Hindawi Publishing CorporationThe Scientific World JournalVolume 2013 Article ID 758750 8 pageshttpdxdoiorg1011552013758750

Research ArticleFailure Behavior and Constitutive Model ofWeakly Consolidated Soft Rock

Wei-ming Wang1 Zeng-hui Zhao2 Yong-ji Wang1 and Xin Gao1

1 College of Civil Engineering of Shandong University of Science and Technology Qingdao Shandong 266590 China2 State Key Laboratory of Mining Disaster Prevention and Control Cofounded by Shandong Province and the Ministry ofScience and Technology Qingdao Shandong 266590 China

Correspondence should be addressed to Zeng-hui Zhao qzzh2004163com

Received 25 August 2013 Accepted 8 October 2013

Academic Editors G Giunta H Moayedi and C-O Ng

Copyright copy 2013 Wei-ming Wang et alThis is an open access article distributed under theCreativeCommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Mining areas in western China are mainly located in soft rock strata with poor bearing capacity In order to make the deformationfailure mechanism and strength behavior of weakly consolidated soft mudstone and coal rock hosted in Ili No 4 mine of Xinjiangarea clear some uniaxial and triaxial compression tests were carried out according to the samples of rocks gathered in the studiedarea respectively Meanwhile a damage constitutive model which considered the initial damage was established by introducinga damage variable and a correction coefficient A linearization process method was introduced according to the characteristicsof the fitting curve and experimental data The results showed that samples under different moisture contents and confiningpressures presented completely different failure mechanismThe given model could accurately describe the elastic and plastic yieldcharacteristics as well as the strain softening behavior of collected samples at postpeak stage Moreover the model could preciselyreflect the relationship between the elastic modulus and confining pressure at prepeak stage

1 Introduction

Coal seams in Chinarsquos western area of Xinjiang InnerMongolia and Ningxia are mainly located in measure stratacomposed of Jurassic and cretaceous soft rock which showpoor characteristics of low strength easy to be weatheredand easy to collapseThese weak mechanical properties bringdifficulties to the construction and stability maintenance ofrock tunnel After excavation of roadway somemine disasterssuch as large range of roof fall terrible floor heave suddencoal bump and landslides may be induced which bring aserious threat to the production safety and economic benefitsTherefore it is urgent to make clear the failure characteristicsand constitutive relations of typical soft rock and coal rockin these engineering sites This subject is of very importantengineering value for the prevention of mining dynamicaldisasters and stability maintenance of roadway

Some research achievements about mechanical behaviorsof coal and rock sample have been reported by domesticand foreign scholars Zuo et al [1] carried out indoor

tests by uniaxial and triaxial compression for sandstonerespectively and came to the conclusion that the type IIcurve during rock damage could be obtained by control ofcircumferential displacement Lu et al [2] and Zhang et al[3] studied the attenuation rules of mechanical parametersof rock mass at postpeak stage by triaxial compression testBased on the laboratory test results of silk screen chloriteschist Li et al [4] established its mechanical model whichconsidered the behavior of dilatancy stain hardening andsoftening The energy dissipation feature of mudstone atfailure was analyzed through triaxial compression test byHan et al [5] which showed that the actual dissipation energyof mudstone at failure was enhanced with the increasingof confining pressure In addition Chong et al [6] studiedthe influences of blasting disturbance water content andsample bedding on the strength behavior of sandstone byuniaxial and triaxial compression tests According to theresearch findings of constitutive model He and Kong [7]Zhang et al [8] and Jiang et al [9] established the constitutiverelation of rock and soil body respectively by employing

2 The Scientific World Journal

the improved nonlinear elastic model of Duncan-ZhangHuang and Subhash [10] Paliwal and Ramesh [11] andGraham-Brady [12] put forward some stress-strain relationsfor rockmass under uniaxial compression through the theoryof fracture mechanics Based on the Weibull probabilitydistribution theory of microunit strength and taking theprobabilistic damage variable Cao et al [13] Liu et al [14]Li et al [15] and Yang and Wang [16] set up severalisotropy statistical damage constitutive models for rock masswhich considered the random distribution of inner defectsin rock mass Actually the strain softening behavior of rockmass at postpeak stage is the result of strain localizationwhich is manifested as non-uniform deformation at failurestage Based on this point Fan et al [17] Wang [18] andChen and Pan [19] presented some different constitutivemodels considering strain localization based on work-energyprinciple

Results of comparison between numerous indoor exper-iments show that rock mass present different mechanicalbehaviors in different regions because of the complexityof geological conditions and the distribution discretenessof rock mass Even in the same area obvious differencesalso exist in rock samples which were taken from differentpositions So the above research results and models are notsuitable for the soft rock in the western mining area In thecurrent research on constitutive model statistical damagemodel which preferably associates the extension of micro-crack with the degradation of mechanical parameters cangive out accurate results for the description of mechanicalbehavior of rock mass However the present researches didnot consider the influence of confining pressure on elasticmodulus at the prepeak stage as well as the effect of stressstate changes on constitutive model In view of this thefailuremechanisms and constitutivemodels of typical weaklyconsolidated mudstone and coal samples in western miningarea under different stress states were studied in this paperThe conclusions obtained may provide theoretical basis forfurther numerical calculation and engineering application

2 Sample Preparation and Test Method

The test samples of mudstone and coal rock were all collectedfrom level transportation entry of NO 4 mine of Xinjiang Iliarea whose formation belongs to the typical weakly consol-idated soft rock In order to maintain the natural moisturecontent the specimens were packaged immediately aftersampling With the poor characteristics of weak cementationand low strength transfixion cracks often appear during thepreparation of mudstone samples and success rate is onlyabout 35 According to the national standard all sampleswere processed into cylinder with a diameter of 50mm and100mm high The group numbers of mudstone and coalsamples is all 3 to 5 for uniaxial and triaxial compressiontests respectively as shown in Figure 1 The sample numberwas recorded as A-B-i where A stands for compression style(among them D denoted uniaxial compression and S wastriaxial compression) B represented sample type (amongthem N denoted mudstone and M stands for coal rock) andi was sample number

Figure 1 Mudstone samples for triaxial compression test

The type of servo triaxial testing machine is TAW-2000Firstly a preload of 02 KN was applied on the sample toensure the close contact between rock specimen and loadingdevice Then axial compression displacement load at rate of02mmmin was applied in both uniaxial and triaxial com-pression tests until the samples were destroyed During thetriaxial compression test confining pressure was increasedto the predetermined value (in triaxial compression teststhe confining pressure was set as 1MPa 3MPa and 5MParesp) first at rate of 15NS and then maintained constant Itsvariation scope was less than 2 of the predetermined value

3 Failure Behavior of Weakly ConsolidatedSoft Rock under Uniaxial Compression

31Mechanical Behavior ofMudstone Therewere two typicalfailure modes for mudstone with different water contentsunder uniaxial compression As shown in Figure 2(a) thesample numbered D-N-1 whose water content (1694) wasclose to saturation state suffered global disintegration failureWhen the load was added to elastic limit the sample becamecrude and short and some local surface showed scale flakingFor the relatively dry samples with low water content asshown in Figure 2(b) which was numbered as D-N-2 and D-N-3 the main failure style was columnar splitting

Figure 3 showed the stress-strain relation of mudstonesamples under uniaxial compression Obviously the threetypical samples were all damaged rapidly at postpeak stageand almost had no residual strength in the process of loadingThe uniaxial compression strength of the three rock speci-mens were 579MPa 814MPa and 845MPa respectivelySo the strength of mudstone had great dispersion and thefundamental cause lied in the weak cementation differentwater content preexisting fractures and the different jointdistribution state and density in different samples

32 Failure Behavior of Coal Rock under Uniaxial Com-pression The failure modes of coal body under uniaxialcompression were illustrated in Figure 4 Overall columnarsplitting was the main failure mode for coal samples inwhich only one primary damage crack appeared and nolocal breakage phenomenon presented The sample mainly

The Scientific World Journal 3

D-N-1

(a) Global disintegration of mudstone with high watercontent

D-N-2 D-N-3

(b) Columnar splitting of mudstone withlow water content

Figure 2 Failure modes of mudstone samples under uniaxial compression with different moisture contents

0 02 04 06 08 10

1

2

3

4

5

6

7

8

9

D-N-1

D-N-2

D-N-3

Axial strain 1205761 ()

Axi

al st

ress1205901

(MPa

)

Figure 3 Stress-strain relations of mudstone samples under uniax-ial compression with different moisture contents

D-M-1 D-M-2

Figure 4 Splitting failure modes of coal samples under uniaxialcompression

suffered tensile damage under axial load and kept relativelyintact after failure The result showed that the pre-existingfractures in coal body were undeveloped

From the stress-strain relations (as shown in Figure 5)the curves of two coal samples were basically in coincidenceand bear the same variation tendency The two kinds of coal

0 02 04 06 08 1 12 140

1

2

3

4

5

6D-M-2

D-M-1

Axial strain 1205761 ()

Axi

al st

ress1205901

(MPa

)

Figure 5 Stress-strain relations of coal samples under uniaxialcompression

samples both suffered splitting failure at postpeak stage andno residual strengthwas retainedHowever the peak strengthof the samples was quite different from each other where D-M-1 was 498MPa and D-M-2 was 523MPa Though therewas difference the difference was not evident Obviouslythe strength of coal rock was lower than that of mudstoneAccording to Figures 4 and 5 the mechanical properties ofcoal body in this area showed small discreteness and relativeintegrity

4 Mechanical Behavior of Soft Rock underTriaxial Compression

41 Failure Behavior of Mudstone The stress-strain relationsof mudstone under confining pressures of 1MPa 3MPa and5MPawere shown in Figure 6 It was obvious that the confin-ing pressure had a significant effect on the shape and defor-mation feature of the curve The peak strength and residualstrengthwere both significantly enhancedwith the increasingof confining pressure Similar to hard rock the complete

4 The Scientific World Journal

0 02 04 06 08 1 12 14 160

3

6

9

12

15

18

21

24

27

30

5MPa

3MPa

1MPa

Axial strain 1205761 ()

Axi

al st

ress1205901

(MPa

)

Figure 6 Stress-strain relations of mudstone samples under triaxialcompression

5MPa3MPa1MPa0MPa

Figure 7 Comparison of failure modes of mudstone samples underdifferent confining pressures

stress-strain curve of weakly consolidated mudstone couldbe subdivided into five stages initial plastic deformationstage elastic deformation stage strain hardening stage strainsoftening stage and residual deformation stage However theultimate strain in each stage was much larger than that ofnormal rock

Because development of internal microcracks in mud-stone samples was inhibited by the effect of confining pres-sure the failure modes of mudstone were changed fromsplitting failure to shear failure under triaxial compressionThe different failure modes of mudstone samples underconfining pressures of 0MPa 1MPa 3MPa and 5MParespectively were presented in Figure 7 When the confiningpressure was set to 1MPa and 3MPa the sample showedsimple shear failure and the width of shear band was reducedwith the increasing of confining pressure which was com-pletely different from the columnar splitting under uniaxialcompression When the confining pressure was increasedto 5MPa no obvious failure band appeared on the samplesurface and the failure pattern of mudstone was changedto global plastic softening damage from shear failure It wasthus clear that the increasing of confining pressure effectively

0 05 1 15 2 25 3 35 40

2

4

6

8

10

12

14

16

18

5MPa

3MPa

1MPa

Axial strain 1205761 ()

Axi

al st

ress1205901

(MPa

)

Figure 8 Stress-strain relations of coal samples under triaxialcompression

5MPa3MPa1MPa0MPa

Figure 9 Changes of failure modes of coal samples under differentconfining pressure

inhabited the propagation of primary fractures and shearfailure in rock samples caused by the slipping of internalgrain

42 Damage Characteristics of Coal Rock in Soft Rock StrataFigure 8 presented the stress-strain relations of coal samplesunder triaxial compression With the increasing of confiningpressure the peak strength and residual strength of coalrock were significantly reinforced but the elastic modulus atprepeak stage was changed littleThe elastic stages at prepeakstage under different confining pressures were basically incoincidence while the peak strengths were moved backwardLarge stress drop was shown under low confining pressureafter peak point

Comparison of failure modes of coal samples underdifferent confining pressures was shown in Figure 9 Thefailuremode of coal sample was changed from brittle fractureto plastic shear failure with the increasing of confiningpressure When the confining pressure was set as 1MPa itwas found that only a single shear band presented whichis coincide with the diagonal line of sample and at a 634degree angle to the horizontal line If the confining pressurewas increased to 3MPa two conjugate shear zones with

The Scientific World Journal 5

0 1 2 3 4 5 6

6

9

12

15

18

21

24

27

30

Coal sample

Stre

ngth

(MPa

) Mudstone sample

Confining pressure 1205903 (MPa)

Figure 10 Changes of strengths ofmudstone and coal with differentconfining pressures

the angle of 599∘ to the horizontal line appeared after thesample damage When the confining pressure continued tobe increased to 5MPa no obvious shear band appeared andthe sample was suffered global plastic softening failure

The variation laws of peak strength (mean value of differ-ent samples) with different confining pressures for mudstoneand coal rock were illustrated in Figure 10 The regressionequation for the mudstone strength was

120590119898

119888

= 746 + 6151205903minus 286120590

2

3

+ 0491205903

3

(1)

The regression equation for the strength was as follows

120590119898

119888

= 556 + 4241205903minus 039120590

2

3

(2)

Obviously the strength of mudstone is more sensitiveto confining pressure than that of coal sample This is theresult of the developed primary fractures inmudstoneWhenthe confining pressure is greater than 3MPa the strengthof mudstone is significant improved with the increasing ofconfining pressure By contrast the strength variation rate ofcoal rock is relatively slow

5 Constitutive Model of Weakly ConsolidatedSoft Rock

In order to accurately establish the relationship betweenmechanical behavior and internal damage the constitutivemodel of soft rock was built on the basis of continuummechanics below A probabilistic damage factor was definedas follows [20]

119863 = 1 minus exp (minus119886120576119887) (3)

where 119886 and 119887 are fitting parameters Considering the initialdamage and residual strength of rock sample after damagethe damage factor should satisfy 0 lt 119863 lt 1 Thus amodification factor 120573 was introduced into (3) which waschanged as

1198631015840

= 1 minus 120573 exp (minus119886120576119887) (0 lt 120573 lt 1) (4)

0 2 4 615

2

25

3

35

4

45

Confining pressure 1205903 (MPa)

Elas

tic m

odul

usE

(GPa

)

Figure 11 Relation of elastic modulus of mudstone with confiningpressure

E

D

Fitting curve 2

BA

Fitting curve 1

Yield point

Dividing point of residual stage

O

C

120573 = 09

120573 = 092

120573 = 094

120573 = 096

120573 = 098

120573 = 1

minus8 minus7 minus6 minus5 minus4

Equivalent strain 120576

minus3

minus2

minus1

0

1

2

Equi

vale

nt st

ress120590

Figure 12 Relation between equivalent stress and equivalent strainwhen confining pressure was set to 3MPa

The following relationwas deduced according to the Lemaitrehypothesis of equivalent strain [21]

1205761= 120576∙

1

=

1

119864

(120590lowast

1

minus ] (120590lowast2

+ 120590lowast

3

)) (5)

where 120590lowast1

120590lowast2

120590lowast3

stand for the effective stress respectivelyand

120590lowast

1

=

1205901

1 minus 1198631015840

120590lowast

2

=

1205902

1 minus 1198631015840

120590lowast

3

=

1205903

1 minus 1198631015840

(6)

The constitutivemodel of weakly consolidated soft rock couldbe established by the comprehensive utilization of (4) to (6)

1205901= 1198641205761120573 exp (minus119886120576119887) + ] (120590

2+ 1205903) (7)

6 The Scientific World Journal

0 02 04 06 08 10

2

4

6

8

10

12

Fitting curves

120573

Axial strain 1205761 ()

Axi

al st

ress1205901

(MPa

)

(a) Confining pressure was set as 1MPa

0 02 04 06 08 1 120

2

4

6

8

10

12

14

Fitting curves

Axial strain 1205761 ()

Axi

al st

ress1205901

(MPa

)

(b) Confining pressure was set as 3MPa

0 03 06 09 12 15 18 210

4

8

12

16

20

24

28

Experimental data

Fitting curves

Axial strain 1205761 ()

Axi

al st

ress1205901

(MPa

)

(c) Confining pressure was set as 5MPa

Figure 13 Comparison of fitting curves with experimental data under different confining pressures

02 04 06 08 10 12 14 16 180

3

6

9

12

15

18

21

24

5MPa

3MPa1MPa

0MPa

Axial strain 1205761 ()

Axi

al st

ress1205901

(MPa

)

Figure 14 Fitting results of mudstone sample when 120573 = 09

The Scientific World Journal 7

Figure 6 indicated that the elastic modulus at prepeakstage ofmudstonewas concernedwith the confining pressureIn order to determine the relation a scatter diagram todescribe the corresponding relation between elastic modulus119864 and confining pressure 120590

3was drawn in Figure 11 where 119909-

coordinate denoted 1205903and 119910-coordinate represented elastic

modulusThe elastic modulus was obtained from the averageslope of elastic stage in complete stress-strain curve

Based on the parabolic equation the data was fitted as

119864 (1205903) = 202 minus 8733120590

3+ 1099120590

2

3

(8)

Therefore the elastic modulus 119864 in (7) should be modified bythe above 119864(120590

3) For the conventional triaxial compression

test the principle stress should meet the following relation1205902= 1205903 After taking logarithm twice for both sides of (7) it

was changed as

1205901= 119860 + 119887120576

1 (9)

in which the equivalent stress was defined as 1205901=

ln[minus ln((1205901minus 2]120590

3)120572119864(120590

3)1205761120573)] the equivalent strain was

defined as 1205761= ln 1205761 and 119860 = minus119887 ln 119886

Evidently (9) was a linear equation and its fitting param-eters could be solved through the series of data points (120590

1 1205761)

under different confining pressures In order to determinethe relationship between equivalent stress and equivalentstrain the data distributions of 120590

1and 120576

1were calculated

as shown in Figure 12 where the range of 120573 must satisfies120573 isin [09 1] and the confining pressure was set to 3MPaFrom the results 120590

1minus 1205761curves under different 120573 had the

same variation trend but the relations between the two werenot simply in linearity According to the analysis results ofexperimental data the turning point C in 120590

1minus 1205761curve just

corresponds to the yield point A at prepeak stage in stress-strain curve and the turning point D just corresponds tothe demarcation point B which lied between upper convexand lower convex at postpeak stage of stress-strain curve Tosimplify the calculation double linear fitting was carried outas shown in Figure 12 where OC and CE stand for the twofitting lines respectively Therefore (9) could be changed as

1205901= 1198601+ 119887112057611205761le 120576119901

1205901= 1198602+ 119887212057611205761gt 120576119901

(10)

where 120576119901was the axial strain at yield point and 119860

1 1198871 1198602 1198872

were the fitting parameters for the two straight fitting linesFigure 13 presented the fitting results according to the

experimental data under different confining pressures and120573 (the values were set as the same in Figure 12) Obviouslythe model could accurately reflect the elastic stage at prepeakstage the plastic yield feature and the strain softeningbehavior of weakly consolidated rockThe confining pressureplayed an important role to affect the strength and stress-strain curve shape of rockmassWhen the confining pressurewas set to 1MPa value of 120573 could determine the peakheight of the fitted curve The peak heights were in goodagreement with the test curve when 120573 = 09 If the confiningpressure was increased to 3MPa and 5MPa the fitting curves

were basically in coincidence and 120573 with different valueshas no effect on the fitting accuracy This implied that theinitial damage had little effect on the ultimate destructionof rock mass because these fractures were closed underhigh confining pressure This was consistent with the failurepattern in Figure 7 under high confining pressures

By comparison of fitting curves with experimental data itis found that the deficiency of thismodel was all lower convexat postpeak stagewhich failed to reflect the residual stagewithapproximate level linear style in Figure 12 The reason wasthat the experimental data at CE stage was fitted only by oneline which would cause error This problem could be solvedby dividing this stage into two parts CD and DE and then fitthem separately Because of the limited paper space this workwill be dealt with in other separate paper

Figure 14 presented the fitting results of experimental dataunder different confining pressures with 120573 = 09 Comparedwith Figure 6 the model could accurately reflect the rela-tionship between elastic modulus and confining pressure atprepeak stage That is the stiffness of test sample would beimproved with the increasing of confining pressure

6 Conclusions

(1) The weakly consolidated soft mudstone showed twofailure modes under uniaxial compression One is horizontalexpansion and disintegration with high water content theother is columnar splitting failure with low water contentUnder triaxial compression with low confining pressure coaland rock samples are all presented shear failures However ifthe confining pressure is increased to 3MPa no obvious sheardilatancy failure surface appears and the samples sufferedglobal plastic softening failure

(2) Similar to hard rock the complete stress-strain curveof soft rock can be divided into the following five stagesinitial plastic stage elastic stage strain hardening stagestrain softening stage and residual stage Nevertheless thecorresponding ultimate strain in each stage is much largerthan that of normal rock

(3) The constitutive model which considers the initialdamage of rock sample can accurately describe the behaviorof elasticity plastic yielding and strain softening as well asthe effect of initial damage on stress-strain curve Further-more the modified elastic modulus can accurately describethe relationship between elastic modulus and confiningpressure at prepeak stage The flaw of this model is thatthe deformation feature in residual phase cannot be fullyreflected

Acknowledgments

This Project is supported by the National Natural ScienceFoundation of China (Grant no 51174128) and the SpecializedResearch Fund for the Doctoral Program of Higher Educa-tion of China (Grant no 20123718110007)

8 The Scientific World Journal

References

[1] J P Zuo H P Xie and A M Wu ldquoInvestigation on failuremechanisms andmechanical behaviors of deep coal-rock singlebody and combined bodyrdquo Chinese Journal of Rock Mechanicsand Engineering vol 30 no 1 pp 84ndash92 2011

[2] Y Lu L Wang F Yang Y Li and H Chen ldquoPost-peak strainsoftening mechanical properties of weak rockrdquo Chinese Journalof Rock Mechanics and Engineering vol 29 no 3 pp 640ndash6482010

[3] F Zhang Q Sheng Z Zhu and Y Zhang ldquoStudy on post-peak mechanical behaviour and strain-softening model ofthree gorges graniterdquo Chinese Journal of Rock Mechanics andEngineering vol 27 supplement 1 pp 2651ndash2655 2008

[4] Z Li H Zhou Y Z Song C Q Zhang andQ Z Hu ldquoMechan-ical model of chlorite schist considering hardening-softeningand dilatancy characteristicsrdquo Rock and Soil Mechanics vol 34no 2 pp 404ndash410 2013

[5] T LHan Y S Chen T Xie Z Yu andMMHe ldquoExperimentalstudy on mechanics characteristics and energy properties ofmudstonerdquo Journal of Water Resources and Architectural Engi-neering vol 10 no 3 pp 116ndash120 2012

[6] Z Chong L Sheng-you Y Xi-jun S Yan and G Pan ldquoExper-imental investigation on strength of sandstone weathered andabsorbed water heavily and its constitutive lawrdquo EngineeringSciences vol 13 no 1 pp 74ndash80 2011

[7] L J He and L W Kong ldquoUniform expression of stress-strainrelationshiop of soilsrdquo Journal of Engineering Geology vol 18no 6 pp 900ndash905 2010

[8] W Z Zhang C X Chen and M Y Yu ldquoStudy of mechanicalproperties and constitutive relation of weathered sandstonerdquoRock and Soil Mechanics vol 30 pp 33ndash36 2009

[9] Y D Jiang X F Xian and J Su ldquoResearch on distortion ofsinglerock and constitutive relationrdquo Rock and Soil Mechanicsvol 26 no 6 pp 941ndash945 2005

[10] C Y Huang and G Subhash ldquoInfluence of lateral confinementon dynamic damage evolution during uniaxial compressiveresponse of brittle solidsrdquo Journal of the Mechanics and Physicsof Solids vol 51 no 6 pp 1089ndash1105 2003

[11] B Paliwal and K T Ramesh ldquoAn interacting micro-crack dam-age model for failure of brittle materials under compressionrdquoJournal of the Mechanics and Physics of Solids vol 56 no 3 pp896ndash923 2008

[12] L Graham-Brady ldquoStatistical characterization of meso-scaleuniaxial compressive strength in brittle materials with ran-domly occurring flawsrdquo International Journal of Solids andStructures vol 47 no 18-19 pp 2398ndash2413 2010

[13] W G Cao M H Zhao and C X Liu ldquoStudy on themodeland its modifying method for rock softening and damagebased onWeibull random distributionrdquo Chinese Journal of RockMechanics and Engineering vol 23 no 19 pp 3226ndash3231 2004

[14] X R Liu J B Wang P Li J Q Gu and Q Q ZhangldquoMechanical properties and constitutive model of thenarditerdquoJournal of PLA University of Science and Tcchnology (NaturalScience Edition) vol 13 no 5 pp 527ndash532 2012

[15] S Li J Xu and K G Li ldquoThe revises damage statisticalconstitutive model for rock based on uniform coefficientrdquoJournal of Sichuan University (Engineering Science Edition) vol39 no 6 pp 41ndash44 2007

[16] W Z Yang and BWang ldquoResearch on compressive constitutivemodel based onmeso-scopic damage of rockmaterialsrdquo Journal

of Zheng ZhouUniversity (Engineering Science) vol 31 no 6 pp6ndash9 2010

[17] H J Fan S X Xiao and K Peng ldquoStudy on constitutive modelof rock sample in uniaxial compression based on strain localiza-tionrdquo Chinese Journal of Rock Mechanics and Engineering vol25 no 1 pp 2634ndash2641 2006

[18] X B Wang ldquoDissipated energies and stabilities of axial and lat-eral deformations of rock specimens in uniaxial compressionrdquoChinese Journal of Rock Mechanics and Engineering vol 24 no5 pp 846ndash853 2005

[19] G Chen and P Y S Pan ldquoTwo-dimentional analytic expressionon strain localization of rock by strain gradient plasticitytheoryrdquoChinese Journal of RockMachanics and Engineering vol23 no 11 pp 1785ndash1791 2004

[20] C A Tang Catastrophe in Rock Unstable Failure China CoalIndustry Publishing House Beijing China 1993

[21] J Lemaitre ldquoHow to use damage mechanicsrdquoNuclear Engineer-ing and Design vol 80 no 2 pp 233ndash245 1984

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2 The Scientific World Journal

the improved nonlinear elastic model of Duncan-ZhangHuang and Subhash [10] Paliwal and Ramesh [11] andGraham-Brady [12] put forward some stress-strain relationsfor rockmass under uniaxial compression through the theoryof fracture mechanics Based on the Weibull probabilitydistribution theory of microunit strength and taking theprobabilistic damage variable Cao et al [13] Liu et al [14]Li et al [15] and Yang and Wang [16] set up severalisotropy statistical damage constitutive models for rock masswhich considered the random distribution of inner defectsin rock mass Actually the strain softening behavior of rockmass at postpeak stage is the result of strain localizationwhich is manifested as non-uniform deformation at failurestage Based on this point Fan et al [17] Wang [18] andChen and Pan [19] presented some different constitutivemodels considering strain localization based on work-energyprinciple

Results of comparison between numerous indoor exper-iments show that rock mass present different mechanicalbehaviors in different regions because of the complexityof geological conditions and the distribution discretenessof rock mass Even in the same area obvious differencesalso exist in rock samples which were taken from differentpositions So the above research results and models are notsuitable for the soft rock in the western mining area In thecurrent research on constitutive model statistical damagemodel which preferably associates the extension of micro-crack with the degradation of mechanical parameters cangive out accurate results for the description of mechanicalbehavior of rock mass However the present researches didnot consider the influence of confining pressure on elasticmodulus at the prepeak stage as well as the effect of stressstate changes on constitutive model In view of this thefailuremechanisms and constitutivemodels of typical weaklyconsolidated mudstone and coal samples in western miningarea under different stress states were studied in this paperThe conclusions obtained may provide theoretical basis forfurther numerical calculation and engineering application

2 Sample Preparation and Test Method

The test samples of mudstone and coal rock were all collectedfrom level transportation entry of NO 4 mine of Xinjiang Iliarea whose formation belongs to the typical weakly consol-idated soft rock In order to maintain the natural moisturecontent the specimens were packaged immediately aftersampling With the poor characteristics of weak cementationand low strength transfixion cracks often appear during thepreparation of mudstone samples and success rate is onlyabout 35 According to the national standard all sampleswere processed into cylinder with a diameter of 50mm and100mm high The group numbers of mudstone and coalsamples is all 3 to 5 for uniaxial and triaxial compressiontests respectively as shown in Figure 1 The sample numberwas recorded as A-B-i where A stands for compression style(among them D denoted uniaxial compression and S wastriaxial compression) B represented sample type (amongthem N denoted mudstone and M stands for coal rock) andi was sample number

Figure 1 Mudstone samples for triaxial compression test

The type of servo triaxial testing machine is TAW-2000Firstly a preload of 02 KN was applied on the sample toensure the close contact between rock specimen and loadingdevice Then axial compression displacement load at rate of02mmmin was applied in both uniaxial and triaxial com-pression tests until the samples were destroyed During thetriaxial compression test confining pressure was increasedto the predetermined value (in triaxial compression teststhe confining pressure was set as 1MPa 3MPa and 5MParesp) first at rate of 15NS and then maintained constant Itsvariation scope was less than 2 of the predetermined value

3 Failure Behavior of Weakly ConsolidatedSoft Rock under Uniaxial Compression

31Mechanical Behavior ofMudstone Therewere two typicalfailure modes for mudstone with different water contentsunder uniaxial compression As shown in Figure 2(a) thesample numbered D-N-1 whose water content (1694) wasclose to saturation state suffered global disintegration failureWhen the load was added to elastic limit the sample becamecrude and short and some local surface showed scale flakingFor the relatively dry samples with low water content asshown in Figure 2(b) which was numbered as D-N-2 and D-N-3 the main failure style was columnar splitting

Figure 3 showed the stress-strain relation of mudstonesamples under uniaxial compression Obviously the threetypical samples were all damaged rapidly at postpeak stageand almost had no residual strength in the process of loadingThe uniaxial compression strength of the three rock speci-mens were 579MPa 814MPa and 845MPa respectivelySo the strength of mudstone had great dispersion and thefundamental cause lied in the weak cementation differentwater content preexisting fractures and the different jointdistribution state and density in different samples

32 Failure Behavior of Coal Rock under Uniaxial Com-pression The failure modes of coal body under uniaxialcompression were illustrated in Figure 4 Overall columnarsplitting was the main failure mode for coal samples inwhich only one primary damage crack appeared and nolocal breakage phenomenon presented The sample mainly

The Scientific World Journal 3

D-N-1

(a) Global disintegration of mudstone with high watercontent

D-N-2 D-N-3

(b) Columnar splitting of mudstone withlow water content

Figure 2 Failure modes of mudstone samples under uniaxial compression with different moisture contents

0 02 04 06 08 10

1

2

3

4

5

6

7

8

9

D-N-1

D-N-2

D-N-3

Axial strain 1205761 ()

Axi

al st

ress1205901

(MPa

)

Figure 3 Stress-strain relations of mudstone samples under uniax-ial compression with different moisture contents

D-M-1 D-M-2

Figure 4 Splitting failure modes of coal samples under uniaxialcompression

suffered tensile damage under axial load and kept relativelyintact after failure The result showed that the pre-existingfractures in coal body were undeveloped

From the stress-strain relations (as shown in Figure 5)the curves of two coal samples were basically in coincidenceand bear the same variation tendency The two kinds of coal

0 02 04 06 08 1 12 140

1

2

3

4

5

6D-M-2

D-M-1

Axial strain 1205761 ()

Axi

al st

ress1205901

(MPa

)

Figure 5 Stress-strain relations of coal samples under uniaxialcompression

samples both suffered splitting failure at postpeak stage andno residual strengthwas retainedHowever the peak strengthof the samples was quite different from each other where D-M-1 was 498MPa and D-M-2 was 523MPa Though therewas difference the difference was not evident Obviouslythe strength of coal rock was lower than that of mudstoneAccording to Figures 4 and 5 the mechanical properties ofcoal body in this area showed small discreteness and relativeintegrity

4 Mechanical Behavior of Soft Rock underTriaxial Compression

41 Failure Behavior of Mudstone The stress-strain relationsof mudstone under confining pressures of 1MPa 3MPa and5MPawere shown in Figure 6 It was obvious that the confin-ing pressure had a significant effect on the shape and defor-mation feature of the curve The peak strength and residualstrengthwere both significantly enhancedwith the increasingof confining pressure Similar to hard rock the complete

4 The Scientific World Journal

0 02 04 06 08 1 12 14 160

3

6

9

12

15

18

21

24

27

30

5MPa

3MPa

1MPa

Axial strain 1205761 ()

Axi

al st

ress1205901

(MPa

)

Figure 6 Stress-strain relations of mudstone samples under triaxialcompression

5MPa3MPa1MPa0MPa

Figure 7 Comparison of failure modes of mudstone samples underdifferent confining pressures

stress-strain curve of weakly consolidated mudstone couldbe subdivided into five stages initial plastic deformationstage elastic deformation stage strain hardening stage strainsoftening stage and residual deformation stage However theultimate strain in each stage was much larger than that ofnormal rock

Because development of internal microcracks in mud-stone samples was inhibited by the effect of confining pres-sure the failure modes of mudstone were changed fromsplitting failure to shear failure under triaxial compressionThe different failure modes of mudstone samples underconfining pressures of 0MPa 1MPa 3MPa and 5MParespectively were presented in Figure 7 When the confiningpressure was set to 1MPa and 3MPa the sample showedsimple shear failure and the width of shear band was reducedwith the increasing of confining pressure which was com-pletely different from the columnar splitting under uniaxialcompression When the confining pressure was increasedto 5MPa no obvious failure band appeared on the samplesurface and the failure pattern of mudstone was changedto global plastic softening damage from shear failure It wasthus clear that the increasing of confining pressure effectively

0 05 1 15 2 25 3 35 40

2

4

6

8

10

12

14

16

18

5MPa

3MPa

1MPa

Axial strain 1205761 ()

Axi

al st

ress1205901

(MPa

)

Figure 8 Stress-strain relations of coal samples under triaxialcompression

5MPa3MPa1MPa0MPa

Figure 9 Changes of failure modes of coal samples under differentconfining pressure

inhabited the propagation of primary fractures and shearfailure in rock samples caused by the slipping of internalgrain

42 Damage Characteristics of Coal Rock in Soft Rock StrataFigure 8 presented the stress-strain relations of coal samplesunder triaxial compression With the increasing of confiningpressure the peak strength and residual strength of coalrock were significantly reinforced but the elastic modulus atprepeak stage was changed littleThe elastic stages at prepeakstage under different confining pressures were basically incoincidence while the peak strengths were moved backwardLarge stress drop was shown under low confining pressureafter peak point

Comparison of failure modes of coal samples underdifferent confining pressures was shown in Figure 9 Thefailuremode of coal sample was changed from brittle fractureto plastic shear failure with the increasing of confiningpressure When the confining pressure was set as 1MPa itwas found that only a single shear band presented whichis coincide with the diagonal line of sample and at a 634degree angle to the horizontal line If the confining pressurewas increased to 3MPa two conjugate shear zones with

The Scientific World Journal 5

0 1 2 3 4 5 6

6

9

12

15

18

21

24

27

30

Coal sample

Stre

ngth

(MPa

) Mudstone sample

Confining pressure 1205903 (MPa)

Figure 10 Changes of strengths ofmudstone and coal with differentconfining pressures

the angle of 599∘ to the horizontal line appeared after thesample damage When the confining pressure continued tobe increased to 5MPa no obvious shear band appeared andthe sample was suffered global plastic softening failure

The variation laws of peak strength (mean value of differ-ent samples) with different confining pressures for mudstoneand coal rock were illustrated in Figure 10 The regressionequation for the mudstone strength was

120590119898

119888

= 746 + 6151205903minus 286120590

2

3

+ 0491205903

3

(1)

The regression equation for the strength was as follows

120590119898

119888

= 556 + 4241205903minus 039120590

2

3

(2)

Obviously the strength of mudstone is more sensitiveto confining pressure than that of coal sample This is theresult of the developed primary fractures inmudstoneWhenthe confining pressure is greater than 3MPa the strengthof mudstone is significant improved with the increasing ofconfining pressure By contrast the strength variation rate ofcoal rock is relatively slow

5 Constitutive Model of Weakly ConsolidatedSoft Rock

In order to accurately establish the relationship betweenmechanical behavior and internal damage the constitutivemodel of soft rock was built on the basis of continuummechanics below A probabilistic damage factor was definedas follows [20]

119863 = 1 minus exp (minus119886120576119887) (3)

where 119886 and 119887 are fitting parameters Considering the initialdamage and residual strength of rock sample after damagethe damage factor should satisfy 0 lt 119863 lt 1 Thus amodification factor 120573 was introduced into (3) which waschanged as

1198631015840

= 1 minus 120573 exp (minus119886120576119887) (0 lt 120573 lt 1) (4)

0 2 4 615

2

25

3

35

4

45

Confining pressure 1205903 (MPa)

Elas

tic m

odul

usE

(GPa

)

Figure 11 Relation of elastic modulus of mudstone with confiningpressure

E

D

Fitting curve 2

BA

Fitting curve 1

Yield point

Dividing point of residual stage

O

C

120573 = 09

120573 = 092

120573 = 094

120573 = 096

120573 = 098

120573 = 1

minus8 minus7 minus6 minus5 minus4

Equivalent strain 120576

minus3

minus2

minus1

0

1

2

Equi

vale

nt st

ress120590

Figure 12 Relation between equivalent stress and equivalent strainwhen confining pressure was set to 3MPa

The following relationwas deduced according to the Lemaitrehypothesis of equivalent strain [21]

1205761= 120576∙

1

=

1

119864

(120590lowast

1

minus ] (120590lowast2

+ 120590lowast

3

)) (5)

where 120590lowast1

120590lowast2

120590lowast3

stand for the effective stress respectivelyand

120590lowast

1

=

1205901

1 minus 1198631015840

120590lowast

2

=

1205902

1 minus 1198631015840

120590lowast

3

=

1205903

1 minus 1198631015840

(6)

The constitutivemodel of weakly consolidated soft rock couldbe established by the comprehensive utilization of (4) to (6)

1205901= 1198641205761120573 exp (minus119886120576119887) + ] (120590

2+ 1205903) (7)

6 The Scientific World Journal

0 02 04 06 08 10

2

4

6

8

10

12

Fitting curves

120573

Axial strain 1205761 ()

Axi

al st

ress1205901

(MPa

)

(a) Confining pressure was set as 1MPa

0 02 04 06 08 1 120

2

4

6

8

10

12

14

Fitting curves

Axial strain 1205761 ()

Axi

al st

ress1205901

(MPa

)

(b) Confining pressure was set as 3MPa

0 03 06 09 12 15 18 210

4

8

12

16

20

24

28

Experimental data

Fitting curves

Axial strain 1205761 ()

Axi

al st

ress1205901

(MPa

)

(c) Confining pressure was set as 5MPa

Figure 13 Comparison of fitting curves with experimental data under different confining pressures

02 04 06 08 10 12 14 16 180

3

6

9

12

15

18

21

24

5MPa

3MPa1MPa

0MPa

Axial strain 1205761 ()

Axi

al st

ress1205901

(MPa

)

Figure 14 Fitting results of mudstone sample when 120573 = 09

The Scientific World Journal 7

Figure 6 indicated that the elastic modulus at prepeakstage ofmudstonewas concernedwith the confining pressureIn order to determine the relation a scatter diagram todescribe the corresponding relation between elastic modulus119864 and confining pressure 120590

3was drawn in Figure 11 where 119909-

coordinate denoted 1205903and 119910-coordinate represented elastic

modulusThe elastic modulus was obtained from the averageslope of elastic stage in complete stress-strain curve

Based on the parabolic equation the data was fitted as

119864 (1205903) = 202 minus 8733120590

3+ 1099120590

2

3

(8)

Therefore the elastic modulus 119864 in (7) should be modified bythe above 119864(120590

3) For the conventional triaxial compression

test the principle stress should meet the following relation1205902= 1205903 After taking logarithm twice for both sides of (7) it

was changed as

1205901= 119860 + 119887120576

1 (9)

in which the equivalent stress was defined as 1205901=

ln[minus ln((1205901minus 2]120590

3)120572119864(120590

3)1205761120573)] the equivalent strain was

defined as 1205761= ln 1205761 and 119860 = minus119887 ln 119886

Evidently (9) was a linear equation and its fitting param-eters could be solved through the series of data points (120590

1 1205761)

under different confining pressures In order to determinethe relationship between equivalent stress and equivalentstrain the data distributions of 120590

1and 120576

1were calculated

as shown in Figure 12 where the range of 120573 must satisfies120573 isin [09 1] and the confining pressure was set to 3MPaFrom the results 120590

1minus 1205761curves under different 120573 had the

same variation trend but the relations between the two werenot simply in linearity According to the analysis results ofexperimental data the turning point C in 120590

1minus 1205761curve just

corresponds to the yield point A at prepeak stage in stress-strain curve and the turning point D just corresponds tothe demarcation point B which lied between upper convexand lower convex at postpeak stage of stress-strain curve Tosimplify the calculation double linear fitting was carried outas shown in Figure 12 where OC and CE stand for the twofitting lines respectively Therefore (9) could be changed as

1205901= 1198601+ 119887112057611205761le 120576119901

1205901= 1198602+ 119887212057611205761gt 120576119901

(10)

where 120576119901was the axial strain at yield point and 119860

1 1198871 1198602 1198872

were the fitting parameters for the two straight fitting linesFigure 13 presented the fitting results according to the

experimental data under different confining pressures and120573 (the values were set as the same in Figure 12) Obviouslythe model could accurately reflect the elastic stage at prepeakstage the plastic yield feature and the strain softeningbehavior of weakly consolidated rockThe confining pressureplayed an important role to affect the strength and stress-strain curve shape of rockmassWhen the confining pressurewas set to 1MPa value of 120573 could determine the peakheight of the fitted curve The peak heights were in goodagreement with the test curve when 120573 = 09 If the confiningpressure was increased to 3MPa and 5MPa the fitting curves

were basically in coincidence and 120573 with different valueshas no effect on the fitting accuracy This implied that theinitial damage had little effect on the ultimate destructionof rock mass because these fractures were closed underhigh confining pressure This was consistent with the failurepattern in Figure 7 under high confining pressures

By comparison of fitting curves with experimental data itis found that the deficiency of thismodel was all lower convexat postpeak stagewhich failed to reflect the residual stagewithapproximate level linear style in Figure 12 The reason wasthat the experimental data at CE stage was fitted only by oneline which would cause error This problem could be solvedby dividing this stage into two parts CD and DE and then fitthem separately Because of the limited paper space this workwill be dealt with in other separate paper

Figure 14 presented the fitting results of experimental dataunder different confining pressures with 120573 = 09 Comparedwith Figure 6 the model could accurately reflect the rela-tionship between elastic modulus and confining pressure atprepeak stage That is the stiffness of test sample would beimproved with the increasing of confining pressure

6 Conclusions

(1) The weakly consolidated soft mudstone showed twofailure modes under uniaxial compression One is horizontalexpansion and disintegration with high water content theother is columnar splitting failure with low water contentUnder triaxial compression with low confining pressure coaland rock samples are all presented shear failures However ifthe confining pressure is increased to 3MPa no obvious sheardilatancy failure surface appears and the samples sufferedglobal plastic softening failure

(2) Similar to hard rock the complete stress-strain curveof soft rock can be divided into the following five stagesinitial plastic stage elastic stage strain hardening stagestrain softening stage and residual stage Nevertheless thecorresponding ultimate strain in each stage is much largerthan that of normal rock

(3) The constitutive model which considers the initialdamage of rock sample can accurately describe the behaviorof elasticity plastic yielding and strain softening as well asthe effect of initial damage on stress-strain curve Further-more the modified elastic modulus can accurately describethe relationship between elastic modulus and confiningpressure at prepeak stage The flaw of this model is thatthe deformation feature in residual phase cannot be fullyreflected

Acknowledgments

This Project is supported by the National Natural ScienceFoundation of China (Grant no 51174128) and the SpecializedResearch Fund for the Doctoral Program of Higher Educa-tion of China (Grant no 20123718110007)

8 The Scientific World Journal

References

[1] J P Zuo H P Xie and A M Wu ldquoInvestigation on failuremechanisms andmechanical behaviors of deep coal-rock singlebody and combined bodyrdquo Chinese Journal of Rock Mechanicsand Engineering vol 30 no 1 pp 84ndash92 2011

[2] Y Lu L Wang F Yang Y Li and H Chen ldquoPost-peak strainsoftening mechanical properties of weak rockrdquo Chinese Journalof Rock Mechanics and Engineering vol 29 no 3 pp 640ndash6482010

[3] F Zhang Q Sheng Z Zhu and Y Zhang ldquoStudy on post-peak mechanical behaviour and strain-softening model ofthree gorges graniterdquo Chinese Journal of Rock Mechanics andEngineering vol 27 supplement 1 pp 2651ndash2655 2008

[4] Z Li H Zhou Y Z Song C Q Zhang andQ Z Hu ldquoMechan-ical model of chlorite schist considering hardening-softeningand dilatancy characteristicsrdquo Rock and Soil Mechanics vol 34no 2 pp 404ndash410 2013

[5] T LHan Y S Chen T Xie Z Yu andMMHe ldquoExperimentalstudy on mechanics characteristics and energy properties ofmudstonerdquo Journal of Water Resources and Architectural Engi-neering vol 10 no 3 pp 116ndash120 2012

[6] Z Chong L Sheng-you Y Xi-jun S Yan and G Pan ldquoExper-imental investigation on strength of sandstone weathered andabsorbed water heavily and its constitutive lawrdquo EngineeringSciences vol 13 no 1 pp 74ndash80 2011

[7] L J He and L W Kong ldquoUniform expression of stress-strainrelationshiop of soilsrdquo Journal of Engineering Geology vol 18no 6 pp 900ndash905 2010

[8] W Z Zhang C X Chen and M Y Yu ldquoStudy of mechanicalproperties and constitutive relation of weathered sandstonerdquoRock and Soil Mechanics vol 30 pp 33ndash36 2009

[9] Y D Jiang X F Xian and J Su ldquoResearch on distortion ofsinglerock and constitutive relationrdquo Rock and Soil Mechanicsvol 26 no 6 pp 941ndash945 2005

[10] C Y Huang and G Subhash ldquoInfluence of lateral confinementon dynamic damage evolution during uniaxial compressiveresponse of brittle solidsrdquo Journal of the Mechanics and Physicsof Solids vol 51 no 6 pp 1089ndash1105 2003

[11] B Paliwal and K T Ramesh ldquoAn interacting micro-crack dam-age model for failure of brittle materials under compressionrdquoJournal of the Mechanics and Physics of Solids vol 56 no 3 pp896ndash923 2008

[12] L Graham-Brady ldquoStatistical characterization of meso-scaleuniaxial compressive strength in brittle materials with ran-domly occurring flawsrdquo International Journal of Solids andStructures vol 47 no 18-19 pp 2398ndash2413 2010

[13] W G Cao M H Zhao and C X Liu ldquoStudy on themodeland its modifying method for rock softening and damagebased onWeibull random distributionrdquo Chinese Journal of RockMechanics and Engineering vol 23 no 19 pp 3226ndash3231 2004

[14] X R Liu J B Wang P Li J Q Gu and Q Q ZhangldquoMechanical properties and constitutive model of thenarditerdquoJournal of PLA University of Science and Tcchnology (NaturalScience Edition) vol 13 no 5 pp 527ndash532 2012

[15] S Li J Xu and K G Li ldquoThe revises damage statisticalconstitutive model for rock based on uniform coefficientrdquoJournal of Sichuan University (Engineering Science Edition) vol39 no 6 pp 41ndash44 2007

[16] W Z Yang and BWang ldquoResearch on compressive constitutivemodel based onmeso-scopic damage of rockmaterialsrdquo Journal

of Zheng ZhouUniversity (Engineering Science) vol 31 no 6 pp6ndash9 2010

[17] H J Fan S X Xiao and K Peng ldquoStudy on constitutive modelof rock sample in uniaxial compression based on strain localiza-tionrdquo Chinese Journal of Rock Mechanics and Engineering vol25 no 1 pp 2634ndash2641 2006

[18] X B Wang ldquoDissipated energies and stabilities of axial and lat-eral deformations of rock specimens in uniaxial compressionrdquoChinese Journal of Rock Mechanics and Engineering vol 24 no5 pp 846ndash853 2005

[19] G Chen and P Y S Pan ldquoTwo-dimentional analytic expressionon strain localization of rock by strain gradient plasticitytheoryrdquoChinese Journal of RockMachanics and Engineering vol23 no 11 pp 1785ndash1791 2004

[20] C A Tang Catastrophe in Rock Unstable Failure China CoalIndustry Publishing House Beijing China 1993

[21] J Lemaitre ldquoHow to use damage mechanicsrdquoNuclear Engineer-ing and Design vol 80 no 2 pp 233ndash245 1984

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The Scientific World Journal 3

D-N-1

(a) Global disintegration of mudstone with high watercontent

D-N-2 D-N-3

(b) Columnar splitting of mudstone withlow water content

Figure 2 Failure modes of mudstone samples under uniaxial compression with different moisture contents

0 02 04 06 08 10

1

2

3

4

5

6

7

8

9

D-N-1

D-N-2

D-N-3

Axial strain 1205761 ()

Axi

al st

ress1205901

(MPa

)

Figure 3 Stress-strain relations of mudstone samples under uniax-ial compression with different moisture contents

D-M-1 D-M-2

Figure 4 Splitting failure modes of coal samples under uniaxialcompression

suffered tensile damage under axial load and kept relativelyintact after failure The result showed that the pre-existingfractures in coal body were undeveloped

From the stress-strain relations (as shown in Figure 5)the curves of two coal samples were basically in coincidenceand bear the same variation tendency The two kinds of coal

0 02 04 06 08 1 12 140

1

2

3

4

5

6D-M-2

D-M-1

Axial strain 1205761 ()

Axi

al st

ress1205901

(MPa

)

Figure 5 Stress-strain relations of coal samples under uniaxialcompression

samples both suffered splitting failure at postpeak stage andno residual strengthwas retainedHowever the peak strengthof the samples was quite different from each other where D-M-1 was 498MPa and D-M-2 was 523MPa Though therewas difference the difference was not evident Obviouslythe strength of coal rock was lower than that of mudstoneAccording to Figures 4 and 5 the mechanical properties ofcoal body in this area showed small discreteness and relativeintegrity

4 Mechanical Behavior of Soft Rock underTriaxial Compression

41 Failure Behavior of Mudstone The stress-strain relationsof mudstone under confining pressures of 1MPa 3MPa and5MPawere shown in Figure 6 It was obvious that the confin-ing pressure had a significant effect on the shape and defor-mation feature of the curve The peak strength and residualstrengthwere both significantly enhancedwith the increasingof confining pressure Similar to hard rock the complete

4 The Scientific World Journal

0 02 04 06 08 1 12 14 160

3

6

9

12

15

18

21

24

27

30

5MPa

3MPa

1MPa

Axial strain 1205761 ()

Axi

al st

ress1205901

(MPa

)

Figure 6 Stress-strain relations of mudstone samples under triaxialcompression

5MPa3MPa1MPa0MPa

Figure 7 Comparison of failure modes of mudstone samples underdifferent confining pressures

stress-strain curve of weakly consolidated mudstone couldbe subdivided into five stages initial plastic deformationstage elastic deformation stage strain hardening stage strainsoftening stage and residual deformation stage However theultimate strain in each stage was much larger than that ofnormal rock

Because development of internal microcracks in mud-stone samples was inhibited by the effect of confining pres-sure the failure modes of mudstone were changed fromsplitting failure to shear failure under triaxial compressionThe different failure modes of mudstone samples underconfining pressures of 0MPa 1MPa 3MPa and 5MParespectively were presented in Figure 7 When the confiningpressure was set to 1MPa and 3MPa the sample showedsimple shear failure and the width of shear band was reducedwith the increasing of confining pressure which was com-pletely different from the columnar splitting under uniaxialcompression When the confining pressure was increasedto 5MPa no obvious failure band appeared on the samplesurface and the failure pattern of mudstone was changedto global plastic softening damage from shear failure It wasthus clear that the increasing of confining pressure effectively

0 05 1 15 2 25 3 35 40

2

4

6

8

10

12

14

16

18

5MPa

3MPa

1MPa

Axial strain 1205761 ()

Axi

al st

ress1205901

(MPa

)

Figure 8 Stress-strain relations of coal samples under triaxialcompression

5MPa3MPa1MPa0MPa

Figure 9 Changes of failure modes of coal samples under differentconfining pressure

inhabited the propagation of primary fractures and shearfailure in rock samples caused by the slipping of internalgrain

42 Damage Characteristics of Coal Rock in Soft Rock StrataFigure 8 presented the stress-strain relations of coal samplesunder triaxial compression With the increasing of confiningpressure the peak strength and residual strength of coalrock were significantly reinforced but the elastic modulus atprepeak stage was changed littleThe elastic stages at prepeakstage under different confining pressures were basically incoincidence while the peak strengths were moved backwardLarge stress drop was shown under low confining pressureafter peak point

Comparison of failure modes of coal samples underdifferent confining pressures was shown in Figure 9 Thefailuremode of coal sample was changed from brittle fractureto plastic shear failure with the increasing of confiningpressure When the confining pressure was set as 1MPa itwas found that only a single shear band presented whichis coincide with the diagonal line of sample and at a 634degree angle to the horizontal line If the confining pressurewas increased to 3MPa two conjugate shear zones with

The Scientific World Journal 5

0 1 2 3 4 5 6

6

9

12

15

18

21

24

27

30

Coal sample

Stre

ngth

(MPa

) Mudstone sample

Confining pressure 1205903 (MPa)

Figure 10 Changes of strengths ofmudstone and coal with differentconfining pressures

the angle of 599∘ to the horizontal line appeared after thesample damage When the confining pressure continued tobe increased to 5MPa no obvious shear band appeared andthe sample was suffered global plastic softening failure

The variation laws of peak strength (mean value of differ-ent samples) with different confining pressures for mudstoneand coal rock were illustrated in Figure 10 The regressionequation for the mudstone strength was

120590119898

119888

= 746 + 6151205903minus 286120590

2

3

+ 0491205903

3

(1)

The regression equation for the strength was as follows

120590119898

119888

= 556 + 4241205903minus 039120590

2

3

(2)

Obviously the strength of mudstone is more sensitiveto confining pressure than that of coal sample This is theresult of the developed primary fractures inmudstoneWhenthe confining pressure is greater than 3MPa the strengthof mudstone is significant improved with the increasing ofconfining pressure By contrast the strength variation rate ofcoal rock is relatively slow

5 Constitutive Model of Weakly ConsolidatedSoft Rock

In order to accurately establish the relationship betweenmechanical behavior and internal damage the constitutivemodel of soft rock was built on the basis of continuummechanics below A probabilistic damage factor was definedas follows [20]

119863 = 1 minus exp (minus119886120576119887) (3)

where 119886 and 119887 are fitting parameters Considering the initialdamage and residual strength of rock sample after damagethe damage factor should satisfy 0 lt 119863 lt 1 Thus amodification factor 120573 was introduced into (3) which waschanged as

1198631015840

= 1 minus 120573 exp (minus119886120576119887) (0 lt 120573 lt 1) (4)

0 2 4 615

2

25

3

35

4

45

Confining pressure 1205903 (MPa)

Elas

tic m

odul

usE

(GPa

)

Figure 11 Relation of elastic modulus of mudstone with confiningpressure

E

D

Fitting curve 2

BA

Fitting curve 1

Yield point

Dividing point of residual stage

O

C

120573 = 09

120573 = 092

120573 = 094

120573 = 096

120573 = 098

120573 = 1

minus8 minus7 minus6 minus5 minus4

Equivalent strain 120576

minus3

minus2

minus1

0

1

2

Equi

vale

nt st

ress120590

Figure 12 Relation between equivalent stress and equivalent strainwhen confining pressure was set to 3MPa

The following relationwas deduced according to the Lemaitrehypothesis of equivalent strain [21]

1205761= 120576∙

1

=

1

119864

(120590lowast

1

minus ] (120590lowast2

+ 120590lowast

3

)) (5)

where 120590lowast1

120590lowast2

120590lowast3

stand for the effective stress respectivelyand

120590lowast

1

=

1205901

1 minus 1198631015840

120590lowast

2

=

1205902

1 minus 1198631015840

120590lowast

3

=

1205903

1 minus 1198631015840

(6)

The constitutivemodel of weakly consolidated soft rock couldbe established by the comprehensive utilization of (4) to (6)

1205901= 1198641205761120573 exp (minus119886120576119887) + ] (120590

2+ 1205903) (7)

6 The Scientific World Journal

0 02 04 06 08 10

2

4

6

8

10

12

Fitting curves

120573

Axial strain 1205761 ()

Axi

al st

ress1205901

(MPa

)

(a) Confining pressure was set as 1MPa

0 02 04 06 08 1 120

2

4

6

8

10

12

14

Fitting curves

Axial strain 1205761 ()

Axi

al st

ress1205901

(MPa

)

(b) Confining pressure was set as 3MPa

0 03 06 09 12 15 18 210

4

8

12

16

20

24

28

Experimental data

Fitting curves

Axial strain 1205761 ()

Axi

al st

ress1205901

(MPa

)

(c) Confining pressure was set as 5MPa

Figure 13 Comparison of fitting curves with experimental data under different confining pressures

02 04 06 08 10 12 14 16 180

3

6

9

12

15

18

21

24

5MPa

3MPa1MPa

0MPa

Axial strain 1205761 ()

Axi

al st

ress1205901

(MPa

)

Figure 14 Fitting results of mudstone sample when 120573 = 09

The Scientific World Journal 7

Figure 6 indicated that the elastic modulus at prepeakstage ofmudstonewas concernedwith the confining pressureIn order to determine the relation a scatter diagram todescribe the corresponding relation between elastic modulus119864 and confining pressure 120590

3was drawn in Figure 11 where 119909-

coordinate denoted 1205903and 119910-coordinate represented elastic

modulusThe elastic modulus was obtained from the averageslope of elastic stage in complete stress-strain curve

Based on the parabolic equation the data was fitted as

119864 (1205903) = 202 minus 8733120590

3+ 1099120590

2

3

(8)

Therefore the elastic modulus 119864 in (7) should be modified bythe above 119864(120590

3) For the conventional triaxial compression

test the principle stress should meet the following relation1205902= 1205903 After taking logarithm twice for both sides of (7) it

was changed as

1205901= 119860 + 119887120576

1 (9)

in which the equivalent stress was defined as 1205901=

ln[minus ln((1205901minus 2]120590

3)120572119864(120590

3)1205761120573)] the equivalent strain was

defined as 1205761= ln 1205761 and 119860 = minus119887 ln 119886

Evidently (9) was a linear equation and its fitting param-eters could be solved through the series of data points (120590

1 1205761)

under different confining pressures In order to determinethe relationship between equivalent stress and equivalentstrain the data distributions of 120590

1and 120576

1were calculated

as shown in Figure 12 where the range of 120573 must satisfies120573 isin [09 1] and the confining pressure was set to 3MPaFrom the results 120590

1minus 1205761curves under different 120573 had the

same variation trend but the relations between the two werenot simply in linearity According to the analysis results ofexperimental data the turning point C in 120590

1minus 1205761curve just

corresponds to the yield point A at prepeak stage in stress-strain curve and the turning point D just corresponds tothe demarcation point B which lied between upper convexand lower convex at postpeak stage of stress-strain curve Tosimplify the calculation double linear fitting was carried outas shown in Figure 12 where OC and CE stand for the twofitting lines respectively Therefore (9) could be changed as

1205901= 1198601+ 119887112057611205761le 120576119901

1205901= 1198602+ 119887212057611205761gt 120576119901

(10)

where 120576119901was the axial strain at yield point and 119860

1 1198871 1198602 1198872

were the fitting parameters for the two straight fitting linesFigure 13 presented the fitting results according to the

experimental data under different confining pressures and120573 (the values were set as the same in Figure 12) Obviouslythe model could accurately reflect the elastic stage at prepeakstage the plastic yield feature and the strain softeningbehavior of weakly consolidated rockThe confining pressureplayed an important role to affect the strength and stress-strain curve shape of rockmassWhen the confining pressurewas set to 1MPa value of 120573 could determine the peakheight of the fitted curve The peak heights were in goodagreement with the test curve when 120573 = 09 If the confiningpressure was increased to 3MPa and 5MPa the fitting curves

were basically in coincidence and 120573 with different valueshas no effect on the fitting accuracy This implied that theinitial damage had little effect on the ultimate destructionof rock mass because these fractures were closed underhigh confining pressure This was consistent with the failurepattern in Figure 7 under high confining pressures

By comparison of fitting curves with experimental data itis found that the deficiency of thismodel was all lower convexat postpeak stagewhich failed to reflect the residual stagewithapproximate level linear style in Figure 12 The reason wasthat the experimental data at CE stage was fitted only by oneline which would cause error This problem could be solvedby dividing this stage into two parts CD and DE and then fitthem separately Because of the limited paper space this workwill be dealt with in other separate paper

Figure 14 presented the fitting results of experimental dataunder different confining pressures with 120573 = 09 Comparedwith Figure 6 the model could accurately reflect the rela-tionship between elastic modulus and confining pressure atprepeak stage That is the stiffness of test sample would beimproved with the increasing of confining pressure

6 Conclusions

(1) The weakly consolidated soft mudstone showed twofailure modes under uniaxial compression One is horizontalexpansion and disintegration with high water content theother is columnar splitting failure with low water contentUnder triaxial compression with low confining pressure coaland rock samples are all presented shear failures However ifthe confining pressure is increased to 3MPa no obvious sheardilatancy failure surface appears and the samples sufferedglobal plastic softening failure

(2) Similar to hard rock the complete stress-strain curveof soft rock can be divided into the following five stagesinitial plastic stage elastic stage strain hardening stagestrain softening stage and residual stage Nevertheless thecorresponding ultimate strain in each stage is much largerthan that of normal rock

(3) The constitutive model which considers the initialdamage of rock sample can accurately describe the behaviorof elasticity plastic yielding and strain softening as well asthe effect of initial damage on stress-strain curve Further-more the modified elastic modulus can accurately describethe relationship between elastic modulus and confiningpressure at prepeak stage The flaw of this model is thatthe deformation feature in residual phase cannot be fullyreflected

Acknowledgments

This Project is supported by the National Natural ScienceFoundation of China (Grant no 51174128) and the SpecializedResearch Fund for the Doctoral Program of Higher Educa-tion of China (Grant no 20123718110007)

8 The Scientific World Journal

References

[1] J P Zuo H P Xie and A M Wu ldquoInvestigation on failuremechanisms andmechanical behaviors of deep coal-rock singlebody and combined bodyrdquo Chinese Journal of Rock Mechanicsand Engineering vol 30 no 1 pp 84ndash92 2011

[2] Y Lu L Wang F Yang Y Li and H Chen ldquoPost-peak strainsoftening mechanical properties of weak rockrdquo Chinese Journalof Rock Mechanics and Engineering vol 29 no 3 pp 640ndash6482010

[3] F Zhang Q Sheng Z Zhu and Y Zhang ldquoStudy on post-peak mechanical behaviour and strain-softening model ofthree gorges graniterdquo Chinese Journal of Rock Mechanics andEngineering vol 27 supplement 1 pp 2651ndash2655 2008

[4] Z Li H Zhou Y Z Song C Q Zhang andQ Z Hu ldquoMechan-ical model of chlorite schist considering hardening-softeningand dilatancy characteristicsrdquo Rock and Soil Mechanics vol 34no 2 pp 404ndash410 2013

[5] T LHan Y S Chen T Xie Z Yu andMMHe ldquoExperimentalstudy on mechanics characteristics and energy properties ofmudstonerdquo Journal of Water Resources and Architectural Engi-neering vol 10 no 3 pp 116ndash120 2012

[6] Z Chong L Sheng-you Y Xi-jun S Yan and G Pan ldquoExper-imental investigation on strength of sandstone weathered andabsorbed water heavily and its constitutive lawrdquo EngineeringSciences vol 13 no 1 pp 74ndash80 2011

[7] L J He and L W Kong ldquoUniform expression of stress-strainrelationshiop of soilsrdquo Journal of Engineering Geology vol 18no 6 pp 900ndash905 2010

[8] W Z Zhang C X Chen and M Y Yu ldquoStudy of mechanicalproperties and constitutive relation of weathered sandstonerdquoRock and Soil Mechanics vol 30 pp 33ndash36 2009

[9] Y D Jiang X F Xian and J Su ldquoResearch on distortion ofsinglerock and constitutive relationrdquo Rock and Soil Mechanicsvol 26 no 6 pp 941ndash945 2005

[10] C Y Huang and G Subhash ldquoInfluence of lateral confinementon dynamic damage evolution during uniaxial compressiveresponse of brittle solidsrdquo Journal of the Mechanics and Physicsof Solids vol 51 no 6 pp 1089ndash1105 2003

[11] B Paliwal and K T Ramesh ldquoAn interacting micro-crack dam-age model for failure of brittle materials under compressionrdquoJournal of the Mechanics and Physics of Solids vol 56 no 3 pp896ndash923 2008

[12] L Graham-Brady ldquoStatistical characterization of meso-scaleuniaxial compressive strength in brittle materials with ran-domly occurring flawsrdquo International Journal of Solids andStructures vol 47 no 18-19 pp 2398ndash2413 2010

[13] W G Cao M H Zhao and C X Liu ldquoStudy on themodeland its modifying method for rock softening and damagebased onWeibull random distributionrdquo Chinese Journal of RockMechanics and Engineering vol 23 no 19 pp 3226ndash3231 2004

[14] X R Liu J B Wang P Li J Q Gu and Q Q ZhangldquoMechanical properties and constitutive model of thenarditerdquoJournal of PLA University of Science and Tcchnology (NaturalScience Edition) vol 13 no 5 pp 527ndash532 2012

[15] S Li J Xu and K G Li ldquoThe revises damage statisticalconstitutive model for rock based on uniform coefficientrdquoJournal of Sichuan University (Engineering Science Edition) vol39 no 6 pp 41ndash44 2007

[16] W Z Yang and BWang ldquoResearch on compressive constitutivemodel based onmeso-scopic damage of rockmaterialsrdquo Journal

of Zheng ZhouUniversity (Engineering Science) vol 31 no 6 pp6ndash9 2010

[17] H J Fan S X Xiao and K Peng ldquoStudy on constitutive modelof rock sample in uniaxial compression based on strain localiza-tionrdquo Chinese Journal of Rock Mechanics and Engineering vol25 no 1 pp 2634ndash2641 2006

[18] X B Wang ldquoDissipated energies and stabilities of axial and lat-eral deformations of rock specimens in uniaxial compressionrdquoChinese Journal of Rock Mechanics and Engineering vol 24 no5 pp 846ndash853 2005

[19] G Chen and P Y S Pan ldquoTwo-dimentional analytic expressionon strain localization of rock by strain gradient plasticitytheoryrdquoChinese Journal of RockMachanics and Engineering vol23 no 11 pp 1785ndash1791 2004

[20] C A Tang Catastrophe in Rock Unstable Failure China CoalIndustry Publishing House Beijing China 1993

[21] J Lemaitre ldquoHow to use damage mechanicsrdquoNuclear Engineer-ing and Design vol 80 no 2 pp 233ndash245 1984

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

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Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

4 The Scientific World Journal

0 02 04 06 08 1 12 14 160

3

6

9

12

15

18

21

24

27

30

5MPa

3MPa

1MPa

Axial strain 1205761 ()

Axi

al st

ress1205901

(MPa

)

Figure 6 Stress-strain relations of mudstone samples under triaxialcompression

5MPa3MPa1MPa0MPa

Figure 7 Comparison of failure modes of mudstone samples underdifferent confining pressures

stress-strain curve of weakly consolidated mudstone couldbe subdivided into five stages initial plastic deformationstage elastic deformation stage strain hardening stage strainsoftening stage and residual deformation stage However theultimate strain in each stage was much larger than that ofnormal rock

Because development of internal microcracks in mud-stone samples was inhibited by the effect of confining pres-sure the failure modes of mudstone were changed fromsplitting failure to shear failure under triaxial compressionThe different failure modes of mudstone samples underconfining pressures of 0MPa 1MPa 3MPa and 5MParespectively were presented in Figure 7 When the confiningpressure was set to 1MPa and 3MPa the sample showedsimple shear failure and the width of shear band was reducedwith the increasing of confining pressure which was com-pletely different from the columnar splitting under uniaxialcompression When the confining pressure was increasedto 5MPa no obvious failure band appeared on the samplesurface and the failure pattern of mudstone was changedto global plastic softening damage from shear failure It wasthus clear that the increasing of confining pressure effectively

0 05 1 15 2 25 3 35 40

2

4

6

8

10

12

14

16

18

5MPa

3MPa

1MPa

Axial strain 1205761 ()

Axi

al st

ress1205901

(MPa

)

Figure 8 Stress-strain relations of coal samples under triaxialcompression

5MPa3MPa1MPa0MPa

Figure 9 Changes of failure modes of coal samples under differentconfining pressure

inhabited the propagation of primary fractures and shearfailure in rock samples caused by the slipping of internalgrain

42 Damage Characteristics of Coal Rock in Soft Rock StrataFigure 8 presented the stress-strain relations of coal samplesunder triaxial compression With the increasing of confiningpressure the peak strength and residual strength of coalrock were significantly reinforced but the elastic modulus atprepeak stage was changed littleThe elastic stages at prepeakstage under different confining pressures were basically incoincidence while the peak strengths were moved backwardLarge stress drop was shown under low confining pressureafter peak point

Comparison of failure modes of coal samples underdifferent confining pressures was shown in Figure 9 Thefailuremode of coal sample was changed from brittle fractureto plastic shear failure with the increasing of confiningpressure When the confining pressure was set as 1MPa itwas found that only a single shear band presented whichis coincide with the diagonal line of sample and at a 634degree angle to the horizontal line If the confining pressurewas increased to 3MPa two conjugate shear zones with

The Scientific World Journal 5

0 1 2 3 4 5 6

6

9

12

15

18

21

24

27

30

Coal sample

Stre

ngth

(MPa

) Mudstone sample

Confining pressure 1205903 (MPa)

Figure 10 Changes of strengths ofmudstone and coal with differentconfining pressures

the angle of 599∘ to the horizontal line appeared after thesample damage When the confining pressure continued tobe increased to 5MPa no obvious shear band appeared andthe sample was suffered global plastic softening failure

The variation laws of peak strength (mean value of differ-ent samples) with different confining pressures for mudstoneand coal rock were illustrated in Figure 10 The regressionequation for the mudstone strength was

120590119898

119888

= 746 + 6151205903minus 286120590

2

3

+ 0491205903

3

(1)

The regression equation for the strength was as follows

120590119898

119888

= 556 + 4241205903minus 039120590

2

3

(2)

Obviously the strength of mudstone is more sensitiveto confining pressure than that of coal sample This is theresult of the developed primary fractures inmudstoneWhenthe confining pressure is greater than 3MPa the strengthof mudstone is significant improved with the increasing ofconfining pressure By contrast the strength variation rate ofcoal rock is relatively slow

5 Constitutive Model of Weakly ConsolidatedSoft Rock

In order to accurately establish the relationship betweenmechanical behavior and internal damage the constitutivemodel of soft rock was built on the basis of continuummechanics below A probabilistic damage factor was definedas follows [20]

119863 = 1 minus exp (minus119886120576119887) (3)

where 119886 and 119887 are fitting parameters Considering the initialdamage and residual strength of rock sample after damagethe damage factor should satisfy 0 lt 119863 lt 1 Thus amodification factor 120573 was introduced into (3) which waschanged as

1198631015840

= 1 minus 120573 exp (minus119886120576119887) (0 lt 120573 lt 1) (4)

0 2 4 615

2

25

3

35

4

45

Confining pressure 1205903 (MPa)

Elas

tic m

odul

usE

(GPa

)

Figure 11 Relation of elastic modulus of mudstone with confiningpressure

E

D

Fitting curve 2

BA

Fitting curve 1

Yield point

Dividing point of residual stage

O

C

120573 = 09

120573 = 092

120573 = 094

120573 = 096

120573 = 098

120573 = 1

minus8 minus7 minus6 minus5 minus4

Equivalent strain 120576

minus3

minus2

minus1

0

1

2

Equi

vale

nt st

ress120590

Figure 12 Relation between equivalent stress and equivalent strainwhen confining pressure was set to 3MPa

The following relationwas deduced according to the Lemaitrehypothesis of equivalent strain [21]

1205761= 120576∙

1

=

1

119864

(120590lowast

1

minus ] (120590lowast2

+ 120590lowast

3

)) (5)

where 120590lowast1

120590lowast2

120590lowast3

stand for the effective stress respectivelyand

120590lowast

1

=

1205901

1 minus 1198631015840

120590lowast

2

=

1205902

1 minus 1198631015840

120590lowast

3

=

1205903

1 minus 1198631015840

(6)

The constitutivemodel of weakly consolidated soft rock couldbe established by the comprehensive utilization of (4) to (6)

1205901= 1198641205761120573 exp (minus119886120576119887) + ] (120590

2+ 1205903) (7)

6 The Scientific World Journal

0 02 04 06 08 10

2

4

6

8

10

12

Fitting curves

120573

Axial strain 1205761 ()

Axi

al st

ress1205901

(MPa

)

(a) Confining pressure was set as 1MPa

0 02 04 06 08 1 120

2

4

6

8

10

12

14

Fitting curves

Axial strain 1205761 ()

Axi

al st

ress1205901

(MPa

)

(b) Confining pressure was set as 3MPa

0 03 06 09 12 15 18 210

4

8

12

16

20

24

28

Experimental data

Fitting curves

Axial strain 1205761 ()

Axi

al st

ress1205901

(MPa

)

(c) Confining pressure was set as 5MPa

Figure 13 Comparison of fitting curves with experimental data under different confining pressures

02 04 06 08 10 12 14 16 180

3

6

9

12

15

18

21

24

5MPa

3MPa1MPa

0MPa

Axial strain 1205761 ()

Axi

al st

ress1205901

(MPa

)

Figure 14 Fitting results of mudstone sample when 120573 = 09

The Scientific World Journal 7

Figure 6 indicated that the elastic modulus at prepeakstage ofmudstonewas concernedwith the confining pressureIn order to determine the relation a scatter diagram todescribe the corresponding relation between elastic modulus119864 and confining pressure 120590

3was drawn in Figure 11 where 119909-

coordinate denoted 1205903and 119910-coordinate represented elastic

modulusThe elastic modulus was obtained from the averageslope of elastic stage in complete stress-strain curve

Based on the parabolic equation the data was fitted as

119864 (1205903) = 202 minus 8733120590

3+ 1099120590

2

3

(8)

Therefore the elastic modulus 119864 in (7) should be modified bythe above 119864(120590

3) For the conventional triaxial compression

test the principle stress should meet the following relation1205902= 1205903 After taking logarithm twice for both sides of (7) it

was changed as

1205901= 119860 + 119887120576

1 (9)

in which the equivalent stress was defined as 1205901=

ln[minus ln((1205901minus 2]120590

3)120572119864(120590

3)1205761120573)] the equivalent strain was

defined as 1205761= ln 1205761 and 119860 = minus119887 ln 119886

Evidently (9) was a linear equation and its fitting param-eters could be solved through the series of data points (120590

1 1205761)

under different confining pressures In order to determinethe relationship between equivalent stress and equivalentstrain the data distributions of 120590

1and 120576

1were calculated

as shown in Figure 12 where the range of 120573 must satisfies120573 isin [09 1] and the confining pressure was set to 3MPaFrom the results 120590

1minus 1205761curves under different 120573 had the

same variation trend but the relations between the two werenot simply in linearity According to the analysis results ofexperimental data the turning point C in 120590

1minus 1205761curve just

corresponds to the yield point A at prepeak stage in stress-strain curve and the turning point D just corresponds tothe demarcation point B which lied between upper convexand lower convex at postpeak stage of stress-strain curve Tosimplify the calculation double linear fitting was carried outas shown in Figure 12 where OC and CE stand for the twofitting lines respectively Therefore (9) could be changed as

1205901= 1198601+ 119887112057611205761le 120576119901

1205901= 1198602+ 119887212057611205761gt 120576119901

(10)

where 120576119901was the axial strain at yield point and 119860

1 1198871 1198602 1198872

were the fitting parameters for the two straight fitting linesFigure 13 presented the fitting results according to the

experimental data under different confining pressures and120573 (the values were set as the same in Figure 12) Obviouslythe model could accurately reflect the elastic stage at prepeakstage the plastic yield feature and the strain softeningbehavior of weakly consolidated rockThe confining pressureplayed an important role to affect the strength and stress-strain curve shape of rockmassWhen the confining pressurewas set to 1MPa value of 120573 could determine the peakheight of the fitted curve The peak heights were in goodagreement with the test curve when 120573 = 09 If the confiningpressure was increased to 3MPa and 5MPa the fitting curves

were basically in coincidence and 120573 with different valueshas no effect on the fitting accuracy This implied that theinitial damage had little effect on the ultimate destructionof rock mass because these fractures were closed underhigh confining pressure This was consistent with the failurepattern in Figure 7 under high confining pressures

By comparison of fitting curves with experimental data itis found that the deficiency of thismodel was all lower convexat postpeak stagewhich failed to reflect the residual stagewithapproximate level linear style in Figure 12 The reason wasthat the experimental data at CE stage was fitted only by oneline which would cause error This problem could be solvedby dividing this stage into two parts CD and DE and then fitthem separately Because of the limited paper space this workwill be dealt with in other separate paper

Figure 14 presented the fitting results of experimental dataunder different confining pressures with 120573 = 09 Comparedwith Figure 6 the model could accurately reflect the rela-tionship between elastic modulus and confining pressure atprepeak stage That is the stiffness of test sample would beimproved with the increasing of confining pressure

6 Conclusions

(1) The weakly consolidated soft mudstone showed twofailure modes under uniaxial compression One is horizontalexpansion and disintegration with high water content theother is columnar splitting failure with low water contentUnder triaxial compression with low confining pressure coaland rock samples are all presented shear failures However ifthe confining pressure is increased to 3MPa no obvious sheardilatancy failure surface appears and the samples sufferedglobal plastic softening failure

(2) Similar to hard rock the complete stress-strain curveof soft rock can be divided into the following five stagesinitial plastic stage elastic stage strain hardening stagestrain softening stage and residual stage Nevertheless thecorresponding ultimate strain in each stage is much largerthan that of normal rock

(3) The constitutive model which considers the initialdamage of rock sample can accurately describe the behaviorof elasticity plastic yielding and strain softening as well asthe effect of initial damage on stress-strain curve Further-more the modified elastic modulus can accurately describethe relationship between elastic modulus and confiningpressure at prepeak stage The flaw of this model is thatthe deformation feature in residual phase cannot be fullyreflected

Acknowledgments

This Project is supported by the National Natural ScienceFoundation of China (Grant no 51174128) and the SpecializedResearch Fund for the Doctoral Program of Higher Educa-tion of China (Grant no 20123718110007)

8 The Scientific World Journal

References

[1] J P Zuo H P Xie and A M Wu ldquoInvestigation on failuremechanisms andmechanical behaviors of deep coal-rock singlebody and combined bodyrdquo Chinese Journal of Rock Mechanicsand Engineering vol 30 no 1 pp 84ndash92 2011

[2] Y Lu L Wang F Yang Y Li and H Chen ldquoPost-peak strainsoftening mechanical properties of weak rockrdquo Chinese Journalof Rock Mechanics and Engineering vol 29 no 3 pp 640ndash6482010

[3] F Zhang Q Sheng Z Zhu and Y Zhang ldquoStudy on post-peak mechanical behaviour and strain-softening model ofthree gorges graniterdquo Chinese Journal of Rock Mechanics andEngineering vol 27 supplement 1 pp 2651ndash2655 2008

[4] Z Li H Zhou Y Z Song C Q Zhang andQ Z Hu ldquoMechan-ical model of chlorite schist considering hardening-softeningand dilatancy characteristicsrdquo Rock and Soil Mechanics vol 34no 2 pp 404ndash410 2013

[5] T LHan Y S Chen T Xie Z Yu andMMHe ldquoExperimentalstudy on mechanics characteristics and energy properties ofmudstonerdquo Journal of Water Resources and Architectural Engi-neering vol 10 no 3 pp 116ndash120 2012

[6] Z Chong L Sheng-you Y Xi-jun S Yan and G Pan ldquoExper-imental investigation on strength of sandstone weathered andabsorbed water heavily and its constitutive lawrdquo EngineeringSciences vol 13 no 1 pp 74ndash80 2011

[7] L J He and L W Kong ldquoUniform expression of stress-strainrelationshiop of soilsrdquo Journal of Engineering Geology vol 18no 6 pp 900ndash905 2010

[8] W Z Zhang C X Chen and M Y Yu ldquoStudy of mechanicalproperties and constitutive relation of weathered sandstonerdquoRock and Soil Mechanics vol 30 pp 33ndash36 2009

[9] Y D Jiang X F Xian and J Su ldquoResearch on distortion ofsinglerock and constitutive relationrdquo Rock and Soil Mechanicsvol 26 no 6 pp 941ndash945 2005

[10] C Y Huang and G Subhash ldquoInfluence of lateral confinementon dynamic damage evolution during uniaxial compressiveresponse of brittle solidsrdquo Journal of the Mechanics and Physicsof Solids vol 51 no 6 pp 1089ndash1105 2003

[11] B Paliwal and K T Ramesh ldquoAn interacting micro-crack dam-age model for failure of brittle materials under compressionrdquoJournal of the Mechanics and Physics of Solids vol 56 no 3 pp896ndash923 2008

[12] L Graham-Brady ldquoStatistical characterization of meso-scaleuniaxial compressive strength in brittle materials with ran-domly occurring flawsrdquo International Journal of Solids andStructures vol 47 no 18-19 pp 2398ndash2413 2010

[13] W G Cao M H Zhao and C X Liu ldquoStudy on themodeland its modifying method for rock softening and damagebased onWeibull random distributionrdquo Chinese Journal of RockMechanics and Engineering vol 23 no 19 pp 3226ndash3231 2004

[14] X R Liu J B Wang P Li J Q Gu and Q Q ZhangldquoMechanical properties and constitutive model of thenarditerdquoJournal of PLA University of Science and Tcchnology (NaturalScience Edition) vol 13 no 5 pp 527ndash532 2012

[15] S Li J Xu and K G Li ldquoThe revises damage statisticalconstitutive model for rock based on uniform coefficientrdquoJournal of Sichuan University (Engineering Science Edition) vol39 no 6 pp 41ndash44 2007

[16] W Z Yang and BWang ldquoResearch on compressive constitutivemodel based onmeso-scopic damage of rockmaterialsrdquo Journal

of Zheng ZhouUniversity (Engineering Science) vol 31 no 6 pp6ndash9 2010

[17] H J Fan S X Xiao and K Peng ldquoStudy on constitutive modelof rock sample in uniaxial compression based on strain localiza-tionrdquo Chinese Journal of Rock Mechanics and Engineering vol25 no 1 pp 2634ndash2641 2006

[18] X B Wang ldquoDissipated energies and stabilities of axial and lat-eral deformations of rock specimens in uniaxial compressionrdquoChinese Journal of Rock Mechanics and Engineering vol 24 no5 pp 846ndash853 2005

[19] G Chen and P Y S Pan ldquoTwo-dimentional analytic expressionon strain localization of rock by strain gradient plasticitytheoryrdquoChinese Journal of RockMachanics and Engineering vol23 no 11 pp 1785ndash1791 2004

[20] C A Tang Catastrophe in Rock Unstable Failure China CoalIndustry Publishing House Beijing China 1993

[21] J Lemaitre ldquoHow to use damage mechanicsrdquoNuclear Engineer-ing and Design vol 80 no 2 pp 233ndash245 1984

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

The Scientific World Journal 5

0 1 2 3 4 5 6

6

9

12

15

18

21

24

27

30

Coal sample

Stre

ngth

(MPa

) Mudstone sample

Confining pressure 1205903 (MPa)

Figure 10 Changes of strengths ofmudstone and coal with differentconfining pressures

the angle of 599∘ to the horizontal line appeared after thesample damage When the confining pressure continued tobe increased to 5MPa no obvious shear band appeared andthe sample was suffered global plastic softening failure

The variation laws of peak strength (mean value of differ-ent samples) with different confining pressures for mudstoneand coal rock were illustrated in Figure 10 The regressionequation for the mudstone strength was

120590119898

119888

= 746 + 6151205903minus 286120590

2

3

+ 0491205903

3

(1)

The regression equation for the strength was as follows

120590119898

119888

= 556 + 4241205903minus 039120590

2

3

(2)

Obviously the strength of mudstone is more sensitiveto confining pressure than that of coal sample This is theresult of the developed primary fractures inmudstoneWhenthe confining pressure is greater than 3MPa the strengthof mudstone is significant improved with the increasing ofconfining pressure By contrast the strength variation rate ofcoal rock is relatively slow

5 Constitutive Model of Weakly ConsolidatedSoft Rock

In order to accurately establish the relationship betweenmechanical behavior and internal damage the constitutivemodel of soft rock was built on the basis of continuummechanics below A probabilistic damage factor was definedas follows [20]

119863 = 1 minus exp (minus119886120576119887) (3)

where 119886 and 119887 are fitting parameters Considering the initialdamage and residual strength of rock sample after damagethe damage factor should satisfy 0 lt 119863 lt 1 Thus amodification factor 120573 was introduced into (3) which waschanged as

1198631015840

= 1 minus 120573 exp (minus119886120576119887) (0 lt 120573 lt 1) (4)

0 2 4 615

2

25

3

35

4

45

Confining pressure 1205903 (MPa)

Elas

tic m

odul

usE

(GPa

)

Figure 11 Relation of elastic modulus of mudstone with confiningpressure

E

D

Fitting curve 2

BA

Fitting curve 1

Yield point

Dividing point of residual stage

O

C

120573 = 09

120573 = 092

120573 = 094

120573 = 096

120573 = 098

120573 = 1

minus8 minus7 minus6 minus5 minus4

Equivalent strain 120576

minus3

minus2

minus1

0

1

2

Equi

vale

nt st

ress120590

Figure 12 Relation between equivalent stress and equivalent strainwhen confining pressure was set to 3MPa

The following relationwas deduced according to the Lemaitrehypothesis of equivalent strain [21]

1205761= 120576∙

1

=

1

119864

(120590lowast

1

minus ] (120590lowast2

+ 120590lowast

3

)) (5)

where 120590lowast1

120590lowast2

120590lowast3

stand for the effective stress respectivelyand

120590lowast

1

=

1205901

1 minus 1198631015840

120590lowast

2

=

1205902

1 minus 1198631015840

120590lowast

3

=

1205903

1 minus 1198631015840

(6)

The constitutivemodel of weakly consolidated soft rock couldbe established by the comprehensive utilization of (4) to (6)

1205901= 1198641205761120573 exp (minus119886120576119887) + ] (120590

2+ 1205903) (7)

6 The Scientific World Journal

0 02 04 06 08 10

2

4

6

8

10

12

Fitting curves

120573

Axial strain 1205761 ()

Axi

al st

ress1205901

(MPa

)

(a) Confining pressure was set as 1MPa

0 02 04 06 08 1 120

2

4

6

8

10

12

14

Fitting curves

Axial strain 1205761 ()

Axi

al st

ress1205901

(MPa

)

(b) Confining pressure was set as 3MPa

0 03 06 09 12 15 18 210

4

8

12

16

20

24

28

Experimental data

Fitting curves

Axial strain 1205761 ()

Axi

al st

ress1205901

(MPa

)

(c) Confining pressure was set as 5MPa

Figure 13 Comparison of fitting curves with experimental data under different confining pressures

02 04 06 08 10 12 14 16 180

3

6

9

12

15

18

21

24

5MPa

3MPa1MPa

0MPa

Axial strain 1205761 ()

Axi

al st

ress1205901

(MPa

)

Figure 14 Fitting results of mudstone sample when 120573 = 09

The Scientific World Journal 7

Figure 6 indicated that the elastic modulus at prepeakstage ofmudstonewas concernedwith the confining pressureIn order to determine the relation a scatter diagram todescribe the corresponding relation between elastic modulus119864 and confining pressure 120590

3was drawn in Figure 11 where 119909-

coordinate denoted 1205903and 119910-coordinate represented elastic

modulusThe elastic modulus was obtained from the averageslope of elastic stage in complete stress-strain curve

Based on the parabolic equation the data was fitted as

119864 (1205903) = 202 minus 8733120590

3+ 1099120590

2

3

(8)

Therefore the elastic modulus 119864 in (7) should be modified bythe above 119864(120590

3) For the conventional triaxial compression

test the principle stress should meet the following relation1205902= 1205903 After taking logarithm twice for both sides of (7) it

was changed as

1205901= 119860 + 119887120576

1 (9)

in which the equivalent stress was defined as 1205901=

ln[minus ln((1205901minus 2]120590

3)120572119864(120590

3)1205761120573)] the equivalent strain was

defined as 1205761= ln 1205761 and 119860 = minus119887 ln 119886

Evidently (9) was a linear equation and its fitting param-eters could be solved through the series of data points (120590

1 1205761)

under different confining pressures In order to determinethe relationship between equivalent stress and equivalentstrain the data distributions of 120590

1and 120576

1were calculated

as shown in Figure 12 where the range of 120573 must satisfies120573 isin [09 1] and the confining pressure was set to 3MPaFrom the results 120590

1minus 1205761curves under different 120573 had the

same variation trend but the relations between the two werenot simply in linearity According to the analysis results ofexperimental data the turning point C in 120590

1minus 1205761curve just

corresponds to the yield point A at prepeak stage in stress-strain curve and the turning point D just corresponds tothe demarcation point B which lied between upper convexand lower convex at postpeak stage of stress-strain curve Tosimplify the calculation double linear fitting was carried outas shown in Figure 12 where OC and CE stand for the twofitting lines respectively Therefore (9) could be changed as

1205901= 1198601+ 119887112057611205761le 120576119901

1205901= 1198602+ 119887212057611205761gt 120576119901

(10)

where 120576119901was the axial strain at yield point and 119860

1 1198871 1198602 1198872

were the fitting parameters for the two straight fitting linesFigure 13 presented the fitting results according to the

experimental data under different confining pressures and120573 (the values were set as the same in Figure 12) Obviouslythe model could accurately reflect the elastic stage at prepeakstage the plastic yield feature and the strain softeningbehavior of weakly consolidated rockThe confining pressureplayed an important role to affect the strength and stress-strain curve shape of rockmassWhen the confining pressurewas set to 1MPa value of 120573 could determine the peakheight of the fitted curve The peak heights were in goodagreement with the test curve when 120573 = 09 If the confiningpressure was increased to 3MPa and 5MPa the fitting curves

were basically in coincidence and 120573 with different valueshas no effect on the fitting accuracy This implied that theinitial damage had little effect on the ultimate destructionof rock mass because these fractures were closed underhigh confining pressure This was consistent with the failurepattern in Figure 7 under high confining pressures

By comparison of fitting curves with experimental data itis found that the deficiency of thismodel was all lower convexat postpeak stagewhich failed to reflect the residual stagewithapproximate level linear style in Figure 12 The reason wasthat the experimental data at CE stage was fitted only by oneline which would cause error This problem could be solvedby dividing this stage into two parts CD and DE and then fitthem separately Because of the limited paper space this workwill be dealt with in other separate paper

Figure 14 presented the fitting results of experimental dataunder different confining pressures with 120573 = 09 Comparedwith Figure 6 the model could accurately reflect the rela-tionship between elastic modulus and confining pressure atprepeak stage That is the stiffness of test sample would beimproved with the increasing of confining pressure

6 Conclusions

(1) The weakly consolidated soft mudstone showed twofailure modes under uniaxial compression One is horizontalexpansion and disintegration with high water content theother is columnar splitting failure with low water contentUnder triaxial compression with low confining pressure coaland rock samples are all presented shear failures However ifthe confining pressure is increased to 3MPa no obvious sheardilatancy failure surface appears and the samples sufferedglobal plastic softening failure

(2) Similar to hard rock the complete stress-strain curveof soft rock can be divided into the following five stagesinitial plastic stage elastic stage strain hardening stagestrain softening stage and residual stage Nevertheless thecorresponding ultimate strain in each stage is much largerthan that of normal rock

(3) The constitutive model which considers the initialdamage of rock sample can accurately describe the behaviorof elasticity plastic yielding and strain softening as well asthe effect of initial damage on stress-strain curve Further-more the modified elastic modulus can accurately describethe relationship between elastic modulus and confiningpressure at prepeak stage The flaw of this model is thatthe deformation feature in residual phase cannot be fullyreflected

Acknowledgments

This Project is supported by the National Natural ScienceFoundation of China (Grant no 51174128) and the SpecializedResearch Fund for the Doctoral Program of Higher Educa-tion of China (Grant no 20123718110007)

8 The Scientific World Journal

References

[1] J P Zuo H P Xie and A M Wu ldquoInvestigation on failuremechanisms andmechanical behaviors of deep coal-rock singlebody and combined bodyrdquo Chinese Journal of Rock Mechanicsand Engineering vol 30 no 1 pp 84ndash92 2011

[2] Y Lu L Wang F Yang Y Li and H Chen ldquoPost-peak strainsoftening mechanical properties of weak rockrdquo Chinese Journalof Rock Mechanics and Engineering vol 29 no 3 pp 640ndash6482010

[3] F Zhang Q Sheng Z Zhu and Y Zhang ldquoStudy on post-peak mechanical behaviour and strain-softening model ofthree gorges graniterdquo Chinese Journal of Rock Mechanics andEngineering vol 27 supplement 1 pp 2651ndash2655 2008

[4] Z Li H Zhou Y Z Song C Q Zhang andQ Z Hu ldquoMechan-ical model of chlorite schist considering hardening-softeningand dilatancy characteristicsrdquo Rock and Soil Mechanics vol 34no 2 pp 404ndash410 2013

[5] T LHan Y S Chen T Xie Z Yu andMMHe ldquoExperimentalstudy on mechanics characteristics and energy properties ofmudstonerdquo Journal of Water Resources and Architectural Engi-neering vol 10 no 3 pp 116ndash120 2012

[6] Z Chong L Sheng-you Y Xi-jun S Yan and G Pan ldquoExper-imental investigation on strength of sandstone weathered andabsorbed water heavily and its constitutive lawrdquo EngineeringSciences vol 13 no 1 pp 74ndash80 2011

[7] L J He and L W Kong ldquoUniform expression of stress-strainrelationshiop of soilsrdquo Journal of Engineering Geology vol 18no 6 pp 900ndash905 2010

[8] W Z Zhang C X Chen and M Y Yu ldquoStudy of mechanicalproperties and constitutive relation of weathered sandstonerdquoRock and Soil Mechanics vol 30 pp 33ndash36 2009

[9] Y D Jiang X F Xian and J Su ldquoResearch on distortion ofsinglerock and constitutive relationrdquo Rock and Soil Mechanicsvol 26 no 6 pp 941ndash945 2005

[10] C Y Huang and G Subhash ldquoInfluence of lateral confinementon dynamic damage evolution during uniaxial compressiveresponse of brittle solidsrdquo Journal of the Mechanics and Physicsof Solids vol 51 no 6 pp 1089ndash1105 2003

[11] B Paliwal and K T Ramesh ldquoAn interacting micro-crack dam-age model for failure of brittle materials under compressionrdquoJournal of the Mechanics and Physics of Solids vol 56 no 3 pp896ndash923 2008

[12] L Graham-Brady ldquoStatistical characterization of meso-scaleuniaxial compressive strength in brittle materials with ran-domly occurring flawsrdquo International Journal of Solids andStructures vol 47 no 18-19 pp 2398ndash2413 2010

[13] W G Cao M H Zhao and C X Liu ldquoStudy on themodeland its modifying method for rock softening and damagebased onWeibull random distributionrdquo Chinese Journal of RockMechanics and Engineering vol 23 no 19 pp 3226ndash3231 2004

[14] X R Liu J B Wang P Li J Q Gu and Q Q ZhangldquoMechanical properties and constitutive model of thenarditerdquoJournal of PLA University of Science and Tcchnology (NaturalScience Edition) vol 13 no 5 pp 527ndash532 2012

[15] S Li J Xu and K G Li ldquoThe revises damage statisticalconstitutive model for rock based on uniform coefficientrdquoJournal of Sichuan University (Engineering Science Edition) vol39 no 6 pp 41ndash44 2007

[16] W Z Yang and BWang ldquoResearch on compressive constitutivemodel based onmeso-scopic damage of rockmaterialsrdquo Journal

of Zheng ZhouUniversity (Engineering Science) vol 31 no 6 pp6ndash9 2010

[17] H J Fan S X Xiao and K Peng ldquoStudy on constitutive modelof rock sample in uniaxial compression based on strain localiza-tionrdquo Chinese Journal of Rock Mechanics and Engineering vol25 no 1 pp 2634ndash2641 2006

[18] X B Wang ldquoDissipated energies and stabilities of axial and lat-eral deformations of rock specimens in uniaxial compressionrdquoChinese Journal of Rock Mechanics and Engineering vol 24 no5 pp 846ndash853 2005

[19] G Chen and P Y S Pan ldquoTwo-dimentional analytic expressionon strain localization of rock by strain gradient plasticitytheoryrdquoChinese Journal of RockMachanics and Engineering vol23 no 11 pp 1785ndash1791 2004

[20] C A Tang Catastrophe in Rock Unstable Failure China CoalIndustry Publishing House Beijing China 1993

[21] J Lemaitre ldquoHow to use damage mechanicsrdquoNuclear Engineer-ing and Design vol 80 no 2 pp 233ndash245 1984

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

6 The Scientific World Journal

0 02 04 06 08 10

2

4

6

8

10

12

Fitting curves

120573

Axial strain 1205761 ()

Axi

al st

ress1205901

(MPa

)

(a) Confining pressure was set as 1MPa

0 02 04 06 08 1 120

2

4

6

8

10

12

14

Fitting curves

Axial strain 1205761 ()

Axi

al st

ress1205901

(MPa

)

(b) Confining pressure was set as 3MPa

0 03 06 09 12 15 18 210

4

8

12

16

20

24

28

Experimental data

Fitting curves

Axial strain 1205761 ()

Axi

al st

ress1205901

(MPa

)

(c) Confining pressure was set as 5MPa

Figure 13 Comparison of fitting curves with experimental data under different confining pressures

02 04 06 08 10 12 14 16 180

3

6

9

12

15

18

21

24

5MPa

3MPa1MPa

0MPa

Axial strain 1205761 ()

Axi

al st

ress1205901

(MPa

)

Figure 14 Fitting results of mudstone sample when 120573 = 09

The Scientific World Journal 7

Figure 6 indicated that the elastic modulus at prepeakstage ofmudstonewas concernedwith the confining pressureIn order to determine the relation a scatter diagram todescribe the corresponding relation between elastic modulus119864 and confining pressure 120590

3was drawn in Figure 11 where 119909-

coordinate denoted 1205903and 119910-coordinate represented elastic

modulusThe elastic modulus was obtained from the averageslope of elastic stage in complete stress-strain curve

Based on the parabolic equation the data was fitted as

119864 (1205903) = 202 minus 8733120590

3+ 1099120590

2

3

(8)

Therefore the elastic modulus 119864 in (7) should be modified bythe above 119864(120590

3) For the conventional triaxial compression

test the principle stress should meet the following relation1205902= 1205903 After taking logarithm twice for both sides of (7) it

was changed as

1205901= 119860 + 119887120576

1 (9)

in which the equivalent stress was defined as 1205901=

ln[minus ln((1205901minus 2]120590

3)120572119864(120590

3)1205761120573)] the equivalent strain was

defined as 1205761= ln 1205761 and 119860 = minus119887 ln 119886

Evidently (9) was a linear equation and its fitting param-eters could be solved through the series of data points (120590

1 1205761)

under different confining pressures In order to determinethe relationship between equivalent stress and equivalentstrain the data distributions of 120590

1and 120576

1were calculated

as shown in Figure 12 where the range of 120573 must satisfies120573 isin [09 1] and the confining pressure was set to 3MPaFrom the results 120590

1minus 1205761curves under different 120573 had the

same variation trend but the relations between the two werenot simply in linearity According to the analysis results ofexperimental data the turning point C in 120590

1minus 1205761curve just

corresponds to the yield point A at prepeak stage in stress-strain curve and the turning point D just corresponds tothe demarcation point B which lied between upper convexand lower convex at postpeak stage of stress-strain curve Tosimplify the calculation double linear fitting was carried outas shown in Figure 12 where OC and CE stand for the twofitting lines respectively Therefore (9) could be changed as

1205901= 1198601+ 119887112057611205761le 120576119901

1205901= 1198602+ 119887212057611205761gt 120576119901

(10)

where 120576119901was the axial strain at yield point and 119860

1 1198871 1198602 1198872

were the fitting parameters for the two straight fitting linesFigure 13 presented the fitting results according to the

experimental data under different confining pressures and120573 (the values were set as the same in Figure 12) Obviouslythe model could accurately reflect the elastic stage at prepeakstage the plastic yield feature and the strain softeningbehavior of weakly consolidated rockThe confining pressureplayed an important role to affect the strength and stress-strain curve shape of rockmassWhen the confining pressurewas set to 1MPa value of 120573 could determine the peakheight of the fitted curve The peak heights were in goodagreement with the test curve when 120573 = 09 If the confiningpressure was increased to 3MPa and 5MPa the fitting curves

were basically in coincidence and 120573 with different valueshas no effect on the fitting accuracy This implied that theinitial damage had little effect on the ultimate destructionof rock mass because these fractures were closed underhigh confining pressure This was consistent with the failurepattern in Figure 7 under high confining pressures

By comparison of fitting curves with experimental data itis found that the deficiency of thismodel was all lower convexat postpeak stagewhich failed to reflect the residual stagewithapproximate level linear style in Figure 12 The reason wasthat the experimental data at CE stage was fitted only by oneline which would cause error This problem could be solvedby dividing this stage into two parts CD and DE and then fitthem separately Because of the limited paper space this workwill be dealt with in other separate paper

Figure 14 presented the fitting results of experimental dataunder different confining pressures with 120573 = 09 Comparedwith Figure 6 the model could accurately reflect the rela-tionship between elastic modulus and confining pressure atprepeak stage That is the stiffness of test sample would beimproved with the increasing of confining pressure

6 Conclusions

(1) The weakly consolidated soft mudstone showed twofailure modes under uniaxial compression One is horizontalexpansion and disintegration with high water content theother is columnar splitting failure with low water contentUnder triaxial compression with low confining pressure coaland rock samples are all presented shear failures However ifthe confining pressure is increased to 3MPa no obvious sheardilatancy failure surface appears and the samples sufferedglobal plastic softening failure

(2) Similar to hard rock the complete stress-strain curveof soft rock can be divided into the following five stagesinitial plastic stage elastic stage strain hardening stagestrain softening stage and residual stage Nevertheless thecorresponding ultimate strain in each stage is much largerthan that of normal rock

(3) The constitutive model which considers the initialdamage of rock sample can accurately describe the behaviorof elasticity plastic yielding and strain softening as well asthe effect of initial damage on stress-strain curve Further-more the modified elastic modulus can accurately describethe relationship between elastic modulus and confiningpressure at prepeak stage The flaw of this model is thatthe deformation feature in residual phase cannot be fullyreflected

Acknowledgments

This Project is supported by the National Natural ScienceFoundation of China (Grant no 51174128) and the SpecializedResearch Fund for the Doctoral Program of Higher Educa-tion of China (Grant no 20123718110007)

8 The Scientific World Journal

References

[1] J P Zuo H P Xie and A M Wu ldquoInvestigation on failuremechanisms andmechanical behaviors of deep coal-rock singlebody and combined bodyrdquo Chinese Journal of Rock Mechanicsand Engineering vol 30 no 1 pp 84ndash92 2011

[2] Y Lu L Wang F Yang Y Li and H Chen ldquoPost-peak strainsoftening mechanical properties of weak rockrdquo Chinese Journalof Rock Mechanics and Engineering vol 29 no 3 pp 640ndash6482010

[3] F Zhang Q Sheng Z Zhu and Y Zhang ldquoStudy on post-peak mechanical behaviour and strain-softening model ofthree gorges graniterdquo Chinese Journal of Rock Mechanics andEngineering vol 27 supplement 1 pp 2651ndash2655 2008

[4] Z Li H Zhou Y Z Song C Q Zhang andQ Z Hu ldquoMechan-ical model of chlorite schist considering hardening-softeningand dilatancy characteristicsrdquo Rock and Soil Mechanics vol 34no 2 pp 404ndash410 2013

[5] T LHan Y S Chen T Xie Z Yu andMMHe ldquoExperimentalstudy on mechanics characteristics and energy properties ofmudstonerdquo Journal of Water Resources and Architectural Engi-neering vol 10 no 3 pp 116ndash120 2012

[6] Z Chong L Sheng-you Y Xi-jun S Yan and G Pan ldquoExper-imental investigation on strength of sandstone weathered andabsorbed water heavily and its constitutive lawrdquo EngineeringSciences vol 13 no 1 pp 74ndash80 2011

[7] L J He and L W Kong ldquoUniform expression of stress-strainrelationshiop of soilsrdquo Journal of Engineering Geology vol 18no 6 pp 900ndash905 2010

[8] W Z Zhang C X Chen and M Y Yu ldquoStudy of mechanicalproperties and constitutive relation of weathered sandstonerdquoRock and Soil Mechanics vol 30 pp 33ndash36 2009

[9] Y D Jiang X F Xian and J Su ldquoResearch on distortion ofsinglerock and constitutive relationrdquo Rock and Soil Mechanicsvol 26 no 6 pp 941ndash945 2005

[10] C Y Huang and G Subhash ldquoInfluence of lateral confinementon dynamic damage evolution during uniaxial compressiveresponse of brittle solidsrdquo Journal of the Mechanics and Physicsof Solids vol 51 no 6 pp 1089ndash1105 2003

[11] B Paliwal and K T Ramesh ldquoAn interacting micro-crack dam-age model for failure of brittle materials under compressionrdquoJournal of the Mechanics and Physics of Solids vol 56 no 3 pp896ndash923 2008

[12] L Graham-Brady ldquoStatistical characterization of meso-scaleuniaxial compressive strength in brittle materials with ran-domly occurring flawsrdquo International Journal of Solids andStructures vol 47 no 18-19 pp 2398ndash2413 2010

[13] W G Cao M H Zhao and C X Liu ldquoStudy on themodeland its modifying method for rock softening and damagebased onWeibull random distributionrdquo Chinese Journal of RockMechanics and Engineering vol 23 no 19 pp 3226ndash3231 2004

[14] X R Liu J B Wang P Li J Q Gu and Q Q ZhangldquoMechanical properties and constitutive model of thenarditerdquoJournal of PLA University of Science and Tcchnology (NaturalScience Edition) vol 13 no 5 pp 527ndash532 2012

[15] S Li J Xu and K G Li ldquoThe revises damage statisticalconstitutive model for rock based on uniform coefficientrdquoJournal of Sichuan University (Engineering Science Edition) vol39 no 6 pp 41ndash44 2007

[16] W Z Yang and BWang ldquoResearch on compressive constitutivemodel based onmeso-scopic damage of rockmaterialsrdquo Journal

of Zheng ZhouUniversity (Engineering Science) vol 31 no 6 pp6ndash9 2010

[17] H J Fan S X Xiao and K Peng ldquoStudy on constitutive modelof rock sample in uniaxial compression based on strain localiza-tionrdquo Chinese Journal of Rock Mechanics and Engineering vol25 no 1 pp 2634ndash2641 2006

[18] X B Wang ldquoDissipated energies and stabilities of axial and lat-eral deformations of rock specimens in uniaxial compressionrdquoChinese Journal of Rock Mechanics and Engineering vol 24 no5 pp 846ndash853 2005

[19] G Chen and P Y S Pan ldquoTwo-dimentional analytic expressionon strain localization of rock by strain gradient plasticitytheoryrdquoChinese Journal of RockMachanics and Engineering vol23 no 11 pp 1785ndash1791 2004

[20] C A Tang Catastrophe in Rock Unstable Failure China CoalIndustry Publishing House Beijing China 1993

[21] J Lemaitre ldquoHow to use damage mechanicsrdquoNuclear Engineer-ing and Design vol 80 no 2 pp 233ndash245 1984

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

The Scientific World Journal 7

Figure 6 indicated that the elastic modulus at prepeakstage ofmudstonewas concernedwith the confining pressureIn order to determine the relation a scatter diagram todescribe the corresponding relation between elastic modulus119864 and confining pressure 120590

3was drawn in Figure 11 where 119909-

coordinate denoted 1205903and 119910-coordinate represented elastic

modulusThe elastic modulus was obtained from the averageslope of elastic stage in complete stress-strain curve

Based on the parabolic equation the data was fitted as

119864 (1205903) = 202 minus 8733120590

3+ 1099120590

2

3

(8)

Therefore the elastic modulus 119864 in (7) should be modified bythe above 119864(120590

3) For the conventional triaxial compression

test the principle stress should meet the following relation1205902= 1205903 After taking logarithm twice for both sides of (7) it

was changed as

1205901= 119860 + 119887120576

1 (9)

in which the equivalent stress was defined as 1205901=

ln[minus ln((1205901minus 2]120590

3)120572119864(120590

3)1205761120573)] the equivalent strain was

defined as 1205761= ln 1205761 and 119860 = minus119887 ln 119886

Evidently (9) was a linear equation and its fitting param-eters could be solved through the series of data points (120590

1 1205761)

under different confining pressures In order to determinethe relationship between equivalent stress and equivalentstrain the data distributions of 120590

1and 120576

1were calculated

as shown in Figure 12 where the range of 120573 must satisfies120573 isin [09 1] and the confining pressure was set to 3MPaFrom the results 120590

1minus 1205761curves under different 120573 had the

same variation trend but the relations between the two werenot simply in linearity According to the analysis results ofexperimental data the turning point C in 120590

1minus 1205761curve just

corresponds to the yield point A at prepeak stage in stress-strain curve and the turning point D just corresponds tothe demarcation point B which lied between upper convexand lower convex at postpeak stage of stress-strain curve Tosimplify the calculation double linear fitting was carried outas shown in Figure 12 where OC and CE stand for the twofitting lines respectively Therefore (9) could be changed as

1205901= 1198601+ 119887112057611205761le 120576119901

1205901= 1198602+ 119887212057611205761gt 120576119901

(10)

where 120576119901was the axial strain at yield point and 119860

1 1198871 1198602 1198872

were the fitting parameters for the two straight fitting linesFigure 13 presented the fitting results according to the

experimental data under different confining pressures and120573 (the values were set as the same in Figure 12) Obviouslythe model could accurately reflect the elastic stage at prepeakstage the plastic yield feature and the strain softeningbehavior of weakly consolidated rockThe confining pressureplayed an important role to affect the strength and stress-strain curve shape of rockmassWhen the confining pressurewas set to 1MPa value of 120573 could determine the peakheight of the fitted curve The peak heights were in goodagreement with the test curve when 120573 = 09 If the confiningpressure was increased to 3MPa and 5MPa the fitting curves

were basically in coincidence and 120573 with different valueshas no effect on the fitting accuracy This implied that theinitial damage had little effect on the ultimate destructionof rock mass because these fractures were closed underhigh confining pressure This was consistent with the failurepattern in Figure 7 under high confining pressures

By comparison of fitting curves with experimental data itis found that the deficiency of thismodel was all lower convexat postpeak stagewhich failed to reflect the residual stagewithapproximate level linear style in Figure 12 The reason wasthat the experimental data at CE stage was fitted only by oneline which would cause error This problem could be solvedby dividing this stage into two parts CD and DE and then fitthem separately Because of the limited paper space this workwill be dealt with in other separate paper

Figure 14 presented the fitting results of experimental dataunder different confining pressures with 120573 = 09 Comparedwith Figure 6 the model could accurately reflect the rela-tionship between elastic modulus and confining pressure atprepeak stage That is the stiffness of test sample would beimproved with the increasing of confining pressure

6 Conclusions

(1) The weakly consolidated soft mudstone showed twofailure modes under uniaxial compression One is horizontalexpansion and disintegration with high water content theother is columnar splitting failure with low water contentUnder triaxial compression with low confining pressure coaland rock samples are all presented shear failures However ifthe confining pressure is increased to 3MPa no obvious sheardilatancy failure surface appears and the samples sufferedglobal plastic softening failure

(2) Similar to hard rock the complete stress-strain curveof soft rock can be divided into the following five stagesinitial plastic stage elastic stage strain hardening stagestrain softening stage and residual stage Nevertheless thecorresponding ultimate strain in each stage is much largerthan that of normal rock

(3) The constitutive model which considers the initialdamage of rock sample can accurately describe the behaviorof elasticity plastic yielding and strain softening as well asthe effect of initial damage on stress-strain curve Further-more the modified elastic modulus can accurately describethe relationship between elastic modulus and confiningpressure at prepeak stage The flaw of this model is thatthe deformation feature in residual phase cannot be fullyreflected

Acknowledgments

This Project is supported by the National Natural ScienceFoundation of China (Grant no 51174128) and the SpecializedResearch Fund for the Doctoral Program of Higher Educa-tion of China (Grant no 20123718110007)

8 The Scientific World Journal

References

[1] J P Zuo H P Xie and A M Wu ldquoInvestigation on failuremechanisms andmechanical behaviors of deep coal-rock singlebody and combined bodyrdquo Chinese Journal of Rock Mechanicsand Engineering vol 30 no 1 pp 84ndash92 2011

[2] Y Lu L Wang F Yang Y Li and H Chen ldquoPost-peak strainsoftening mechanical properties of weak rockrdquo Chinese Journalof Rock Mechanics and Engineering vol 29 no 3 pp 640ndash6482010

[3] F Zhang Q Sheng Z Zhu and Y Zhang ldquoStudy on post-peak mechanical behaviour and strain-softening model ofthree gorges graniterdquo Chinese Journal of Rock Mechanics andEngineering vol 27 supplement 1 pp 2651ndash2655 2008

[4] Z Li H Zhou Y Z Song C Q Zhang andQ Z Hu ldquoMechan-ical model of chlorite schist considering hardening-softeningand dilatancy characteristicsrdquo Rock and Soil Mechanics vol 34no 2 pp 404ndash410 2013

[5] T LHan Y S Chen T Xie Z Yu andMMHe ldquoExperimentalstudy on mechanics characteristics and energy properties ofmudstonerdquo Journal of Water Resources and Architectural Engi-neering vol 10 no 3 pp 116ndash120 2012

[6] Z Chong L Sheng-you Y Xi-jun S Yan and G Pan ldquoExper-imental investigation on strength of sandstone weathered andabsorbed water heavily and its constitutive lawrdquo EngineeringSciences vol 13 no 1 pp 74ndash80 2011

[7] L J He and L W Kong ldquoUniform expression of stress-strainrelationshiop of soilsrdquo Journal of Engineering Geology vol 18no 6 pp 900ndash905 2010

[8] W Z Zhang C X Chen and M Y Yu ldquoStudy of mechanicalproperties and constitutive relation of weathered sandstonerdquoRock and Soil Mechanics vol 30 pp 33ndash36 2009

[9] Y D Jiang X F Xian and J Su ldquoResearch on distortion ofsinglerock and constitutive relationrdquo Rock and Soil Mechanicsvol 26 no 6 pp 941ndash945 2005

[10] C Y Huang and G Subhash ldquoInfluence of lateral confinementon dynamic damage evolution during uniaxial compressiveresponse of brittle solidsrdquo Journal of the Mechanics and Physicsof Solids vol 51 no 6 pp 1089ndash1105 2003

[11] B Paliwal and K T Ramesh ldquoAn interacting micro-crack dam-age model for failure of brittle materials under compressionrdquoJournal of the Mechanics and Physics of Solids vol 56 no 3 pp896ndash923 2008

[12] L Graham-Brady ldquoStatistical characterization of meso-scaleuniaxial compressive strength in brittle materials with ran-domly occurring flawsrdquo International Journal of Solids andStructures vol 47 no 18-19 pp 2398ndash2413 2010

[13] W G Cao M H Zhao and C X Liu ldquoStudy on themodeland its modifying method for rock softening and damagebased onWeibull random distributionrdquo Chinese Journal of RockMechanics and Engineering vol 23 no 19 pp 3226ndash3231 2004

[14] X R Liu J B Wang P Li J Q Gu and Q Q ZhangldquoMechanical properties and constitutive model of thenarditerdquoJournal of PLA University of Science and Tcchnology (NaturalScience Edition) vol 13 no 5 pp 527ndash532 2012

[15] S Li J Xu and K G Li ldquoThe revises damage statisticalconstitutive model for rock based on uniform coefficientrdquoJournal of Sichuan University (Engineering Science Edition) vol39 no 6 pp 41ndash44 2007

[16] W Z Yang and BWang ldquoResearch on compressive constitutivemodel based onmeso-scopic damage of rockmaterialsrdquo Journal

of Zheng ZhouUniversity (Engineering Science) vol 31 no 6 pp6ndash9 2010

[17] H J Fan S X Xiao and K Peng ldquoStudy on constitutive modelof rock sample in uniaxial compression based on strain localiza-tionrdquo Chinese Journal of Rock Mechanics and Engineering vol25 no 1 pp 2634ndash2641 2006

[18] X B Wang ldquoDissipated energies and stabilities of axial and lat-eral deformations of rock specimens in uniaxial compressionrdquoChinese Journal of Rock Mechanics and Engineering vol 24 no5 pp 846ndash853 2005

[19] G Chen and P Y S Pan ldquoTwo-dimentional analytic expressionon strain localization of rock by strain gradient plasticitytheoryrdquoChinese Journal of RockMachanics and Engineering vol23 no 11 pp 1785ndash1791 2004

[20] C A Tang Catastrophe in Rock Unstable Failure China CoalIndustry Publishing House Beijing China 1993

[21] J Lemaitre ldquoHow to use damage mechanicsrdquoNuclear Engineer-ing and Design vol 80 no 2 pp 233ndash245 1984

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

8 The Scientific World Journal

References

[1] J P Zuo H P Xie and A M Wu ldquoInvestigation on failuremechanisms andmechanical behaviors of deep coal-rock singlebody and combined bodyrdquo Chinese Journal of Rock Mechanicsand Engineering vol 30 no 1 pp 84ndash92 2011

[2] Y Lu L Wang F Yang Y Li and H Chen ldquoPost-peak strainsoftening mechanical properties of weak rockrdquo Chinese Journalof Rock Mechanics and Engineering vol 29 no 3 pp 640ndash6482010

[3] F Zhang Q Sheng Z Zhu and Y Zhang ldquoStudy on post-peak mechanical behaviour and strain-softening model ofthree gorges graniterdquo Chinese Journal of Rock Mechanics andEngineering vol 27 supplement 1 pp 2651ndash2655 2008

[4] Z Li H Zhou Y Z Song C Q Zhang andQ Z Hu ldquoMechan-ical model of chlorite schist considering hardening-softeningand dilatancy characteristicsrdquo Rock and Soil Mechanics vol 34no 2 pp 404ndash410 2013

[5] T LHan Y S Chen T Xie Z Yu andMMHe ldquoExperimentalstudy on mechanics characteristics and energy properties ofmudstonerdquo Journal of Water Resources and Architectural Engi-neering vol 10 no 3 pp 116ndash120 2012

[6] Z Chong L Sheng-you Y Xi-jun S Yan and G Pan ldquoExper-imental investigation on strength of sandstone weathered andabsorbed water heavily and its constitutive lawrdquo EngineeringSciences vol 13 no 1 pp 74ndash80 2011

[7] L J He and L W Kong ldquoUniform expression of stress-strainrelationshiop of soilsrdquo Journal of Engineering Geology vol 18no 6 pp 900ndash905 2010

[8] W Z Zhang C X Chen and M Y Yu ldquoStudy of mechanicalproperties and constitutive relation of weathered sandstonerdquoRock and Soil Mechanics vol 30 pp 33ndash36 2009

[9] Y D Jiang X F Xian and J Su ldquoResearch on distortion ofsinglerock and constitutive relationrdquo Rock and Soil Mechanicsvol 26 no 6 pp 941ndash945 2005

[10] C Y Huang and G Subhash ldquoInfluence of lateral confinementon dynamic damage evolution during uniaxial compressiveresponse of brittle solidsrdquo Journal of the Mechanics and Physicsof Solids vol 51 no 6 pp 1089ndash1105 2003

[11] B Paliwal and K T Ramesh ldquoAn interacting micro-crack dam-age model for failure of brittle materials under compressionrdquoJournal of the Mechanics and Physics of Solids vol 56 no 3 pp896ndash923 2008

[12] L Graham-Brady ldquoStatistical characterization of meso-scaleuniaxial compressive strength in brittle materials with ran-domly occurring flawsrdquo International Journal of Solids andStructures vol 47 no 18-19 pp 2398ndash2413 2010

[13] W G Cao M H Zhao and C X Liu ldquoStudy on themodeland its modifying method for rock softening and damagebased onWeibull random distributionrdquo Chinese Journal of RockMechanics and Engineering vol 23 no 19 pp 3226ndash3231 2004

[14] X R Liu J B Wang P Li J Q Gu and Q Q ZhangldquoMechanical properties and constitutive model of thenarditerdquoJournal of PLA University of Science and Tcchnology (NaturalScience Edition) vol 13 no 5 pp 527ndash532 2012

[15] S Li J Xu and K G Li ldquoThe revises damage statisticalconstitutive model for rock based on uniform coefficientrdquoJournal of Sichuan University (Engineering Science Edition) vol39 no 6 pp 41ndash44 2007

[16] W Z Yang and BWang ldquoResearch on compressive constitutivemodel based onmeso-scopic damage of rockmaterialsrdquo Journal

of Zheng ZhouUniversity (Engineering Science) vol 31 no 6 pp6ndash9 2010

[17] H J Fan S X Xiao and K Peng ldquoStudy on constitutive modelof rock sample in uniaxial compression based on strain localiza-tionrdquo Chinese Journal of Rock Mechanics and Engineering vol25 no 1 pp 2634ndash2641 2006

[18] X B Wang ldquoDissipated energies and stabilities of axial and lat-eral deformations of rock specimens in uniaxial compressionrdquoChinese Journal of Rock Mechanics and Engineering vol 24 no5 pp 846ndash853 2005

[19] G Chen and P Y S Pan ldquoTwo-dimentional analytic expressionon strain localization of rock by strain gradient plasticitytheoryrdquoChinese Journal of RockMachanics and Engineering vol23 no 11 pp 1785ndash1791 2004

[20] C A Tang Catastrophe in Rock Unstable Failure China CoalIndustry Publishing House Beijing China 1993

[21] J Lemaitre ldquoHow to use damage mechanicsrdquoNuclear Engineer-ing and Design vol 80 no 2 pp 233ndash245 1984

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of