrength of jointed rock massesrdquo Geotechniquevol 33 no 3 pp 187ndash223 1983

[3] E Hoek and Z T Bieniawski ldquoBrittle fracture propagation inrock under compressionrdquo International Journal of Fracturevol 26 no 4 pp 276ndash294 1984

[4] M Basista and D Gross ldquoe sliding crack model of brittledeformation an internal variable approachrdquo InternationalJournal of Solids and Structures vol 35 no 5-6 pp 487ndash5091998

[5] I Vardoulakis J F Labuz E Papamichos and J TronvollldquoContinuum fracture mechanics of uniaxial compression onbrittle materialsrdquo International Journal of Solids and Struc-tures vol 35 no 31-32 pp 4313ndash4335 1998

[6] A Paluszny and S K Matthai ldquoNumerical modeling ofdiscrete multi-crack growth applied to pattern formation ingeological brittle mediardquo International Journal of Solids andStructures vol 46 no 18-19 pp 3383ndash3397 2009

[7] R-H Cao P Cao H Lin C-Z Pu and K Ou ldquoMechanicalbehavior of brittle rock-like specimens with pre-existingfissures under uniaxial loading experimental studies andparticle mechanics approachrdquo Rock Mechanics and RockEngineering vol 49 no 3 pp 763ndash783 2016

[8] X Fan R Chen H Lin H Lai C Zhang and Q ZhaoldquoCracking and failure in rock specimen containing combinedflaw and hole under uniaxial compressionrdquo Advances in CivilEngineering vol 2018 Article ID 9818250 15 pages 2018

[9] D Bing H Guicheng and Z Zhijun ldquoA numerical researchon crack process of gypsum containing single flaw withdifferent angle and length in uniaxial loadingrdquo Shock andVibration vol 2018 Article ID 2968205 16 pages 2018

[10] J Zhang ldquoInvestigation of relation between fracture scale andacoustic emission time-frequency parameters in rocksrdquo Shockand Vibration vol 2018 Article ID 3057628 14 pages 2018

[11] X Meng and W Liu ldquoStudy on failure process and perme-ation evolution of single-cracked rockrdquo Advances in MaterialsScience and Engineering vol 2018 Article ID 79156529 pages 2018

[12] M Sagong D Park J Yoo and J S Lee ldquoExperimental andnumerical analyses of an opening in a jointed rockmass underbiaxial compressionrdquo International Journal of RockMechanicsand Mining Sciences vol 48 no 7 pp 1055ndash1067 2011

[13] W Chen H Konietzky X Tan and T Fruhwirt ldquoPre-failuredamage analysis for brittle rocks under triaxial compressionrdquoComputers and Geotechnics vol 74 pp 45ndash55 2016

[14] S-Q Yang P G Ranjith Y-H Huang et al ldquoExperimentalinvestigation on mechanical damage characteristics ofsandstone under triaxial cyclic loadingrdquo Geophysical JournalInternational vol 201 no 2 pp 662ndash682 2015

[15] N Erarslan ldquoMicrostructural investigation of subcriticalcrack propagation and fracture process zone (FPZ) by the

Advances in Materials Science and Engineering 13

0 20 40 60 80 1000

1

2

3

4

5

6

7

8E 5

0 (G

Pa)

∆σ3 ()

10 MPa8 MPa6 MPa

4 MPa2 MPa

(a)

E 50 (

GPa

)

∆σ3 ()0 20 40 60 80 100

0

1

2

3

4

5

6

7

8

10 MPa8 MPa6 MPa

4 MPa2 MPa

(b)

E 50 (

GPa

)

∆σ3 ()

10 MPa8 MPa6 MPa

4 MPa2 MPa

0 20 40 60 80 1003

4

5

6

7

8

(c)

E 50 (

GPa

)

∆σ3 ()

10 MPa8 MPa6 MPa

4 MPa2 MPa

0 20 40 60 80 1000

4

8

2

16

20

(d)

E 50 (

GPa

)

∆σ3 ()

10 MPa8 MPa6 MPa

4 MPa2 MPa

0 20 40 60 80 1000

2

4

6

8

10

(e)

E 50 (

GPa

)

∆σ3 ()

10 MPa8 MPa6 MPa

4 MPa2 MPa

0 20 40 60 80 1000

2

4

6

8

0

(f )

Figure 5 Continued

Advances in Materials Science and Engineering 7

E 50 (

GPa

)

∆σ3 ()

10 MPa8 MPa6 MPa

4 MPa2 MPa

0 20 40 60 80 1000

2

4

6

8

10

12

14

(g)

E 50 (

GPa

)

∆σ3 ()

10 MPa8 MPa6 MPa

4 MPa2 MPa

0 20 40 60 80 1000

2

4

6

(h)

E 50 (

GPa

)

∆σ3 ()

10 MPa8 MPa6 MPa

4 MPa2 MPa

0 20 40 60 80 1000

2

4

6

8

(i)

Figure 5 Relationship between deformation modulus and unloading percentage (joint orientation defined as β joint connectivity ratedefined as k) (a) β 30deg k 03 (b) β 30deg k 045 (c) β 30deg k 06 (d) β 45deg k 03 (e) β 45deg k 045 (f ) β 45deg k 06 (g) β 60degk 03 (h) β 60deg k 045 (i) β 60deg k 06

ndash

Failure modes

Shear failure

Splitting failure

Mixed failure (tension and shear)

Slide along the joint planeCompletely

Partly

Does not slide along the joint plane

ndash

Figure 6 Failure modes

8 Advances in Materials Science and Engineering

2 MPa 6 MPa4 MPa 8 MPa 10 MPa

(a)

2 MPa 6 MPa4 MPa 8 MPa 10 MPa

(b)

2 MPa 6 MPa4 MPa 8 MPa 10 MPa

(c)

2 MPa 6 MPa4 MPa 8 MPa 10 MPa

(d)

2 MPa 6 MPa4 MPa 8 MPa 10 MPa

(e)

Figure 7 Continued

Advances in Materials Science and Engineering 9

cannot be reconstructed to form intact specimens(Figure 7(d)) e amount of cracks initiated during the testwas also considered as a parameter to characterize the effectof confining pressures Note that the amount of crack isinversely proportional to the confining pressure Here theconfining pressure was designated by σ3 e fractureexhibited a continuous line-type crack at higher confiningpressures (ie 8 and 10megapascals) However the crackshowed a fork-like pattern when the confining pressure was

less than 4MPa At lower confining pressures samples weremore vulnerable for damages during the unloading test ephenomena were explained in terms of the restraint effect ofconfining pressure on the lateral deformation of the spec-imens According to MohrndashCoulomb failure criterion thedeviatoric stress (σ1 minus σ3) will increase when the confiningpressure (σ3) decreases and the failure envelop can be easilyreached Meanwhile internal defects were rapidly propa-gating as the deviatoric stress increased Lastly perforated

2 MPa 6 MPa4 MPa 8 MPa 10 MPa

(f )

2 MPa 6 MPa4 MPa 8 MPa 10 MPa

(g)

2 MPa 6 MPa4 MPa 8 MPa 10 MPa

(h)

2 MPa 6 MPa4 MPa 8 MPa 10 MPa

(i)

Figure 7 Sample failures under different pressures (joint orientation defined as β joint connectivity rate defined as k) (a) β 30deg k 03 (b)β 30deg k 045 (c) β 30deg k 06 (d) β 45deg k 03 (e) β 45deg k 045 (f ) β 45deg k 06 (g) β 60deg k 03 (h) β 60deg k 045 (i)β 60deg k 06

10 Advances in Materials Science and Engineering

cracks emerged On the contrary internal defects weresqueezed by the confining pressure Hence higher values ofconfining pressures can impose a significant restraint againstthe lateral deformations Similarly excavation can beregarded as unloading condition in rock slope engineering

62 Effect of Orientation on the Failure Modes In naturerocks have different orientations of joints which affect theirphysical and mechanical properties In this study the re-lationships between joint orientations and crack propaga-tions were studied Obviously the failure modes for differentjoint orientations were divergent As shown in Figure 6splitting failure was the predominant failure type forspecimens with 30deg joint orientations (Figures 7(a)ndash7(c))Hence joints with mild slopes have relatively small impacton the properties of rocks On the contrary mixed types offailures have been noticed on the specimens with 45deg jointsDuring testing the specimens were fragmented into piecesIn fact it was difficult to trace the failure paths However at60deg joint orientation shear failure was the predominant oneA flat surface was formed as a result of flaw cuttings Henceat 60deg joint orientation dislocations of pieces occurred dueto a noticeable sliding

63 Effect of Joint Connectivity Rate on the Failure ModesFor different joint angles the effect of connectivity rate (k) isdistinct It had a considerable effect when the joint angleswere 45deg or 60deg However at 30deg joint angle the effect of jointconnectivity was relatively small e path of crack prop-agation was twisted when the value of kwas 03 (Figures 7(a)7(d) and 7(g)) is is because of the restraint action ofuncracked section of the specimens against sliding erupture face was rough since it is a result of multiple factorsOn the one hand joint and rock bridge alter or replace eachother in accordance with the connectivity rate On the otherhand the nonlinear variation of tension stress at the tip offlaws exists As is shown in Figure 6 the crack path wasbecoming edge shaped when the value of k increasedMeanwhile the failure modes gradually changed by thejoints Higher values of k yielded the straight plane ofruptures Nonetheless the concentrations of stresses werenoticeable as shown in the development of secondary cracks(Figures 7(e) and 7(f )) e cracks were extended between

the tips of the joints and the edges of the specimens when thejoint angle and the connectivity rate were 45deg and 045respectively For small values of k the crack pattern becamepolygonal However the crack pattern was transforminginto an arc shape when the value of k increased Besides thesecondary cracks became normal to the edge of the speci-mens during shearing At 60deg joint orientation there was noremarkable effect on the failure modes due to the variationsof joint connectivity rate (k) All surfaces of failures werenearly flat Nonetheless there was a slight discrepancy be-tween them Similar failure modes were developed whenjoint orientation and connectivity rate were 45deg and 03respectively e significant effect of joint connectivity ratewas obvious Hence shear failures can easily develop andtheir probability of occurrence depends on the values of kShear failures were developed at higher values of k(Figures 7(h) and 7(i))

Table 3 summarizes different types of failure modes forspecimens with diverse joint orientations and connectivityrates e failure mode was governed by the joint param-eters Fundamentally the failure modes depend on the jointorientations initially For instance splitting failure is un-likely to occur when the joint angle is above 45deg Meanwhileconnectivity rate determines the extent of shear failuree kvalues control the transformation between mixed andcomplete shear failures At 60deg joint orientation both typesof failures were observed e critical value for k was 03 inthis case

7 Discussions of Crack Force Analysis Based onGriffithrsquos Theory

MohrndashCoulomb and Griffith criteria are conventionallyused to interpret the failure of rock specimens It is no-ticeable that the MohrndashCoulomb criterion cannot clarify theinfluence of inside flaws on rock sample failure Howevermaximum tensile stress in the perimeter of Griffithrsquos crack(Figure 8) can be analyzed according to Griffithrsquos theory forthe sake of exploring the effectiveness of the internal defects

Han et al [26] deduced equation (2) a function of α andc considering primary stress of equation (1) and Inglisrsquosformula

σy σ1 + σ3

2+σ1 minus σ3

2cos 2c

σx σ1 + σ3

2minusσ1 minus σ3

2cos 2c

τxy minusσ1 minus σ3

2sin 2c

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(1)

σb σ1

m2 + α2m 1 + α21113872 1113873(1 + λ) +(1 + m) mminus α21113872 1113873(1minus α)cos 2c + α(1 + m)

2(1minus λ)sin 2c1113966 1113967 (2)

Advances in Materials Science and Engineering 11

As shown in the specimen schematic diagram α and thecrack width can be valued as 0degand 1mm respectively sincethe crack is long and thin Hence the following equation isattained from equation (2)

σb σ1m

(1 + λ) +(1 + m)(1minus λ)cos 2c1113864 1113865 (3)

e stress extremum hinges on σ1 λ and m at the tips ofthe crack in the condition of c minus(π2) e equation isobtained as follows

σ0 σ1m

2λminusm(1minus λ) (4)

According to equation (4) the maximum tensile stress atthe tips of flaw was calculated Meanwhile the influences onthe maximum tensile stress were analyzed considering jointinclined angles joint connectivity and confining pressures(Figures 9 and 10) Obviously the maximum tensile stressincreases promptly in accordance with the confining pres-sures Simultaneously higher joint connectivity rates lead toa large stress extremum And the confining pressure enlargesthis trend As it is shown in Figure 10 the escalation of jointorientation boosts the maximum tensile stress at the tips offlaw erefore the alteration of confining pressure is ofmost significant on the variation of the tensile stress whilethe joint angle contributes the minimal impact

With confining pressure (ie λ) declining gradually inthe unloading stage the longitudinal crack may arise due toextremum of tensile stress on the tips and when it satisfiesthe fracture strength the rupture of the longitudinal crackappears along the plane of the crack and may propagatecontinuously to the specimenrsquos failure Considering the axialstress stops expansion in the unloading stage the specimencannot be broken by some slanting cracks or crossed cracks

directly during the compression erefore the splittingfailure coupled with shear failure emerges finally

8 Conclusions

In the preceding pages the crack propagation was studied onthe specimens with partially cut-through joints e primeobjective of this study was to examine the effect of jointorientations and joint connectivity rates on the mechanicalcharacteristics of specimens Even though nonlinear me-chanical characteristics were observed the macroscopicbehavior was studied based upon the different values of jointgeometries Furthermore nine types of specimens were usedto investigate the failure mechanisms at different confiningpressures e confining pressure joint orientation andconnectivity rate were considered as key parameters toanalyze different failure modes Finally Griffithrsquos theory wasused to analyze the maximum tensile stress Nevertheless afurther study is important to extend the application of thisstudy in real engineering problemsemain findings of thispaper are summarized below

A nonlinear correlation was observed among the elas-ticity modulus joint orientation and joint connectivity rateMeanwhile the confining pressure was arithmetically

Y

σy

σx

Xα

γ

σx

σy

σ3σ1

τxy τxyτyx

τyx

Figure 8 Stress state of the crack based on Griffithrsquos theory [26]

0

100

200

300

400

500

600

2 4 6 8 10

σ 0 (M

Pa)

σ3 (MPa)

β = 30deg k = 03β = 30deg k = 045β = 30deg k = 06

Figure 9 e stress extremum with 30deg joint angle under differentconfining pressures

0

200

400

600

800

1000

30 45 60

σ 0 (M

Pa)

β (deg)k = 03k = 045k = 06

Figure 10 e stress extremum under different joint orientationsand connectivity rates

Table 3 Failure modes for different types of samples

Connectivity rateOrientation 03 045 0630 Splitting Splitting Splitting45 Mixed Mixed Mixed60 Mixed Shear Shear

12 Advances in Materials Science and Engineering

increasing During the unloading stage the reduction ofdeformation modulus can be used to predict the rockspecimen failure And the critical value of decrease per-centage is 80 e deformation modulus is less sensitive tothe change of joint connectivity whereas it is more sensitiveto the change of the inclination angle

ree different types (shear split and mixed) of failuremodes were observed e higher values of confiningpressure can restraint the lateral deformation At 30deg jointorientation the predominant failure was splitting failureSimilarly mixed failures were observed at 45deg joint ori-entations because of the concentration of stresses at the tipof the joints On the contrary shear failure was quitecommon due to sufficient length of 60deg joint orientationComparatively the effect of joint connectivity rate wassmaller than that of the joint orientation However therewas a clear variation in the crack propagation when thejoint angle was 60deg e position of joints plays a significantrole for anisotropic failures

A few equations were introduced based on Griffithrsquostheory to evaluate the maximum tensile stress at the tips ofthe crack e results based on the calculation reveal theimpaction of joint characteristics on the stress extremum

Abbreviations

c e angle of principal stress σ1 and Y axisα e central angle of the ellipsem e ratio of the minor axis (b) to major axis (a) of the

ellipseλ e ratio of two principal stresses between σ3 and σ1σ0 e maximum tensile stressσ1 e maximum principal stressσ3 e minimum principal stressσb Tangential stress on the perimeter of a slender ellipse

Data Availability

e data used to support the findings of this study may bereleased upon application to the China ree GorgesUniversity Review Board which can be done by contactingthe author ldquoGuoyong Duan (dgyhhueducn)rdquo

Conflicts of Interest

e authors wish to confirm that there are no knownconflicts of interest associated with this publication andthere has been no significant financial support for this workthat could have influenced its outcome Accordingly thealteration of confining pressure is of most significance onthe variation of the tensile stress while the joint anglecontributes the minimal impact

Acknowledgments

e authors gratefully acknowledge the financial supportfrom the National Natural Science Foundation of China(Grant nos 51439003 and 51279091) the Fundamental

Research Funds for the Central Universities (Hohai Uni-versity) (Grant no 2016B43014) Open Fund of Key Lab-oratory of Geological Hazards on ree Gorges ReservoirArea (China ree Gorges University) (Grant no2015KDZ12) and Key Project of Scientific Research Plan ofHubei Provincial Department of Education (Grant noD20181202)

References

[1] T Liu P Cao and H Lin ldquoEvolution procedure of multiplerock cracks under seepage pressurerdquo Mathematical Problemsin Engineering vol 2013 Article ID 738013 11 pages 2013

[2] E Hoek ldquoStrength of jointed rock massesrdquo Geotechniquevol 33 no 3 pp 187ndash223 1983

[3] E Hoek and Z T Bieniawski ldquoBrittle fracture propagation inrock under compressionrdquo International Journal of Fracturevol 26 no 4 pp 276ndash294 1984

[4] M Basista and D Gross ldquoe sliding crack model of brittledeformation an internal variable approachrdquo InternationalJournal of Solids and Structures vol 35 no 5-6 pp 487ndash5091998

[5] I Vardoulakis J F Labuz E Papamichos and J TronvollldquoContinuum fracture mechanics of uniaxial compression onbrittle materialsrdquo International Journal of Solids and Struc-tures vol 35 no 31-32 pp 4313ndash4335 1998

[6] A Paluszny and S K Matthai ldquoNumerical modeling ofdiscrete multi-crack growth applied to pattern formation ingeological brittle mediardquo International Journal of Solids andStructures vol 46 no 18-19 pp 3383ndash3397 2009

[7] R-H Cao P Cao H Lin C-Z Pu and K Ou ldquoMechanicalbehavior of brittle rock-like specimens with pre-existingfissures under uniaxial loading experimental studies andparticle mechanics approachrdquo Rock Mechanics and RockEngineering vol 49 no 3 pp 763ndash783 2016

[8] X Fan R Chen H Lin H Lai C Zhang and Q ZhaoldquoCracking and failure in rock specimen containing combinedflaw and hole under uniaxial compressionrdquo Advances in CivilEngineering vol 2018 Article ID 9818250 15 pages 2018

[9] D Bing H Guicheng and Z Zhijun ldquoA numerical researchon crack process of gypsum containing single flaw withdifferent angle and length in uniaxial loadingrdquo Shock andVibration vol 2018 Article ID 2968205 16 pages 2018

[10] J Zhang ldquoInvestigation of relation between fracture scale andacoustic emission time-frequency parameters in rocksrdquo Shockand Vibration vol 2018 Article ID 3057628 14 pages 2018

[11] X Meng and W Liu ldquoStudy on failure process and perme-ation evolution of single-cracked rockrdquo Advances in MaterialsScience and Engineering vol 2018 Article ID 79156529 pages 2018

[12] M Sagong D Park J Yoo and J S Lee ldquoExperimental andnumerical analyses of an opening in a jointed rockmass underbiaxial compressionrdquo International Journal of RockMechanicsand Mining Sciences vol 48 no 7 pp 1055ndash1067 2011

[13] W Chen H Konietzky X Tan and T Fruhwirt ldquoPre-failuredamage analysis for brittle rocks under triaxial compressionrdquoComputers and Geotechnics vol 74 pp 45ndash55 2016

[14] S-Q Yang P G Ranjith Y-H Huang et al ldquoExperimentalinvestigation on mechanical damage characteristics ofsandstone under triaxial cyclic loadingrdquo Geophysical JournalInternational vol 201 no 2 pp 662ndash682 2015

[15] N Erarslan ldquoMicrostructural investigation of subcriticalcrack propagation and fracture process zone (FPZ) by the

Advances in Materials Science and Engineering 13

E 50 (

GPa

)

∆σ3 ()

10 MPa8 MPa6 MPa

4 MPa2 MPa

0 20 40 60 80 1000

2

4

6

8

10

12

14

(g)

E 50 (

GPa

)

∆σ3 ()

10 MPa8 MPa6 MPa

4 MPa2 MPa

0 20 40 60 80 1000

2

4

6

(h)

E 50 (

GPa

)

∆σ3 ()

10 MPa8 MPa6 MPa

4 MPa2 MPa

0 20 40 60 80 1000

2

4

6

8

(i)

Figure 5 Relationship between deformation modulus and unloading percentage (joint orientation defined as β joint connectivity ratedefined as k) (a) β 30deg k 03 (b) β 30deg k 045 (c) β 30deg k 06 (d) β 45deg k 03 (e) β 45deg k 045 (f ) β 45deg k 06 (g) β 60degk 03 (h) β 60deg k 045 (i) β 60deg k 06

ndash

Failure modes

Shear failure

Splitting failure

Mixed failure (tension and shear)

Slide along the joint planeCompletely

Partly

Does not slide along the joint plane

ndash

Figure 6 Failure modes

8 Advances in Materials Science and Engineering

2 MPa 6 MPa4 MPa 8 MPa 10 MPa

(a)

2 MPa 6 MPa4 MPa 8 MPa 10 MPa

(b)

2 MPa 6 MPa4 MPa 8 MPa 10 MPa

(c)

2 MPa 6 MPa4 MPa 8 MPa 10 MPa

(d)

2 MPa 6 MPa4 MPa 8 MPa 10 MPa

(e)

Figure 7 Continued

Advances in Materials Science and Engineering 9

cannot be reconstructed to form intact specimens(Figure 7(d)) e amount of cracks initiated during the testwas also considered as a parameter to characterize the effectof confining pressures Note that the amount of crack isinversely proportional to the confining pressure Here theconfining pressure was designated by σ3 e fractureexhibited a continuous line-type crack at higher confiningpressures (ie 8 and 10megapascals) However the crackshowed a fork-like pattern when the confining pressure was

less than 4MPa At lower confining pressures samples weremore vulnerable for damages during the unloading test ephenomena were explained in terms of the restraint effect ofconfining pressure on the lateral deformation of the spec-imens According to MohrndashCoulomb failure criterion thedeviatoric stress (σ1 minus σ3) will increase when the confiningpressure (σ3) decreases and the failure envelop can be easilyreached Meanwhile internal defects were rapidly propa-gating as the deviatoric stress increased Lastly perforated

2 MPa 6 MPa4 MPa 8 MPa 10 MPa

(f )

2 MPa 6 MPa4 MPa 8 MPa 10 MPa

(g)

2 MPa 6 MPa4 MPa 8 MPa 10 MPa

(h)

2 MPa 6 MPa4 MPa 8 MPa 10 MPa

(i)

Figure 7 Sample failures under different pressures (joint orientation defined as β joint connectivity rate defined as k) (a) β 30deg k 03 (b)β 30deg k 045 (c) β 30deg k 06 (d) β 45deg k 03 (e) β 45deg k 045 (f ) β 45deg k 06 (g) β 60deg k 03 (h) β 60deg k 045 (i)β 60deg k 06

10 Advances in Materials Science and Engineering

cracks emerged On the contrary internal defects weresqueezed by the confining pressure Hence higher values ofconfining pressures can impose a significant restraint againstthe lateral deformations Similarly excavation can beregarded as unloading condition in rock slope engineering

62 Effect of Orientation on the Failure Modes In naturerocks have different orientations of joints which affect theirphysical and mechanical properties In this study the re-lationships between joint orientations and crack propaga-tions were studied Obviously the failure modes for differentjoint orientations were divergent As shown in Figure 6splitting failure was the predominant failure type forspecimens with 30deg joint orientations (Figures 7(a)ndash7(c))Hence joints with mild slopes have relatively small impacton the properties of rocks On the contrary mixed types offailures have been noticed on the specimens with 45deg jointsDuring testing the specimens were fragmented into piecesIn fact it was difficult to trace the failure paths However at60deg joint orientation shear failure was the predominant oneA flat surface was formed as a result of flaw cuttings Henceat 60deg joint orientation dislocations of pieces occurred dueto a noticeable sliding

63 Effect of Joint Connectivity Rate on the Failure ModesFor different joint angles the effect of connectivity rate (k) isdistinct It had a considerable effect when the joint angleswere 45deg or 60deg However at 30deg joint angle the effect of jointconnectivity was relatively small e path of crack prop-agation was twisted when the value of kwas 03 (Figures 7(a)7(d) and 7(g)) is is because of the restraint action ofuncracked section of the specimens against sliding erupture face was rough since it is a result of multiple factorsOn the one hand joint and rock bridge alter or replace eachother in accordance with the connectivity rate On the otherhand the nonlinear variation of tension stress at the tip offlaws exists As is shown in Figure 6 the crack path wasbecoming edge shaped when the value of k increasedMeanwhile the failure modes gradually changed by thejoints Higher values of k yielded the straight plane ofruptures Nonetheless the concentrations of stresses werenoticeable as shown in the development of secondary cracks(Figures 7(e) and 7(f )) e cracks were extended between

the tips of the joints and the edges of the specimens when thejoint angle and the connectivity rate were 45deg and 045respectively For small values of k the crack pattern becamepolygonal However the crack pattern was transforminginto an arc shape when the value of k increased Besides thesecondary cracks became normal to the edge of the speci-mens during shearing At 60deg joint orientation there was noremarkable effect on the failure modes due to the variationsof joint connectivity rate (k) All surfaces of failures werenearly flat Nonetheless there was a slight discrepancy be-tween them Similar failure modes were developed whenjoint orientation and connectivity rate were 45deg and 03respectively e significant effect of joint connectivity ratewas obvious Hence shear failures can easily develop andtheir probability of occurrence depends on the values of kShear failures were developed at higher values of k(Figures 7(h) and 7(i))

Table 3 summarizes different types of failure modes forspecimens with diverse joint orientations and connectivityrates e failure mode was governed by the joint param-eters Fundamentally the failure modes depend on the jointorientations initially For instance splitting failure is un-likely to occur when the joint angle is above 45deg Meanwhileconnectivity rate determines the extent of shear failuree kvalues control the transformation between mixed andcomplete shear failures At 60deg joint orientation both typesof failures were observed e critical value for k was 03 inthis case

7 Discussions of Crack Force Analysis Based onGriffithrsquos Theory

MohrndashCoulomb and Griffith criteria are conventionallyused to interpret the failure of rock specimens It is no-ticeable that the MohrndashCoulomb criterion cannot clarify theinfluence of inside flaws on rock sample failure Howevermaximum tensile stress in the perimeter of Griffithrsquos crack(Figure 8) can be analyzed according to Griffithrsquos theory forthe sake of exploring the effectiveness of the internal defects

Han et al [26] deduced equation (2) a function of α andc considering primary stress of equation (1) and Inglisrsquosformula

σy σ1 + σ3

2+σ1 minus σ3

2cos 2c

σx σ1 + σ3

2minusσ1 minus σ3

2cos 2c

τxy minusσ1 minus σ3

2sin 2c

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(1)

σb σ1

m2 + α2m 1 + α21113872 1113873(1 + λ) +(1 + m) mminus α21113872 1113873(1minus α)cos 2c + α(1 + m)

2(1minus λ)sin 2c1113966 1113967 (2)

Advances in Materials Science and Engineering 11

As shown in the specimen schematic diagram α and thecrack width can be valued as 0degand 1mm respectively sincethe crack is long and thin Hence the following equation isattained from equation (2)

σb σ1m

(1 + λ) +(1 + m)(1minus λ)cos 2c1113864 1113865 (3)

e stress extremum hinges on σ1 λ and m at the tips ofthe crack in the condition of c minus(π2) e equation isobtained as follows

σ0 σ1m

2λminusm(1minus λ) (4)

According to equation (4) the maximum tensile stress atthe tips of flaw was calculated Meanwhile the influences onthe maximum tensile stress were analyzed considering jointinclined angles joint connectivity and confining pressures(Figures 9 and 10) Obviously the maximum tensile stressincreases promptly in accordance with the confining pres-sures Simultaneously higher joint connectivity rates lead toa large stress extremum And the confining pressure enlargesthis trend As it is shown in Figure 10 the escalation of jointorientation boosts the maximum tensile stress at the tips offlaw erefore the alteration of confining pressure is ofmost significant on the variation of the tensile stress whilethe joint angle contributes the minimal impact

With confining pressure (ie λ) declining gradually inthe unloading stage the longitudinal crack may arise due toextremum of tensile stress on the tips and when it satisfiesthe fracture strength the rupture of the longitudinal crackappears along the plane of the crack and may propagatecontinuously to the specimenrsquos failure Considering the axialstress stops expansion in the unloading stage the specimencannot be broken by some slanting cracks or crossed cracks

directly during the compression erefore the splittingfailure coupled with shear failure emerges finally

8 Conclusions

In the preceding pages the crack propagation was studied onthe specimens with partially cut-through joints e primeobjective of this study was to examine the effect of jointorientations and joint connectivity rates on the mechanicalcharacteristics of specimens Even though nonlinear me-chanical characteristics were observed the macroscopicbehavior was studied based upon the different values of jointgeometries Furthermore nine types of specimens were usedto investigate the failure mechanisms at different confiningpressures e confining pressure joint orientation andconnectivity rate were considered as key parameters toanalyze different failure modes Finally Griffithrsquos theory wasused to analyze the maximum tensile stress Nevertheless afurther study is important to extend the application of thisstudy in real engineering problemsemain findings of thispaper are summarized below

A nonlinear correlation was observed among the elas-ticity modulus joint orientation and joint connectivity rateMeanwhile the confining pressure was arithmetically

Y

σy

σx

Xα

γ

σx

σy

σ3σ1

τxy τxyτyx

τyx

Figure 8 Stress state of the crack based on Griffithrsquos theory [26]

0

100

200

300

400

500

600

2 4 6 8 10

σ 0 (M

Pa)

σ3 (MPa)

β = 30deg k = 03β = 30deg k = 045β = 30deg k = 06

Figure 9 e stress extremum with 30deg joint angle under differentconfining pressures

0

200

400

600

800

1000

30 45 60

σ 0 (M

Pa)

β (deg)k = 03k = 045k = 06

Figure 10 e stress extremum under different joint orientationsand connectivity rates

Table 3 Failure modes for different types of samples

Connectivity rateOrientation 03 045 0630 Splitting Splitting Splitting45 Mixed Mixed Mixed60 Mixed Shear Shear

12 Advances in Materials Science and Engineering

increasing During the unloading stage the reduction ofdeformation modulus can be used to predict the rockspecimen failure And the critical value of decrease per-centage is 80 e deformation modulus is less sensitive tothe change of joint connectivity whereas it is more sensitiveto the change of the inclination angle

ree different types (shear split and mixed) of failuremodes were observed e higher values of confiningpressure can restraint the lateral deformation At 30deg jointorientation the predominant failure was splitting failureSimilarly mixed failures were observed at 45deg joint ori-entations because of the concentration of stresses at the tipof the joints On the contrary shear failure was quitecommon due to sufficient length of 60deg joint orientationComparatively the effect of joint connectivity rate wassmaller than that of the joint orientation However therewas a clear variation in the crack propagation when thejoint angle was 60deg e position of joints plays a significantrole for anisotropic failures

A few equations were introduced based on Griffithrsquostheory to evaluate the maximum tensile stress at the tips ofthe crack e results based on the calculation reveal theimpaction of joint characteristics on the stress extremum

Abbreviations

c e angle of principal stress σ1 and Y axisα e central angle of the ellipsem e ratio of the minor axis (b) to major axis (a) of the

ellipseλ e ratio of two principal stresses between σ3 and σ1σ0 e maximum tensile stressσ1 e maximum principal stressσ3 e minimum principal stressσb Tangential stress on the perimeter of a slender ellipse

Data Availability

e data used to support the findings of this study may bereleased upon application to the China ree GorgesUniversity Review Board which can be done by contactingthe author ldquoGuoyong Duan (dgyhhueducn)rdquo

Conflicts of Interest

e authors wish to confirm that there are no knownconflicts of interest associated with this publication andthere has been no significant financial support for this workthat could have influenced its outcome Accordingly thealteration of confining pressure is of most significance onthe variation of the tensile stress while the joint anglecontributes the minimal impact

Acknowledgments

e authors gratefully acknowledge the financial supportfrom the National Natural Science Foundation of China(Grant nos 51439003 and 51279091) the Fundamental

Research Funds for the Central Universities (Hohai Uni-versity) (Grant no 2016B43014) Open Fund of Key Lab-oratory of Geological Hazards on ree Gorges ReservoirArea (China ree Gorges University) (Grant no2015KDZ12) and Key Project of Scientific Research Plan ofHubei Provincial Department of Education (Grant noD20181202)

References

[1] T Liu P Cao and H Lin ldquoEvolution procedure of multiplerock cracks under seepage pressurerdquo Mathematical Problemsin Engineering vol 2013 Article ID 738013 11 pages 2013

[2] E Hoek ldquoStrength of jointed rock massesrdquo Geotechniquevol 33 no 3 pp 187ndash223 1983

[3] E Hoek and Z T Bieniawski ldquoBrittle fracture propagation inrock under compressionrdquo International Journal of Fracturevol 26 no 4 pp 276ndash294 1984

[4] M Basista and D Gross ldquoe sliding crack model of brittledeformation an internal variable approachrdquo InternationalJournal of Solids and Structures vol 35 no 5-6 pp 487ndash5091998

[5] I Vardoulakis J F Labuz E Papamichos and J TronvollldquoContinuum fracture mechanics of uniaxial compression onbrittle materialsrdquo International Journal of Solids and Struc-tures vol 35 no 31-32 pp 4313ndash4335 1998

[6] A Paluszny and S K Matthai ldquoNumerical modeling ofdiscrete multi-crack growth applied to pattern formation ingeological brittle mediardquo International Journal of Solids andStructures vol 46 no 18-19 pp 3383ndash3397 2009

[7] R-H Cao P Cao H Lin C-Z Pu and K Ou ldquoMechanicalbehavior of brittle rock-like specimens with pre-existingfissures under uniaxial loading experimental studies andparticle mechanics approachrdquo Rock Mechanics and RockEngineering vol 49 no 3 pp 763ndash783 2016

[8] X Fan R Chen H Lin H Lai C Zhang and Q ZhaoldquoCracking and failure in rock specimen containing combinedflaw and hole under uniaxial compressionrdquo Advances in CivilEngineering vol 2018 Article ID 9818250 15 pages 2018

[9] D Bing H Guicheng and Z Zhijun ldquoA numerical researchon crack process of gypsum containing single flaw withdifferent angle and length in uniaxial loadingrdquo Shock andVibration vol 2018 Article ID 2968205 16 pages 2018

[10] J Zhang ldquoInvestigation of relation between fracture scale andacoustic emission time-frequency parameters in rocksrdquo Shockand Vibration vol 2018 Article ID 3057628 14 pages 2018

[11] X Meng and W Liu ldquoStudy on failure process and perme-ation evolution of single-cracked rockrdquo Advances in MaterialsScience and Engineering vol 2018 Article ID 79156529 pages 2018

[12] M Sagong D Park J Yoo and J S Lee ldquoExperimental andnumerical analyses of an opening in a jointed rockmass underbiaxial compressionrdquo International Journal of RockMechanicsand Mining Sciences vol 48 no 7 pp 1055ndash1067 2011

[13] W Chen H Konietzky X Tan and T Fruhwirt ldquoPre-failuredamage analysis for brittle rocks under triaxial compressionrdquoComputers and Geotechnics vol 74 pp 45ndash55 2016

[14] S-Q Yang P G Ranjith Y-H Huang et al ldquoExperimentalinvestigation on mechanical damage characteristics ofsandstone under triaxial cyclic loadingrdquo Geophysical JournalInternational vol 201 no 2 pp 662ndash682 2015

[15] N Erarslan ldquoMicrostructural investigation of subcriticalcrack propagation and fracture process zone (FPZ) by the

Advances in Materials Science and Engineering 13

2 MPa 6 MPa4 MPa 8 MPa 10 MPa

(a)

2 MPa 6 MPa4 MPa 8 MPa 10 MPa

(b)

2 MPa 6 MPa4 MPa 8 MPa 10 MPa

(c)

2 MPa 6 MPa4 MPa 8 MPa 10 MPa

(d)

2 MPa 6 MPa4 MPa 8 MPa 10 MPa

(e)

Figure 7 Continued

Advances in Materials Science and Engineering 9

cannot be reconstructed to form intact specimens(Figure 7(d)) e amount of cracks initiated during the testwas also considered as a parameter to characterize the effectof confining pressures Note that the amount of crack isinversely proportional to the confining pressure Here theconfining pressure was designated by σ3 e fractureexhibited a continuous line-type crack at higher confiningpressures (ie 8 and 10megapascals) However the crackshowed a fork-like pattern when the confining pressure was

less than 4MPa At lower confining pressures samples weremore vulnerable for damages during the unloading test ephenomena were explained in terms of the restraint effect ofconfining pressure on the lateral deformation of the spec-imens According to MohrndashCoulomb failure criterion thedeviatoric stress (σ1 minus σ3) will increase when the confiningpressure (σ3) decreases and the failure envelop can be easilyreached Meanwhile internal defects were rapidly propa-gating as the deviatoric stress increased Lastly perforated

2 MPa 6 MPa4 MPa 8 MPa 10 MPa

(f )

2 MPa 6 MPa4 MPa 8 MPa 10 MPa

(g)

2 MPa 6 MPa4 MPa 8 MPa 10 MPa

(h)

2 MPa 6 MPa4 MPa 8 MPa 10 MPa

(i)

Figure 7 Sample failures under different pressures (joint orientation defined as β joint connectivity rate defined as k) (a) β 30deg k 03 (b)β 30deg k 045 (c) β 30deg k 06 (d) β 45deg k 03 (e) β 45deg k 045 (f ) β 45deg k 06 (g) β 60deg k 03 (h) β 60deg k 045 (i)β 60deg k 06

10 Advances in Materials Science and Engineering

cracks emerged On the contrary internal defects weresqueezed by the confining pressure Hence higher values ofconfining pressures can impose a significant restraint againstthe lateral deformations Similarly excavation can beregarded as unloading condition in rock slope engineering

62 Effect of Orientation on the Failure Modes In naturerocks have different orientations of joints which affect theirphysical and mechanical properties In this study the re-lationships between joint orientations and crack propaga-tions were studied Obviously the failure modes for differentjoint orientations were divergent As shown in Figure 6splitting failure was the predominant failure type forspecimens with 30deg joint orientations (Figures 7(a)ndash7(c))Hence joints with mild slopes have relatively small impacton the properties of rocks On the contrary mixed types offailures have been noticed on the specimens with 45deg jointsDuring testing the specimens were fragmented into piecesIn fact it was difficult to trace the failure paths However at60deg joint orientation shear failure was the predominant oneA flat surface was formed as a result of flaw cuttings Henceat 60deg joint orientation dislocations of pieces occurred dueto a noticeable sliding

63 Effect of Joint Connectivity Rate on the Failure ModesFor different joint angles the effect of connectivity rate (k) isdistinct It had a considerable effect when the joint angleswere 45deg or 60deg However at 30deg joint angle the effect of jointconnectivity was relatively small e path of crack prop-agation was twisted when the value of kwas 03 (Figures 7(a)7(d) and 7(g)) is is because of the restraint action ofuncracked section of the specimens against sliding erupture face was rough since it is a result of multiple factorsOn the one hand joint and rock bridge alter or replace eachother in accordance with the connectivity rate On the otherhand the nonlinear variation of tension stress at the tip offlaws exists As is shown in Figure 6 the crack path wasbecoming edge shaped when the value of k increasedMeanwhile the failure modes gradually changed by thejoints Higher values of k yielded the straight plane ofruptures Nonetheless the concentrations of stresses werenoticeable as shown in the development of secondary cracks(Figures 7(e) and 7(f )) e cracks were extended between

the tips of the joints and the edges of the specimens when thejoint angle and the connectivity rate were 45deg and 045respectively For small values of k the crack pattern becamepolygonal However the crack pattern was transforminginto an arc shape when the value of k increased Besides thesecondary cracks became normal to the edge of the speci-mens during shearing At 60deg joint orientation there was noremarkable effect on the failure modes due to the variationsof joint connectivity rate (k) All surfaces of failures werenearly flat Nonetheless there was a slight discrepancy be-tween them Similar failure modes were developed whenjoint orientation and connectivity rate were 45deg and 03respectively e significant effect of joint connectivity ratewas obvious Hence shear failures can easily develop andtheir probability of occurrence depends on the values of kShear failures were developed at higher values of k(Figures 7(h) and 7(i))

Table 3 summarizes different types of failure modes forspecimens with diverse joint orientations and connectivityrates e failure mode was governed by the joint param-eters Fundamentally the failure modes depend on the jointorientations initially For instance splitting failure is un-likely to occur when the joint angle is above 45deg Meanwhileconnectivity rate determines the extent of shear failuree kvalues control the transformation between mixed andcomplete shear failures At 60deg joint orientation both typesof failures were observed e critical value for k was 03 inthis case

7 Discussions of Crack Force Analysis Based onGriffithrsquos Theory

MohrndashCoulomb and Griffith criteria are conventionallyused to interpret the failure of rock specimens It is no-ticeable that the MohrndashCoulomb criterion cannot clarify theinfluence of inside flaws on rock sample failure Howevermaximum tensile stress in the perimeter of Griffithrsquos crack(Figure 8) can be analyzed according to Griffithrsquos theory forthe sake of exploring the effectiveness of the internal defects

Han et al [26] deduced equation (2) a function of α andc considering primary stress of equation (1) and Inglisrsquosformula

σy σ1 + σ3

2+σ1 minus σ3

2cos 2c

σx σ1 + σ3

2minusσ1 minus σ3

2cos 2c

τxy minusσ1 minus σ3

2sin 2c

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(1)

σb σ1

m2 + α2m 1 + α21113872 1113873(1 + λ) +(1 + m) mminus α21113872 1113873(1minus α)cos 2c + α(1 + m)

2(1minus λ)sin 2c1113966 1113967 (2)

Advances in Materials Science and Engineering 11

As shown in the specimen schematic diagram α and thecrack width can be valued as 0degand 1mm respectively sincethe crack is long and thin Hence the following equation isattained from equation (2)

σb σ1m

(1 + λ) +(1 + m)(1minus λ)cos 2c1113864 1113865 (3)

e stress extremum hinges on σ1 λ and m at the tips ofthe crack in the condition of c minus(π2) e equation isobtained as follows

σ0 σ1m

2λminusm(1minus λ) (4)

According to equation (4) the maximum tensile stress atthe tips of flaw was calculated Meanwhile the influences onthe maximum tensile stress were analyzed considering jointinclined angles joint connectivity and confining pressures(Figures 9 and 10) Obviously the maximum tensile stressincreases promptly in accordance with the confining pres-sures Simultaneously higher joint connectivity rates lead toa large stress extremum And the confining pressure enlargesthis trend As it is shown in Figure 10 the escalation of jointorientation boosts the maximum tensile stress at the tips offlaw erefore the alteration of confining pressure is ofmost significant on the variation of the tensile stress whilethe joint angle contributes the minimal impact

With confining pressure (ie λ) declining gradually inthe unloading stage the longitudinal crack may arise due toextremum of tensile stress on the tips and when it satisfiesthe fracture strength the rupture of the longitudinal crackappears along the plane of the crack and may propagatecontinuously to the specimenrsquos failure Considering the axialstress stops expansion in the unloading stage the specimencannot be broken by some slanting cracks or crossed cracks

directly during the compression erefore the splittingfailure coupled with shear failure emerges finally

8 Conclusions

In the preceding pages the crack propagation was studied onthe specimens with partially cut-through joints e primeobjective of this study was to examine the effect of jointorientations and joint connectivity rates on the mechanicalcharacteristics of specimens Even though nonlinear me-chanical characteristics were observed the macroscopicbehavior was studied based upon the different values of jointgeometries Furthermore nine types of specimens were usedto investigate the failure mechanisms at different confiningpressures e confining pressure joint orientation andconnectivity rate were considered as key parameters toanalyze different failure modes Finally Griffithrsquos theory wasused to analyze the maximum tensile stress Nevertheless afurther study is important to extend the application of thisstudy in real engineering problemsemain findings of thispaper are summarized below

A nonlinear correlation was observed among the elas-ticity modulus joint orientation and joint connectivity rateMeanwhile the confining pressure was arithmetically

Y

σy

σx

Xα

γ

σx

σy

σ3σ1

τxy τxyτyx

τyx

Figure 8 Stress state of the crack based on Griffithrsquos theory [26]

0

100

200

300

400

500

600

2 4 6 8 10

σ 0 (M

Pa)

σ3 (MPa)

β = 30deg k = 03β = 30deg k = 045β = 30deg k = 06

Figure 9 e stress extremum with 30deg joint angle under differentconfining pressures

0

200

400

600

800

1000

30 45 60

σ 0 (M

Pa)

β (deg)k = 03k = 045k = 06

Figure 10 e stress extremum under different joint orientationsand connectivity rates

Table 3 Failure modes for different types of samples

Connectivity rateOrientation 03 045 0630 Splitting Splitting Splitting45 Mixed Mixed Mixed60 Mixed Shear Shear

12 Advances in Materials Science and Engineering

increasing During the unloading stage the reduction ofdeformation modulus can be used to predict the rockspecimen failure And the critical value of decrease per-centage is 80 e deformation modulus is less sensitive tothe change of joint connectivity whereas it is more sensitiveto the change of the inclination angle

ree different types (shear split and mixed) of failuremodes were observed e higher values of confiningpressure can restraint the lateral deformation At 30deg jointorientation the predominant failure was splitting failureSimilarly mixed failures were observed at 45deg joint ori-entations because of the concentration of stresses at the tipof the joints On the contrary shear failure was quitecommon due to sufficient length of 60deg joint orientationComparatively the effect of joint connectivity rate wassmaller than that of the joint orientation However therewas a clear variation in the crack propagation when thejoint angle was 60deg e position of joints plays a significantrole for anisotropic failures

A few equations were introduced based on Griffithrsquostheory to evaluate the maximum tensile stress at the tips ofthe crack e results based on the calculation reveal theimpaction of joint characteristics on the stress extremum

Abbreviations

c e angle of principal stress σ1 and Y axisα e central angle of the ellipsem e ratio of the minor axis (b) to major axis (a) of the

ellipseλ e ratio of two principal stresses between σ3 and σ1σ0 e maximum tensile stressσ1 e maximum principal stressσ3 e minimum principal stressσb Tangential stress on the perimeter of a slender ellipse

Data Availability

e data used to support the findings of this study may bereleased upon application to the China ree GorgesUniversity Review Board which can be done by contactingthe author ldquoGuoyong Duan (dgyhhueducn)rdquo

Conflicts of Interest

e authors wish to confirm that there are no knownconflicts of interest associated with this publication andthere has been no significant financial support for this workthat could have influenced its outcome Accordingly thealteration of confining pressure is of most significance onthe variation of the tensile stress while the joint anglecontributes the minimal impact

Acknowledgments

e authors gratefully acknowledge the financial supportfrom the National Natural Science Foundation of China(Grant nos 51439003 and 51279091) the Fundamental

Research Funds for the Central Universities (Hohai Uni-versity) (Grant no 2016B43014) Open Fund of Key Lab-oratory of Geological Hazards on ree Gorges ReservoirArea (China ree Gorges University) (Grant no2015KDZ12) and Key Project of Scientific Research Plan ofHubei Provincial Department of Education (Grant noD20181202)

References

[1] T Liu P Cao and H Lin ldquoEvolution procedure of multiplerock cracks under seepage pressurerdquo Mathematical Problemsin Engineering vol 2013 Article ID 738013 11 pages 2013

[2] E Hoek ldquoStrength of jointed rock massesrdquo Geotechniquevol 33 no 3 pp 187ndash223 1983

[3] E Hoek and Z T Bieniawski ldquoBrittle fracture propagation inrock under compressionrdquo International Journal of Fracturevol 26 no 4 pp 276ndash294 1984

[4] M Basista and D Gross ldquoe sliding crack model of brittledeformation an internal variable approachrdquo InternationalJournal of Solids and Structures vol 35 no 5-6 pp 487ndash5091998

[5] I Vardoulakis J F Labuz E Papamichos and J TronvollldquoContinuum fracture mechanics of uniaxial compression onbrittle materialsrdquo International Journal of Solids and Struc-tures vol 35 no 31-32 pp 4313ndash4335 1998

[6] A Paluszny and S K Matthai ldquoNumerical modeling ofdiscrete multi-crack growth applied to pattern formation ingeological brittle mediardquo International Journal of Solids andStructures vol 46 no 18-19 pp 3383ndash3397 2009

[7] R-H Cao P Cao H Lin C-Z Pu and K Ou ldquoMechanicalbehavior of brittle rock-like specimens with pre-existingfissures under uniaxial loading experimental studies andparticle mechanics approachrdquo Rock Mechanics and RockEngineering vol 49 no 3 pp 763ndash783 2016

[8] X Fan R Chen H Lin H Lai C Zhang and Q ZhaoldquoCracking and failure in rock specimen containing combinedflaw and hole under uniaxial compressionrdquo Advances in CivilEngineering vol 2018 Article ID 9818250 15 pages 2018

[9] D Bing H Guicheng and Z Zhijun ldquoA numerical researchon crack process of gypsum containing single flaw withdifferent angle and length in uniaxial loadingrdquo Shock andVibration vol 2018 Article ID 2968205 16 pages 2018

[10] J Zhang ldquoInvestigation of relation between fracture scale andacoustic emission time-frequency parameters in rocksrdquo Shockand Vibration vol 2018 Article ID 3057628 14 pages 2018

[11] X Meng and W Liu ldquoStudy on failure process and perme-ation evolution of single-cracked rockrdquo Advances in MaterialsScience and Engineering vol 2018 Article ID 79156529 pages 2018

[12] M Sagong D Park J Yoo and J S Lee ldquoExperimental andnumerical analyses of an opening in a jointed rockmass underbiaxial compressionrdquo International Journal of RockMechanicsand Mining Sciences vol 48 no 7 pp 1055ndash1067 2011

[13] W Chen H Konietzky X Tan and T Fruhwirt ldquoPre-failuredamage analysis for brittle rocks under triaxial compressionrdquoComputers and Geotechnics vol 74 pp 45ndash55 2016

[14] S-Q Yang P G Ranjith Y-H Huang et al ldquoExperimentalinvestigation on mechanical damage characteristics ofsandstone under triaxial cyclic loadingrdquo Geophysical JournalInternational vol 201 no 2 pp 662ndash682 2015

[15] N Erarslan ldquoMicrostructural investigation of subcriticalcrack propagation and fracture process zone (FPZ) by the

Advances in Materials Science and Engineering 13

cannot be reconstructed to form intact specimens(Figure 7(d)) e amount of cracks initiated during the testwas also considered as a parameter to characterize the effectof confining pressures Note that the amount of crack isinversely proportional to the confining pressure Here theconfining pressure was designated by σ3 e fractureexhibited a continuous line-type crack at higher confiningpressures (ie 8 and 10megapascals) However the crackshowed a fork-like pattern when the confining pressure was

less than 4MPa At lower confining pressures samples weremore vulnerable for damages during the unloading test ephenomena were explained in terms of the restraint effect ofconfining pressure on the lateral deformation of the spec-imens According to MohrndashCoulomb failure criterion thedeviatoric stress (σ1 minus σ3) will increase when the confiningpressure (σ3) decreases and the failure envelop can be easilyreached Meanwhile internal defects were rapidly propa-gating as the deviatoric stress increased Lastly perforated

2 MPa 6 MPa4 MPa 8 MPa 10 MPa

(f )

2 MPa 6 MPa4 MPa 8 MPa 10 MPa

(g)

2 MPa 6 MPa4 MPa 8 MPa 10 MPa

(h)

2 MPa 6 MPa4 MPa 8 MPa 10 MPa

(i)

Figure 7 Sample failures under different pressures (joint orientation defined as β joint connectivity rate defined as k) (a) β 30deg k 03 (b)β 30deg k 045 (c) β 30deg k 06 (d) β 45deg k 03 (e) β 45deg k 045 (f ) β 45deg k 06 (g) β 60deg k 03 (h) β 60deg k 045 (i)β 60deg k 06

10 Advances in Materials Science and Engineering

cracks emerged On the contrary internal defects weresqueezed by the confining pressure Hence higher values ofconfining pressures can impose a significant restraint againstthe lateral deformations Similarly excavation can beregarded as unloading condition in rock slope engineering

62 Effect of Orientation on the Failure Modes In naturerocks have different orientations of joints which affect theirphysical and mechanical properties In this study the re-lationships between joint orientations and crack propaga-tions were studied Obviously the failure modes for differentjoint orientations were divergent As shown in Figure 6splitting failure was the predominant failure type forspecimens with 30deg joint orientations (Figures 7(a)ndash7(c))Hence joints with mild slopes have relatively small impacton the properties of rocks On the contrary mixed types offailures have been noticed on the specimens with 45deg jointsDuring testing the specimens were fragmented into piecesIn fact it was difficult to trace the failure paths However at60deg joint orientation shear failure was the predominant oneA flat surface was formed as a result of flaw cuttings Henceat 60deg joint orientation dislocations of pieces occurred dueto a noticeable sliding

63 Effect of Joint Connectivity Rate on the Failure ModesFor different joint angles the effect of connectivity rate (k) isdistinct It had a considerable effect when the joint angleswere 45deg or 60deg However at 30deg joint angle the effect of jointconnectivity was relatively small e path of crack prop-agation was twisted when the value of kwas 03 (Figures 7(a)7(d) and 7(g)) is is because of the restraint action ofuncracked section of the specimens against sliding erupture face was rough since it is a result of multiple factorsOn the one hand joint and rock bridge alter or replace eachother in accordance with the connectivity rate On the otherhand the nonlinear variation of tension stress at the tip offlaws exists As is shown in Figure 6 the crack path wasbecoming edge shaped when the value of k increasedMeanwhile the failure modes gradually changed by thejoints Higher values of k yielded the straight plane ofruptures Nonetheless the concentrations of stresses werenoticeable as shown in the development of secondary cracks(Figures 7(e) and 7(f )) e cracks were extended between

the tips of the joints and the edges of the specimens when thejoint angle and the connectivity rate were 45deg and 045respectively For small values of k the crack pattern becamepolygonal However the crack pattern was transforminginto an arc shape when the value of k increased Besides thesecondary cracks became normal to the edge of the speci-mens during shearing At 60deg joint orientation there was noremarkable effect on the failure modes due to the variationsof joint connectivity rate (k) All surfaces of failures werenearly flat Nonetheless there was a slight discrepancy be-tween them Similar failure modes were developed whenjoint orientation and connectivity rate were 45deg and 03respectively e significant effect of joint connectivity ratewas obvious Hence shear failures can easily develop andtheir probability of occurrence depends on the values of kShear failures were developed at higher values of k(Figures 7(h) and 7(i))

Table 3 summarizes different types of failure modes forspecimens with diverse joint orientations and connectivityrates e failure mode was governed by the joint param-eters Fundamentally the failure modes depend on the jointorientations initially For instance splitting failure is un-likely to occur when the joint angle is above 45deg Meanwhileconnectivity rate determines the extent of shear failuree kvalues control the transformation between mixed andcomplete shear failures At 60deg joint orientation both typesof failures were observed e critical value for k was 03 inthis case

7 Discussions of Crack Force Analysis Based onGriffithrsquos Theory

MohrndashCoulomb and Griffith criteria are conventionallyused to interpret the failure of rock specimens It is no-ticeable that the MohrndashCoulomb criterion cannot clarify theinfluence of inside flaws on rock sample failure Howevermaximum tensile stress in the perimeter of Griffithrsquos crack(Figure 8) can be analyzed according to Griffithrsquos theory forthe sake of exploring the effectiveness of the internal defects

Han et al [26] deduced equation (2) a function of α andc considering primary stress of equation (1) and Inglisrsquosformula

σy σ1 + σ3

2+σ1 minus σ3

2cos 2c

σx σ1 + σ3

2minusσ1 minus σ3

2cos 2c

τxy minusσ1 minus σ3

2sin 2c

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(1)

σb σ1

m2 + α2m 1 + α21113872 1113873(1 + λ) +(1 + m) mminus α21113872 1113873(1minus α)cos 2c + α(1 + m)

2(1minus λ)sin 2c1113966 1113967 (2)

Advances in Materials Science and Engineering 11

As shown in the specimen schematic diagram α and thecrack width can be valued as 0degand 1mm respectively sincethe crack is long and thin Hence the following equation isattained from equation (2)

σb σ1m

(1 + λ) +(1 + m)(1minus λ)cos 2c1113864 1113865 (3)

e stress extremum hinges on σ1 λ and m at the tips ofthe crack in the condition of c minus(π2) e equation isobtained as follows

σ0 σ1m

2λminusm(1minus λ) (4)

According to equation (4) the maximum tensile stress atthe tips of flaw was calculated Meanwhile the influences onthe maximum tensile stress were analyzed considering jointinclined angles joint connectivity and confining pressures(Figures 9 and 10) Obviously the maximum tensile stressincreases promptly in accordance with the confining pres-sures Simultaneously higher joint connectivity rates lead toa large stress extremum And the confining pressure enlargesthis trend As it is shown in Figure 10 the escalation of jointorientation boosts the maximum tensile stress at the tips offlaw erefore the alteration of confining pressure is ofmost significant on the variation of the tensile stress whilethe joint angle contributes the minimal impact

With confining pressure (ie λ) declining gradually inthe unloading stage the longitudinal crack may arise due toextremum of tensile stress on the tips and when it satisfiesthe fracture strength the rupture of the longitudinal crackappears along the plane of the crack and may propagatecontinuously to the specimenrsquos failure Considering the axialstress stops expansion in the unloading stage the specimencannot be broken by some slanting cracks or crossed cracks

directly during the compression erefore the splittingfailure coupled with shear failure emerges finally

8 Conclusions

In the preceding pages the crack propagation was studied onthe specimens with partially cut-through joints e primeobjective of this study was to examine the effect of jointorientations and joint connectivity rates on the mechanicalcharacteristics of specimens Even though nonlinear me-chanical characteristics were observed the macroscopicbehavior was studied based upon the different values of jointgeometries Furthermore nine types of specimens were usedto investigate the failure mechanisms at different confiningpressures e confining pressure joint orientation andconnectivity rate were considered as key parameters toanalyze different failure modes Finally Griffithrsquos theory wasused to analyze the maximum tensile stress Nevertheless afurther study is important to extend the application of thisstudy in real engineering problemsemain findings of thispaper are summarized below

A nonlinear correlation was observed among the elas-ticity modulus joint orientation and joint connectivity rateMeanwhile the confining pressure was arithmetically

Y

σy

σx

Xα

γ

σx

σy

σ3σ1

τxy τxyτyx

τyx

Figure 8 Stress state of the crack based on Griffithrsquos theory [26]

0

100

200

300

400

500

600

2 4 6 8 10

σ 0 (M

Pa)

σ3 (MPa)

β = 30deg k = 03β = 30deg k = 045β = 30deg k = 06

Figure 9 e stress extremum with 30deg joint angle under differentconfining pressures

0

200

400

600

800

1000

30 45 60

σ 0 (M

Pa)

β (deg)k = 03k = 045k = 06

Figure 10 e stress extremum under different joint orientationsand connectivity rates

Table 3 Failure modes for different types of samples

Connectivity rateOrientation 03 045 0630 Splitting Splitting Splitting45 Mixed Mixed Mixed60 Mixed Shear Shear

12 Advances in Materials Science and Engineering

increasing During the unloading stage the reduction ofdeformation modulus can be used to predict the rockspecimen failure And the critical value of decrease per-centage is 80 e deformation modulus is less sensitive tothe change of joint connectivity whereas it is more sensitiveto the change of the inclination angle

ree different types (shear split and mixed) of failuremodes were observed e higher values of confiningpressure can restraint the lateral deformation At 30deg jointorientation the predominant failure was splitting failureSimilarly mixed failures were observed at 45deg joint ori-entations because of the concentration of stresses at the tipof the joints On the contrary shear failure was quitecommon due to sufficient length of 60deg joint orientationComparatively the effect of joint connectivity rate wassmaller than that of the joint orientation However therewas a clear variation in the crack propagation when thejoint angle was 60deg e position of joints plays a significantrole for anisotropic failures

A few equations were introduced based on Griffithrsquostheory to evaluate the maximum tensile stress at the tips ofthe crack e results based on the calculation reveal theimpaction of joint characteristics on the stress extremum

Abbreviations

c e angle of principal stress σ1 and Y axisα e central angle of the ellipsem e ratio of the minor axis (b) to major axis (a) of the

ellipseλ e ratio of two principal stresses between σ3 and σ1σ0 e maximum tensile stressσ1 e maximum principal stressσ3 e minimum principal stressσb Tangential stress on the perimeter of a slender ellipse

Data Availability

e data used to support the findings of this study may bereleased upon application to the China ree GorgesUniversity Review Board which can be done by contactingthe author ldquoGuoyong Duan (dgyhhueducn)rdquo

Conflicts of Interest

e authors wish to confirm that there are no knownconflicts of interest associated with this publication andthere has been no significant financial support for this workthat could have influenced its outcome Accordingly thealteration of confining pressure is of most significance onthe variation of the tensile stress while the joint anglecontributes the minimal impact

Acknowledgments

e authors gratefully acknowledge the financial supportfrom the National Natural Science Foundation of China(Grant nos 51439003 and 51279091) the Fundamental

Research Funds for the Central Universities (Hohai Uni-versity) (Grant no 2016B43014) Open Fund of Key Lab-oratory of Geological Hazards on ree Gorges ReservoirArea (China ree Gorges University) (Grant no2015KDZ12) and Key Project of Scientific Research Plan ofHubei Provincial Department of Education (Grant noD20181202)

References

[1] T Liu P Cao and H Lin ldquoEvolution procedure of multiplerock cracks under seepage pressurerdquo Mathematical Problemsin Engineering vol 2013 Article ID 738013 11 pages 2013

[2] E Hoek ldquoStrength of jointed rock massesrdquo Geotechniquevol 33 no 3 pp 187ndash223 1983

[3] E Hoek and Z T Bieniawski ldquoBrittle fracture propagation inrock under compressionrdquo International Journal of Fracturevol 26 no 4 pp 276ndash294 1984

[4] M Basista and D Gross ldquoe sliding crack model of brittledeformation an internal variable approachrdquo InternationalJournal of Solids and Structures vol 35 no 5-6 pp 487ndash5091998

[5] I Vardoulakis J F Labuz E Papamichos and J TronvollldquoContinuum fracture mechanics of uniaxial compression onbrittle materialsrdquo International Journal of Solids and Struc-tures vol 35 no 31-32 pp 4313ndash4335 1998

[6] A Paluszny and S K Matthai ldquoNumerical modeling ofdiscrete multi-crack growth applied to pattern formation ingeological brittle mediardquo International Journal of Solids andStructures vol 46 no 18-19 pp 3383ndash3397 2009

[7] R-H Cao P Cao H Lin C-Z Pu and K Ou ldquoMechanicalbehavior of brittle rock-like specimens with pre-existingfissures under uniaxial loading experimental studies andparticle mechanics approachrdquo Rock Mechanics and RockEngineering vol 49 no 3 pp 763ndash783 2016

[8] X Fan R Chen H Lin H Lai C Zhang and Q ZhaoldquoCracking and failure in rock specimen containing combinedflaw and hole under uniaxial compressionrdquo Advances in CivilEngineering vol 2018 Article ID 9818250 15 pages 2018

[9] D Bing H Guicheng and Z Zhijun ldquoA numerical researchon crack process of gypsum containing single flaw withdifferent angle and length in uniaxial loadingrdquo Shock andVibration vol 2018 Article ID 2968205 16 pages 2018

[10] J Zhang ldquoInvestigation of relation between fracture scale andacoustic emission time-frequency parameters in rocksrdquo Shockand Vibration vol 2018 Article ID 3057628 14 pages 2018

[11] X Meng and W Liu ldquoStudy on failure process and perme-ation evolution of single-cracked rockrdquo Advances in MaterialsScience and Engineering vol 2018 Article ID 79156529 pages 2018

[12] M Sagong D Park J Yoo and J S Lee ldquoExperimental andnumerical analyses of an opening in a jointed rockmass underbiaxial compressionrdquo International Journal of RockMechanicsand Mining Sciences vol 48 no 7 pp 1055ndash1067 2011

[13] W Chen H Konietzky X Tan and T Fruhwirt ldquoPre-failuredamage analysis for brittle rocks under triaxial compressionrdquoComputers and Geotechnics vol 74 pp 45ndash55 2016

[14] S-Q Yang P G Ranjith Y-H Huang et al ldquoExperimentalinvestigation on mechanical damage characteristics ofsandstone under triaxial cyclic loadingrdquo Geophysical JournalInternational vol 201 no 2 pp 662ndash682 2015

[15] N Erarslan ldquoMicrostructural investigation of subcriticalcrack propagation and fracture process zone (FPZ) by the

Advances in Materials Science and Engineering 13

cracks emerged On the contrary internal defects weresqueezed by the confining pressure Hence higher values ofconfining pressures can impose a significant restraint againstthe lateral deformations Similarly excavation can beregarded as unloading condition in rock slope engineering

62 Effect of Orientation on the Failure Modes In naturerocks have different orientations of joints which affect theirphysical and mechanical properties In this study the re-lationships between joint orientations and crack propaga-tions were studied Obviously the failure modes for differentjoint orientations were divergent As shown in Figure 6splitting failure was the predominant failure type forspecimens with 30deg joint orientations (Figures 7(a)ndash7(c))Hence joints with mild slopes have relatively small impacton the properties of rocks On the contrary mixed types offailures have been noticed on the specimens with 45deg jointsDuring testing the specimens were fragmented into piecesIn fact it was difficult to trace the failure paths However at60deg joint orientation shear failure was the predominant oneA flat surface was formed as a result of flaw cuttings Henceat 60deg joint orientation dislocations of pieces occurred dueto a noticeable sliding

63 Effect of Joint Connectivity Rate on the Failure ModesFor different joint angles the effect of connectivity rate (k) isdistinct It had a considerable effect when the joint angleswere 45deg or 60deg However at 30deg joint angle the effect of jointconnectivity was relatively small e path of crack prop-agation was twisted when the value of kwas 03 (Figures 7(a)7(d) and 7(g)) is is because of the restraint action ofuncracked section of the specimens against sliding erupture face was rough since it is a result of multiple factorsOn the one hand joint and rock bridge alter or replace eachother in accordance with the connectivity rate On the otherhand the nonlinear variation of tension stress at the tip offlaws exists As is shown in Figure 6 the crack path wasbecoming edge shaped when the value of k increasedMeanwhile the failure modes gradually changed by thejoints Higher values of k yielded the straight plane ofruptures Nonetheless the concentrations of stresses werenoticeable as shown in the development of secondary cracks(Figures 7(e) and 7(f )) e cracks were extended between

the tips of the joints and the edges of the specimens when thejoint angle and the connectivity rate were 45deg and 045respectively For small values of k the crack pattern becamepolygonal However the crack pattern was transforminginto an arc shape when the value of k increased Besides thesecondary cracks became normal to the edge of the speci-mens during shearing At 60deg joint orientation there was noremarkable effect on the failure modes due to the variationsof joint connectivity rate (k) All surfaces of failures werenearly flat Nonetheless there was a slight discrepancy be-tween them Similar failure modes were developed whenjoint orientation and connectivity rate were 45deg and 03respectively e significant effect of joint connectivity ratewas obvious Hence shear failures can easily develop andtheir probability of occurrence depends on the values of kShear failures were developed at higher values of k(Figures 7(h) and 7(i))

Table 3 summarizes different types of failure modes forspecimens with diverse joint orientations and connectivityrates e failure mode was governed by the joint param-eters Fundamentally the failure modes depend on the jointorientations initially For instance splitting failure is un-likely to occur when the joint angle is above 45deg Meanwhileconnectivity rate determines the extent of shear failuree kvalues control the transformation between mixed andcomplete shear failures At 60deg joint orientation both typesof failures were observed e critical value for k was 03 inthis case

7 Discussions of Crack Force Analysis Based onGriffithrsquos Theory

MohrndashCoulomb and Griffith criteria are conventionallyused to interpret the failure of rock specimens It is no-ticeable that the MohrndashCoulomb criterion cannot clarify theinfluence of inside flaws on rock sample failure Howevermaximum tensile stress in the perimeter of Griffithrsquos crack(Figure 8) can be analyzed according to Griffithrsquos theory forthe sake of exploring the effectiveness of the internal defects

Han et al [26] deduced equation (2) a function of α andc considering primary stress of equation (1) and Inglisrsquosformula

σy σ1 + σ3

2+σ1 minus σ3

2cos 2c

σx σ1 + σ3

2minusσ1 minus σ3

2cos 2c

τxy minusσ1 minus σ3

2sin 2c

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(1)

σb σ1

m2 + α2m 1 + α21113872 1113873(1 + λ) +(1 + m) mminus α21113872 1113873(1minus α)cos 2c + α(1 + m)

2(1minus λ)sin 2c1113966 1113967 (2)

Advances in Materials Science and Engineering 11

As shown in the specimen schematic diagram α and thecrack width can be valued as 0degand 1mm respectively sincethe crack is long and thin Hence the following equation isattained from equation (2)

σb σ1m

(1 + λ) +(1 + m)(1minus λ)cos 2c1113864 1113865 (3)

e stress extremum hinges on σ1 λ and m at the tips ofthe crack in the condition of c minus(π2) e equation isobtained as follows

σ0 σ1m

2λminusm(1minus λ) (4)

According to equation (4) the maximum tensile stress atthe tips of flaw was calculated Meanwhile the influences onthe maximum tensile stress were analyzed considering jointinclined angles joint connectivity and confining pressures(Figures 9 and 10) Obviously the maximum tensile stressincreases promptly in accordance with the confining pres-sures Simultaneously higher joint connectivity rates lead toa large stress extremum And the confining pressure enlargesthis trend As it is shown in Figure 10 the escalation of jointorientation boosts the maximum tensile stress at the tips offlaw erefore the alteration of confining pressure is ofmost significant on the variation of the tensile stress whilethe joint angle contributes the minimal impact

With confining pressure (ie λ) declining gradually inthe unloading stage the longitudinal crack may arise due toextremum of tensile stress on the tips and when it satisfiesthe fracture strength the rupture of the longitudinal crackappears along the plane of the crack and may propagatecontinuously to the specimenrsquos failure Considering the axialstress stops expansion in the unloading stage the specimencannot be broken by some slanting cracks or crossed cracks

directly during the compression erefore the splittingfailure coupled with shear failure emerges finally

8 Conclusions

In the preceding pages the crack propagation was studied onthe specimens with partially cut-through joints e primeobjective of this study was to examine the effect of jointorientations and joint connectivity rates on the mechanicalcharacteristics of specimens Even though nonlinear me-chanical characteristics were observed the macroscopicbehavior was studied based upon the different values of jointgeometries Furthermore nine types of specimens were usedto investigate the failure mechanisms at different confiningpressures e confining pressure joint orientation andconnectivity rate were considered as key parameters toanalyze different failure modes Finally Griffithrsquos theory wasused to analyze the maximum tensile stress Nevertheless afurther study is important to extend the application of thisstudy in real engineering problemsemain findings of thispaper are summarized below

A nonlinear correlation was observed among the elas-ticity modulus joint orientation and joint connectivity rateMeanwhile the confining pressure was arithmetically

Y

σy

σx

Xα

γ

σx

σy

σ3σ1

τxy τxyτyx

τyx

Figure 8 Stress state of the crack based on Griffithrsquos theory [26]

0

100

200

300

400

500

600

2 4 6 8 10

σ 0 (M

Pa)

σ3 (MPa)

β = 30deg k = 03β = 30deg k = 045β = 30deg k = 06

Figure 9 e stress extremum with 30deg joint angle under differentconfining pressures

0

200

400

600

800

1000

30 45 60

σ 0 (M

Pa)

β (deg)k = 03k = 045k = 06

Figure 10 e stress extremum under different joint orientationsand connectivity rates

Table 3 Failure modes for different types of samples

Connectivity rateOrientation 03 045 0630 Splitting Splitting Splitting45 Mixed Mixed Mixed60 Mixed Shear Shear

12 Advances in Materials Science and Engineering

increasing During the unloading stage the reduction ofdeformation modulus can be used to predict the rockspecimen failure And the critical value of decrease per-centage is 80 e deformation modulus is less sensitive tothe change of joint connectivity whereas it is more sensitiveto the change of the inclination angle

ree different types (shear split and mixed) of failuremodes were observed e higher values of confiningpressure can restraint the lateral deformation At 30deg jointorientation the predominant failure was splitting failureSimilarly mixed failures were observed at 45deg joint ori-entations because of the concentration of stresses at the tipof the joints On the contrary shear failure was quitecommon due to sufficient length of 60deg joint orientationComparatively the effect of joint connectivity rate wassmaller than that of the joint orientation However therewas a clear variation in the crack propagation when thejoint angle was 60deg e position of joints plays a significantrole for anisotropic failures

A few equations were introduced based on Griffithrsquostheory to evaluate the maximum tensile stress at the tips ofthe crack e results based on the calculation reveal theimpaction of joint characteristics on the stress extremum

Abbreviations

c e angle of principal stress σ1 and Y axisα e central angle of the ellipsem e ratio of the minor axis (b) to major axis (a) of the

ellipseλ e ratio of two principal stresses between σ3 and σ1σ0 e maximum tensile stressσ1 e maximum principal stressσ3 e minimum principal stressσb Tangential stress on the perimeter of a slender ellipse

Data Availability

e data used to support the findings of this study may bereleased upon application to the China ree GorgesUniversity Review Board which can be done by contactingthe author ldquoGuoyong Duan (dgyhhueducn)rdquo

Conflicts of Interest

e authors wish to confirm that there are no knownconflicts of interest associated with this publication andthere has been no significant financial support for this workthat could have influenced its outcome Accordingly thealteration of confining pressure is of most significance onthe variation of the tensile stress while the joint anglecontributes the minimal impact

Acknowledgments

e authors gratefully acknowledge the financial supportfrom the National Natural Science Foundation of China(Grant nos 51439003 and 51279091) the Fundamental

Research Funds for the Central Universities (Hohai Uni-versity) (Grant no 2016B43014) Open Fund of Key Lab-oratory of Geological Hazards on ree Gorges ReservoirArea (China ree Gorges University) (Grant no2015KDZ12) and Key Project of Scientific Research Plan ofHubei Provincial Department of Education (Grant noD20181202)

References

[1] T Liu P Cao and H Lin ldquoEvolution procedure of multiplerock cracks under seepage pressurerdquo Mathematical Problemsin Engineering vol 2013 Article ID 738013 11 pages 2013

[2] E Hoek ldquoStrength of jointed rock massesrdquo Geotechniquevol 33 no 3 pp 187ndash223 1983

[3] E Hoek and Z T Bieniawski ldquoBrittle fracture propagation inrock under compressionrdquo International Journal of Fracturevol 26 no 4 pp 276ndash294 1984

[4] M Basista and D Gross ldquoe sliding crack model of brittledeformation an internal variable approachrdquo InternationalJournal of Solids and Structures vol 35 no 5-6 pp 487ndash5091998

[5] I Vardoulakis J F Labuz E Papamichos and J TronvollldquoContinuum fracture mechanics of uniaxial compression onbrittle materialsrdquo International Journal of Solids and Struc-tures vol 35 no 31-32 pp 4313ndash4335 1998

[6] A Paluszny and S K Matthai ldquoNumerical modeling ofdiscrete multi-crack growth applied to pattern formation ingeological brittle mediardquo International Journal of Solids andStructures vol 46 no 18-19 pp 3383ndash3397 2009

[7] R-H Cao P Cao H Lin C-Z Pu and K Ou ldquoMechanicalbehavior of brittle rock-like specimens with pre-existingfissures under uniaxial loading experimental studies andparticle mechanics approachrdquo Rock Mechanics and RockEngineering vol 49 no 3 pp 763ndash783 2016

[8] X Fan R Chen H Lin H Lai C Zhang and Q ZhaoldquoCracking and failure in rock specimen containing combinedflaw and hole under uniaxial compressionrdquo Advances in CivilEngineering vol 2018 Article ID 9818250 15 pages 2018

[9] D Bing H Guicheng and Z Zhijun ldquoA numerical researchon crack process of gypsum containing single flaw withdifferent angle and length in uniaxial loadingrdquo Shock andVibration vol 2018 Article ID 2968205 16 pages 2018

[10] J Zhang ldquoInvestigation of relation between fracture scale andacoustic emission time-frequency parameters in rocksrdquo Shockand Vibration vol 2018 Article ID 3057628 14 pages 2018

[11] X Meng and W Liu ldquoStudy on failure process and perme-ation evolution of single-cracked rockrdquo Advances in MaterialsScience and Engineering vol 2018 Article ID 79156529 pages 2018

[12] M Sagong D Park J Yoo and J S Lee ldquoExperimental andnumerical analyses of an opening in a jointed rockmass underbiaxial compressionrdquo International Journal of RockMechanicsand Mining Sciences vol 48 no 7 pp 1055ndash1067 2011

[13] W Chen H Konietzky X Tan and T Fruhwirt ldquoPre-failuredamage analysis for brittle rocks under triaxial compressionrdquoComputers and Geotechnics vol 74 pp 45ndash55 2016

[14] S-Q Yang P G Ranjith Y-H Huang et al ldquoExperimentalinvestigation on mechanical damage characteristics ofsandstone under triaxial cyclic loadingrdquo Geophysical JournalInternational vol 201 no 2 pp 662ndash682 2015

[15] N Erarslan ldquoMicrostructural investigation of subcriticalcrack propagation and fracture process zone (FPZ) by the

Advances in Materials Science and Engineering 13

As shown in the specimen schematic diagram α and thecrack width can be valued as 0degand 1mm respectively sincethe crack is long and thin Hence the following equation isattained from equation (2)

σb σ1m

(1 + λ) +(1 + m)(1minus λ)cos 2c1113864 1113865 (3)

e stress extremum hinges on σ1 λ and m at the tips ofthe crack in the condition of c minus(π2) e equation isobtained as follows

σ0 σ1m

2λminusm(1minus λ) (4)

According to equation (4) the maximum tensile stress atthe tips of flaw was calculated Meanwhile the influences onthe maximum tensile stress were analyzed considering jointinclined angles joint connectivity and confining pressures(Figures 9 and 10) Obviously the maximum tensile stressincreases promptly in accordance with the confining pres-sures Simultaneously higher joint connectivity rates lead toa large stress extremum And the confining pressure enlargesthis trend As it is shown in Figure 10 the escalation of jointorientation boosts the maximum tensile stress at the tips offlaw erefore the alteration of confining pressure is ofmost significant on the variation of the tensile stress whilethe joint angle contributes the minimal impact

With confining pressure (ie λ) declining gradually inthe unloading stage the longitudinal crack may arise due toextremum of tensile stress on the tips and when it satisfiesthe fracture strength the rupture of the longitudinal crackappears along the plane of the crack and may propagatecontinuously to the specimenrsquos failure Considering the axialstress stops expansion in the unloading stage the specimencannot be broken by some slanting cracks or crossed cracks

directly during the compression erefore the splittingfailure coupled with shear failure emerges finally

8 Conclusions

In the preceding pages the crack propagation was studied onthe specimens with partially cut-through joints e primeobjective of this study was to examine the effect of jointorientations and joint connectivity rates on the mechanicalcharacteristics of specimens Even though nonlinear me-chanical characteristics were observed the macroscopicbehavior was studied based upon the different values of jointgeometries Furthermore nine types of specimens were usedto investigate the failure mechanisms at different confiningpressures e confining pressure joint orientation andconnectivity rate were considered as key parameters toanalyze different failure modes Finally Griffithrsquos theory wasused to analyze the maximum tensile stress Nevertheless afurther study is important to extend the application of thisstudy in real engineering problemsemain findings of thispaper are summarized below

A nonlinear correlation was observed among the elas-ticity modulus joint orientation and joint connectivity rateMeanwhile the confining pressure was arithmetically

Y

σy

σx

Xα

γ

σx

σy

σ3σ1

τxy τxyτyx

τyx

Figure 8 Stress state of the crack based on Griffithrsquos theory [26]

0

100

200

300

400

500

600

2 4 6 8 10

σ 0 (M

Pa)

σ3 (MPa)

β = 30deg k = 03β = 30deg k = 045β = 30deg k = 06

Figure 9 e stress extremum with 30deg joint angle under differentconfining pressures

0

200

400

600

800

1000

30 45 60

σ 0 (M

Pa)

β (deg)k = 03k = 045k = 06

Figure 10 e stress extremum under different joint orientationsand connectivity rates

Table 3 Failure modes for different types of samples

Connectivity rateOrientation 03 045 0630 Splitting Splitting Splitting45 Mixed Mixed Mixed60 Mixed Shear Shear

12 Advances in Materials Science and Engineering

increasing During the unloading stage the reduction ofdeformation modulus can be used to predict the rockspecimen failure And the critical value of decrease per-centage is 80 e deformation modulus is less sensitive tothe change of joint connectivity whereas it is more sensitiveto the change of the inclination angle

ree different types (shear split and mixed) of failuremodes were observed e higher values of confiningpressure can restraint the lateral deformation At 30deg jointorientation the predominant failure was splitting failureSimilarly mixed failures were observed at 45deg joint ori-entations because of the concentration of stresses at the tipof the joints On the contrary shear failure was quitecommon due to sufficient length of 60deg joint orientationComparatively the effect of joint connectivity rate wassmaller than that of the joint orientation However therewas a clear variation in the crack propagation when thejoint angle was 60deg e position of joints plays a significantrole for anisotropic failures

A few equations were introduced based on Griffithrsquostheory to evaluate the maximum tensile stress at the tips ofthe crack e results based on the calculation reveal theimpaction of joint characteristics on the stress extremum

Abbreviations

c e angle of principal stress σ1 and Y axisα e central angle of the ellipsem e ratio of the minor axis (b) to major axis (a) of the

ellipseλ e ratio of two principal stresses between σ3 and σ1σ0 e maximum tensile stressσ1 e maximum principal stressσ3 e minimum principal stressσb Tangential stress on the perimeter of a slender ellipse

Data Availability

e data used to support the findings of this study may bereleased upon application to the China ree GorgesUniversity Review Board which can be done by contactingthe author ldquoGuoyong Duan (dgyhhueducn)rdquo

Conflicts of Interest

e authors wish to confirm that there are no knownconflicts of interest associated with this publication andthere has been no significant financial support for this workthat could have influenced its outcome Accordingly thealteration of confining pressure is of most significance onthe variation of the tensile stress while the joint anglecontributes the minimal impact

Acknowledgments

e authors gratefully acknowledge the financial supportfrom the National Natural Science Foundation of China(Grant nos 51439003 and 51279091) the Fundamental

Research Funds for the Central Universities (Hohai Uni-versity) (Grant no 2016B43014) Open Fund of Key Lab-oratory of Geological Hazards on ree Gorges ReservoirArea (China ree Gorges University) (Grant no2015KDZ12) and Key Project of Scientific Research Plan ofHubei Provincial Department of Education (Grant noD20181202)

References

[1] T Liu P Cao and H Lin ldquoEvolution procedure of multiplerock cracks under seepage pressurerdquo Mathematical Problemsin Engineering vol 2013 Article ID 738013 11 pages 2013

[2] E Hoek ldquoStrength of jointed rock massesrdquo Geotechniquevol 33 no 3 pp 187ndash223 1983

[3] E Hoek and Z T Bieniawski ldquoBrittle fracture propagation inrock under compressionrdquo International Journal of Fracturevol 26 no 4 pp 276ndash294 1984

[4] M Basista and D Gross ldquoe sliding crack model of brittledeformation an internal variable approachrdquo InternationalJournal of Solids and Structures vol 35 no 5-6 pp 487ndash5091998

[5] I Vardoulakis J F Labuz E Papamichos and J TronvollldquoContinuum fracture mechanics of uniaxial compression onbrittle materialsrdquo International Journal of Solids and Struc-tures vol 35 no 31-32 pp 4313ndash4335 1998

[6] A Paluszny and S K Matthai ldquoNumerical modeling ofdiscrete multi-crack growth applied to pattern formation ingeological brittle mediardquo International Journal of Solids andStructures vol 46 no 18-19 pp 3383ndash3397 2009

[7] R-H Cao P Cao H Lin C-Z Pu and K Ou ldquoMechanicalbehavior of brittle rock-like specimens with pre-existingfissures under uniaxial loading experimental studies andparticle mechanics approachrdquo Rock Mechanics and RockEngineering vol 49 no 3 pp 763ndash783 2016

[8] X Fan R Chen H Lin H Lai C Zhang and Q ZhaoldquoCracking and failure in rock specimen containing combinedflaw and hole under uniaxial compressionrdquo Advances in CivilEngineering vol 2018 Article ID 9818250 15 pages 2018

[9] D Bing H Guicheng and Z Zhijun ldquoA numerical researchon crack process of gypsum containing single flaw withdifferent angle and length in uniaxial loadingrdquo Shock andVibration vol 2018 Article ID 2968205 16 pages 2018

[10] J Zhang ldquoInvestigation of relation between fracture scale andacoustic emission time-frequency parameters in rocksrdquo Shockand Vibration vol 2018 Article ID 3057628 14 pages 2018

[11] X Meng and W Liu ldquoStudy on failure process and perme-ation evolution of single-cracked rockrdquo Advances in MaterialsScience and Engineering vol 2018 Article ID 79156529 pages 2018

[12] M Sagong D Park J Yoo and J S Lee ldquoExperimental andnumerical analyses of an opening in a jointed rockmass underbiaxial compressionrdquo International Journal of RockMechanicsand Mining Sciences vol 48 no 7 pp 1055ndash1067 2011

[13] W Chen H Konietzky X Tan and T Fruhwirt ldquoPre-failuredamage analysis for brittle rocks under triaxial compressionrdquoComputers and Geotechnics vol 74 pp 45ndash55 2016

[14] S-Q Yang P G Ranjith Y-H Huang et al ldquoExperimentalinvestigation on mechanical damage characteristics ofsandstone under triaxial cyclic loadingrdquo Geophysical JournalInternational vol 201 no 2 pp 662ndash682 2015

[15] N Erarslan ldquoMicrostructural investigation of subcriticalcrack propagation and fracture process zone (FPZ) by the

Advances in Materials Science and Engineering 13

increasing During the unloading stage the reduction ofdeformation modulus can be used to predict the rockspecimen failure And the critical value of decrease per-centage is 80 e deformation modulus is less sensitive tothe change of joint connectivity whereas it is more sensitiveto the change of the inclination angle

ree different types (shear split and mixed) of failuremodes were observed e higher values of confiningpressure can restraint the lateral deformation At 30deg jointorientation the predominant failure was splitting failureSimilarly mixed failures were observed at 45deg joint ori-entations because of the concentration of stresses at the tipof the joints On the contrary shear failure was quitecommon due to sufficient length of 60deg joint orientationComparatively the effect of joint connectivity rate wassmaller than that of the joint orientation However therewas a clear variation in the crack propagation when thejoint angle was 60deg e position of joints plays a significantrole for anisotropic failures

A few equations were introduced based on Griffithrsquostheory to evaluate the maximum tensile stress at the tips ofthe crack e results based on the calculation reveal theimpaction of joint characteristics on the stress extremum

Abbreviations

c e angle of principal stress σ1 and Y axisα e central angle of the ellipsem e ratio of the minor axis (b) to major axis (a) of the

ellipseλ e ratio of two principal stresses between σ3 and σ1σ0 e maximum tensile stressσ1 e maximum principal stressσ3 e minimum principal stressσb Tangential stress on the perimeter of a slender ellipse

Data Availability

e data used to support the findings of this study may bereleased upon application to the China ree GorgesUniversity Review Board which can be done by contactingthe author ldquoGuoyong Duan (dgyhhueducn)rdquo

Conflicts of Interest

e authors wish to confirm that there are no knownconflicts of interest associated with this publication andthere has been no significant financial support for this workthat could have influenced its outcome Accordingly thealteration of confining pressure is of most significance onthe variation of the tensile stress while the joint anglecontributes the minimal impact

Acknowledgments

e authors gratefully acknowledge the financial supportfrom the National Natural Science Foundation of China(Grant nos 51439003 and 51279091) the Fundamental

Research Funds for the Central Universities (Hohai Uni-versity) (Grant no 2016B43014) Open Fund of Key Lab-oratory of Geological Hazards on ree Gorges ReservoirArea (China ree Gorges University) (Grant no2015KDZ12) and Key Project of Scientific Research Plan ofHubei Provincial Department of Education (Grant noD20181202)

References

[1] T Liu P Cao and H Lin ldquoEvolution procedure of multiplerock cracks under seepage pressurerdquo Mathematical Problemsin Engineering vol 2013 Article ID 738013 11 pages 2013

[2] E Hoek ldquoStrength of jointed rock massesrdquo Geotechniquevol 33 no 3 pp 187ndash223 1983

[3] E Hoek and Z T Bieniawski ldquoBrittle fracture propagation inrock under compressionrdquo International Journal of Fracturevol 26 no 4 pp 276ndash294 1984

[4] M Basista and D Gross ldquoe sliding crack model of brittledeformation an internal variable approachrdquo InternationalJournal of Solids and Structures vol 35 no 5-6 pp 487ndash5091998

[5] I Vardoulakis J F Labuz E Papamichos and J TronvollldquoContinuum fracture mechanics of uniaxial compression onbrittle materialsrdquo International Journal of Solids and Struc-tures vol 35 no 31-32 pp 4313ndash4335 1998

[6] A Paluszny and S K Matthai ldquoNumerical modeling ofdiscrete multi-crack growth applied to pattern formation ingeological brittle mediardquo International Journal of Solids andStructures vol 46 no 18-19 pp 3383ndash3397 2009

[7] R-H Cao P Cao H Lin C-Z Pu and K Ou ldquoMechanicalbehavior of brittle rock-like specimens with pre-existingfissures under uniaxial loading experimental studies andparticle mechanics approachrdquo Rock Mechanics and RockEngineering vol 49 no 3 pp 763ndash783 2016

[8] X Fan R Chen H Lin H Lai C Zhang and Q ZhaoldquoCracking and failure in rock specimen containing combinedflaw and hole under uniaxial compressionrdquo Advances in CivilEngineering vol 2018 Article ID 9818250 15 pages 2018

[9] D Bing H Guicheng and Z Zhijun ldquoA numerical researchon crack process of gypsum containing single flaw withdifferent angle and length in uniaxial loadingrdquo Shock andVibration vol 2018 Article ID 2968205 16 pages 2018

[10] J Zhang ldquoInvestigation of relation between fracture scale andacoustic emission time-frequency parameters in rocksrdquo Shockand Vibration vol 2018 Article ID 3057628 14 pages 2018

[11] X Meng and W Liu ldquoStudy on failure process and perme-ation evolution of single-cracked rockrdquo Advances in MaterialsScience and Engineering vol 2018 Article ID 79156529 pages 2018

[12] M Sagong D Park J Yoo and J S Lee ldquoExperimental andnumerical analyses of an opening in a jointed rockmass underbiaxial compressionrdquo International Journal of RockMechanicsand Mining Sciences vol 48 no 7 pp 1055ndash1067 2011

[13] W Chen H Konietzky X Tan and T Fruhwirt ldquoPre-failuredamage analysis for brittle rocks under triaxial compressionrdquoComputers and Geotechnics vol 74 pp 45ndash55 2016

[14] S-Q Yang P G Ranjith Y-H Huang et al ldquoExperimentalinvestigation on mechanical damage characteristics ofsandstone under triaxial cyclic loadingrdquo Geophysical JournalInternational vol 201 no 2 pp 662ndash682 2015

[15] N Erarslan ldquoMicrostructural investigation of subcriticalcrack propagation and fracture process zone (FPZ) by the

Advances in Materials Science and Engineering 13

reduction of rock fracture toughness under cyclic loadingrdquoEngineering Geology vol 208 pp 181ndash190 2016

[16] X Wang Y Jiang and B Li ldquoExperimental and numericalstudy on crack propagation and deformation around un-derground opening in jointed rock massesrdquo GeosciencesJournal vol 21 no 2 pp 291ndash304 2017

[17] X P Zhou and H Yang ldquoDamage by cracking of rock massinduced by dynamic unloading in the high slopes of the threegorges project permanent shiplockrdquo International Journal ofTerraspace Science amp Engineering vol 1 no 1 pp 47ndash582009

[18] J-H Hu T Lei K-P Zhou X-W Luo and N-G YangldquoMechanical response of roof rock mass unloading duringcontinuous mining process in underground minerdquo Trans-actions of Nonferrous Metals Society of China vol 21 no 12pp 2727ndash2733 2011

[19] S Zhou X Zhuang H Zhu and T Rabczuk ldquoPhase fieldmodelling of crack propagation branching and coalescence inrocksrdquo +eoretical and Applied Fracture Mechanics vol 96pp 174ndash192 2018

[20] X Z Li Z S Shao and L F Fan ldquoA micro-macro method forpredicting the shear strength of brittle rock under com-pressive loadingrdquo Mechanics Research Communicationsvol 75 pp 13ndash19 2016

[21] J Xu J Jiang L Zuo and Y Gao ldquoAcoustic emissionmonitoring and failure precursors of sandstone samplesunder various loading and unloading pathsrdquo Shock and Vi-bration vol 2017 Article ID 9760940 11 pages 2017

[22] Q Tan R Zhang M Gao Q Liu Z Zhang and Z Jia ldquoCoalpermeability and crack distribution characteristics inunloading confining pressure experiments under differentwater pressuresrdquo +ermal Science vol 21 no 1 pp 241ndash2492017

[23] H Yang J Liu and X Zhou ldquoEffects of the loading andunloading conditions on the stress relaxation behavior of pre-cracked graniterdquo Rock Mechanics and Rock Engineeringvol 50 no 5 pp 1157ndash1169 2017

[24] D Li Z Sun T Xie X Li and P G Ranjith ldquoEnergyevolution characteristics of hard rock during triaxial failurewith different loading and unloading pathsrdquo EngineeringGeology vol 228 pp 270ndash281 2017

[25] H Liu and L Zhang ldquoA damage constitutive model for rockmass with nonpersistently closed joints under uniaxialcompressionrdquo Arabian Journal for Science and Engineeringvol 40 no 11 pp 3107ndash3117 2015

[26] L Han Y He and H Zhang ldquoStudy of rock splitting failurebased on griffith strength theoryrdquo International Journal ofRock Mechanics and Mining Sciences vol 83 pp 116ndash1212016

14 Advances in Materials Science and Engineering

CorrosionInternational Journal of

Hindawiwwwhindawicom Volume 2018

Advances in

Materials Science and EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Journal of

Chemistry

Analytical ChemistryInternational Journal of

Hindawiwwwhindawicom Volume 2018

ScienticaHindawiwwwhindawicom Volume 2018

Polymer ScienceInternational Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Advances in Condensed Matter Physics

Hindawiwwwhindawicom Volume 2018

International Journal of

BiomaterialsHindawiwwwhindawicom

Journal ofEngineeringVolume 2018

Applied ChemistryJournal of

Hindawiwwwhindawicom Volume 2018

NanotechnologyHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

High Energy PhysicsAdvances in

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

TribologyAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

ChemistryAdvances in

Hindawiwwwhindawicom Volume 2018

Advances inPhysical Chemistry

Hindawiwwwhindawicom Volume 2018

BioMed Research InternationalMaterials

Journal of

Hindawiwwwhindawicom Volume 2018

Na

nom

ate

ria

ls

Hindawiwwwhindawicom Volume 2018

Journal ofNanomaterials

Submit your manuscripts atwwwhindawicom

CorrosionInternational Journal of

Hindawiwwwhindawicom Volume 2018

Advances in

Materials Science and EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Journal of

Chemistry

Analytical ChemistryInternational Journal of

Hindawiwwwhindawicom Volume 2018

ScienticaHindawiwwwhindawicom Volume 2018

Polymer ScienceInternational Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Advances in Condensed Matter Physics

Hindawiwwwhindawicom Volume 2018

International Journal of

BiomaterialsHindawiwwwhindawicom

Journal ofEngineeringVolume 2018

Applied ChemistryJournal of

Hindawiwwwhindawicom Volume 2018

NanotechnologyHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

High Energy PhysicsAdvances in

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

TribologyAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

ChemistryAdvances in

Hindawiwwwhindawicom Volume 2018

Advances inPhysical Chemistry

Hindawiwwwhindawicom Volume 2018

BioMed Research InternationalMaterials

Journal of

Hindawiwwwhindawicom Volume 2018

Na

nom

ate

ria

ls

Hindawiwwwhindawicom Volume 2018

Journal ofNanomaterials

Submit your manuscripts atwwwhindawicom