Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 738013 11 pageshttpdxdoiorg1011552013738013

Research ArticleEvolution Procedure of Multiple Rock Cracks underSeepage Pressure

Taoying Liu1 Ping Cao1 and Hang Lin12

1 School of Resources amp Safety Engineering Central South University Changsha Hunan 410083 China2 State Key Laboratory of Coal Resources and Safe Mining China University of Mining and Technology Xuzhou Jiangsu 221116 China

Correspondence should be addressed to Hang Lin linhangabc126com

Received 20 January 2013 Revised 23 April 2013 Accepted 1 May 2013

Academic Editor Gianluca Ranzi

Copyright copy 2013 Taoying Liu et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

In practical geotechnical engineering most of rock masses with multiple cracks exist in water environment Under suchcircumstance these adjacent cracks could interact with each other Moreover the seepage pressure produced by the high waterpressure can change cracksrsquo status and have an impact on the stress state of fragile rocks According to the theory of fracturemechanics this paper discusses the law of crack initiation and the evolution law of stress intensity factor at the tip of a wing crackcaused by compression-shear stress and seepage pressure Subsequently considering the interaction of the wing cracks and theadditional stress caused by rock bridge damage this paper proposes the intensity factor evolution equation under the combinedaction of compression-shear stress and seepage pressure In addition this paper analyzes the propagation of cracks under differentseepage pressure which reveals that the existence of seepage pressure facilitates the wing crackrsquos growth The result indicates thatthe high seepage pressure converts wing crack growth from stable form to unstable form Meanwhile based on the criterion andmechanism for crack initiation and propagation this paper puts forward the mechanical model for different fracture transfixionfailure modes of the crag bridge under the combined action of seepage pressure and compression-shear stress At the last part thispaper through investigating the flexibility tensor of the rock massrsquos initial damage and its damage evolution in terms of jointedrockmassrsquos damagemechanics deduces the damage evolution equation for the rockmass with multiple cracks under the combinedaction of compression-shear stress and seepage pressure The achievement of this investigation provides a reliable theoreticalprinciple for quantitative research of the fractured rock mass failure under seepage pressure

1 Introduction

In recent years rock mechanics is supposed to considerthe most striking feature of rock masses that their blockystructures are caused by the discontinuity surfaces such asjoints cracks and faults As frictional force is overcomeby shear stress induced by far-field stresses near the cracksurface the crack surface is inclined to slide over each otherwhich induces stress concentration on tip of crack and leadsto the initiation and splitting propagation of the wing crackat last [1ndash3] Worse the existence of seepage pressure couldreinforce the trend of themicrofracture and fault degradationto fractured rock masses With the development of rockmechanics engineering increasing number of situations is

involved in seepage pressure which results in numerousengineering accidents around the world [4 5] Henceresearches on rock mechanics referring to seepage pressureare gradually becoming a hot spot in the area of geotechnicalengineering A vast number of scholars havemade great effortin developing crack propagation theories [6ndash8] or developingtechniques to calculate the stress intensity factors of crack tips[9ndash12] or studying the relationship between microdamagedevelopment and macrodeformation of rock under uniaxialcompression [13ndash15] Nevertheless the former studies mainlyfocus onmechanicalmechanism of special single crack ratherthan the effect of seepage pressure Besides there are fewreasonable models for evolution laws of stress intensity factorfor the branch crack tip in the multicrack rock mass as well

2 Mathematical Problems in Engineering

1205901

1205901

2119886

1198791198901205903 1205903

119897

119897

Figure 1 Sketch of wing cracks seeding and propagation

as studies of the damage and fracture evolutionmechanism ofthe fractured rock mass with multiple cracks under seepagepressure

As for the multicrack rock mass the mechanical stateon the crack surface could be changed by the action ofseepage pressure As the branch crack expands the inter-action of cracks leads to the continuous degradation of themacroscopic mechanical properties of the rock mass [8] Theinteraction theory of multiple cracks in rock masses is a keyfactor in the analysis of microdamage mechanismThereforebased on previous studies and the rock fracture mechanicscriterion this paper proposes the mechanical model of themulticrack rock mass under the action of seepage pressureand investigates traits of gradual fracture and damage evo-lution of the multicrack rock under seepage pressure on thebasis of exploring the law of the compression-shear crackinitiation branch crack growth and rock bridge connectionThen this paper applies the self-consistent theory to settingup the constitutive and damage evolution equation for themulticrack rock mass under seepage pressure

2 Analysis of Compression-Shear MulticrackDamage Fracture Models

21 Crack Initiation Theunderground rock usually exists in acompression stress state and a lot of testing results and theo-retical calculation prove that cracks expand approximately inthe direction perpendicular to the maximum principal stress[16 17] as shown in Figure 1

Firstly this paper assumes that the rock mass is catego-rized as crisp flexible meeting the theory of linear elasticfracture mechanics and the seepage pressure is equal inevery directions along the crack The fractured rock massis under the remote field stress 120590

1and 120590

3 where 120590

1is the

maximum principal stress 1205901ge 1205903 The angle between

the crack and vertical stress 1205901is 120595 and there is a seepage

pressure119901 in the crackThenormal stress on the crack surfaceis compressive stress The shear stress forces the crack toslide which generates a friction 120583120590

119899119890+ 119862 owing to the part

closure of the crack where 120583 is the friction coefficient onthe crack surface and 119862 is the cohesion on the crack surfaceMeanwhile introducing the coefficient 120573 which presents theratio of the connected area to the total area the seepagepressure contributes120573119901 to the surfaceTherefore the effectiveshear driving force 120591eff and effective normal stress 120590

119899119890appear

in following formulas [18] (here the compressive stress ispositive)

120590119899119890= 120590119899minus 120573119901 = (1 minus 119862

119899) (1205901sin2120595 + 120590

3cos2120595) minus 120573119901 (1a)

120591eff = (1 minus 119862V)1205901minus 1205903

2

sin2120595 minus 120583120590119899119890minus 119862 (1b)

119862119899=

120587119886

120587119886 + (1198640 (1 minus V2

0)119870119899)

119862V =120587119886

120587119886 + (1198640 (1 minus V2

0)119870119904)

(1c)

where119862119899and119862V are the compression transmitting factor and

shearing transmitting factor respectively which are resultedfrom the part closure of the crack

According to the maximum circumferential stress cri-terion the initial crack extends along the direction of themaximum normal stress Thus the cracking angle 120579

3can be

obtained (1205793= 705

∘) [19] Then the stress intensity factor atthe wing crack initiation can be concluded as [20]

119870119868=

2

radic3

120591effradic120587119886 (2)

Making 119870119868= 119870119868119862

in (2) 119870119868119862

is the fracture toughnessthe biggest crack stress intensity factor It is easy to get theultimate strength 120590

11of the fracture rock under seepage

pressure

12059011=

radic3119870119868119862radic120587119886 + 2119862 minus 2120583120573119901 + 119861120590

3

119860

(3a)

119860 = minus120583 (1 minus 119862119899) (1 minus cos2120595) + (1 minus 119862V) sin

2120595

119861 = 120583 (1 minus 119862119899) (1 + cos2120595) + (1 minus 119862V) sin

2120595

(3b)

Mathematical Problems in Engineering 3

1

1205901

1205901

1205903 12059031radic119873119860 1205909984003119905

119901 119901

119901 119901

1

119897

119865 = 119879119890 sin 120595

119865 = 119879119890 sin 120595

119886cos120595

Figure 2 Schematic drawing ofmultiple interacting cracks119865 is the horizontal stress in the crack 12059010158403is the horizontal stress in the rock bridge

and 119905 is the length of the rock bridge

From (3a) it is clear to see that the ultimate strength 12059011

decreases linearly with the increase of the seepage pressure119901

As for multicrack rock masses when at the initial stage ofcrack expansion or the crack space is relatively large it canbe treated as the wing crack propagation of a single crackUnder such circumstance the stress intensity factor119870

119868at the

crack tip could be defined through the revised wing crackcalculation model [21] Considering the additional seepagepressure in the crack the stress intensity factor 119870

119868consists

of the stress intensity factor produced by the effective sheardriving force 119879

119890and the remote field stress 120590

1and 120590

3 and it

can be shown as in (4)

119870119868= 3120591119899119890radic119886119897119905119910

120587

sinminus1 ( 1

119897119905119910

) sin 120579 cos 1205792

minus

1

2

[(1205901+ 1205903) + (120590

1minus 1205903) cos2 (120579 + 120573)]radic120587119897 + 119901radic120587119897

(4)

where 119897119905119910is the impact factor which is the function of thewing

crack length 119897 wing crack azimuth 120579 and main crack length119886 as shown in (5)

119897119905119910= [1 +

9119897

4119886

cos2 (1205792

)] (1 minus 119890minus119897119886

) + 0667 sec2 (1205792

) 119890minus119897119886

(5)

The wing crack extends approximately in the directionperpendicular to the maximum principal stress until 119870

119868=

119870119868119862 and then the wing crack propagation length 119897 under

the combined action of compression-shear stress and seepagepressure can be calculated as shown in (6)

119897 =

6120591119899119890radic(119886119897119905119910120587)sinminus1 (1119897

119905119910) sin 120579 cos (1205792) minus 2119870

119868119862

[(1205901+ 1205903) + (120590

1minus 1205903) cos2 (120579 + 120573) minus 2119901]radic120587

(6)

22 The Multicrack Interaction Models As the crack expandsor when the crack space is relatively small the interactionbetween cracks leads to a damaged connection and unstablebreak of the rock bridge [22] The rock bridge interactionmechanical model of the multicrack rock mass with the wingcrack expansion is shown in Figure 2

Assuming that the number of the compression-shearcracks per unit area is 119873

119860 the length between main crack

centers and the rock bridge lengths between wing cracksrespectively are presented as follows

119878 =

1

radic119873119860

119879 = 119878 minus 2 (119897 + 119886 cos120595)

(7)

In Figure 2 119865 = 119879119890sin120595 is balanced by the tensile stress

1205901015840

3in the rock bridge together with seepage pressure 119901in the

wing crack

1205901015840

3=

119879119890sin120595 + 2119901119897

119878 minus 2 (119897 + 119886 cos120595) (8)

where 119879119890= 2119886120591

119899119890

Acting on the wing crack 12059010158403produces an additional

intensity factor at the crack tip [23]

1198701015840

119868= minus1205901015840

3radic120587119897 =

119886 (1205901minus 1205903) sin2120595 sin120595 + 2119901119897

119873minus12

119860minus 2 (119897 + 119886 cos120595)

radic120587119897 (9)

4 Mathematical Problems in Engineering

Considering the damage to the rock bridge caused by thewing crack interaction the stress intensity factor at the wingcrack tip is composed of (4) and (9) as follows

119870119868= 119870119868+ 1198701015840

119868= 3120591119899119890radic119886119897119905119910

120587

sinminus1 ( 1

119897119905119910

) sin 120579 cos 1205792

minus

1

2

[(1205901+ 1205903) + (120590

1minus 1205903) cos2 (120579 + 120573)]radic120587119897

+ 119901radic120587119897 +

119886 (1205901minus 1205903) sin2120595 sin120595 + 2119901119897

119873minus12

119860minus 2 (119897 + 119886 cos120595)

radic120587119897

(10)

From (10) it is clear to see that the interaction of multiplewing cracks in rock bridge damage makes the stress intensityfactor at the crack tip larger than that at a single wing crack

The process of rock fracture is the one of the increasingdamage and inherent fracture in rock and is the couplingdamage result of the microdamage and macrodamage Thereare many ways to define the damage variable such as thecracks quantity crack length crack area and crack densityHere the cracks quantity is adopted to define119863

0for the initial

damage of the rockmass and119863 for the damagewhen thewingcrack expands to 119897

1198630= 120587(119886 cos120595)2119873

119860

119863 = 120587(119897 + 119886 cos120595)2119873119860

(11)

Substituting (11) into (8) the fellow equation can be easilyobtained

1205901015840

3= minus

(120591119899119890tan120595 + 2119901119897) (119863

0120587)12

119886 (1 minus 2(119863120587)12)

(12)

Combining (10) (11) and (12) the relation curvesbetween the damage variables 119863 and dimensionless stressintensity factor (119870

1198681205901radic120587119886) at the wing crack tip with

different crack density are shown in Figure 3The graph reveals that with the increase of the damage

variables the dimensionless stress intensity factor1198701198681205901radic120587119886

at the crack tip under seepage pressure decreases at thevery beginning but rises gradually when the equivalent cracklength is 05 (119897119886) The graph also illustrates that the sparserof the cracks are the higher of the stress intensity at the wingcrack will be when the rock bridge is connected During thewing crack expansion process the damage variable 119863 variesfrom119863

0to 1 When119863 = 1 and the wing crack stress intensity

factor119870119868⩾ 119870119868119862

at the crack tip the wing crack connects therock bridge cracks and the rock mass loses bearing capacity

Figure 4 shows the relationship between the stress inten-sity factor at the crack tip and the equivalent crack prop-agation length under different seepage pressure It canbe concluded that when the seepage pressure is rela-tively low (119901 = 0MPa) the wing crack expands stably

0 015 03 045 06 075 09 105

036

038

04

042

044

Damage variables (119863)

119873119860 = 100119873119860 = 225

119873119860 = 400

Dim

ensio

nles

s stre

ss in

tens

ity119870119868120590 1(120587119886)

12

Figure 3 The relationship between damage variables and intensityfactor at the crack tip of the branch stress

(120597119870119868120597119871 lt 0) While when the seepage pressure is relatively

high (119901 = 15MPa) the wing crack tends to expand unstablyBesides there is the expansion stage that 120597119870

119868120597119871 gt 0 under

such condition and the growth rate of wing crack stressintensity factor becomes greater with the increase of seepagepressure 119901 It indicates that under high seepage pressure thecompressive-shear rock crack expands at a high speed as longas it cracks namely and the high seepage pressure contributesto unstable expansion

3 Wing Crack Connection Model andDamage Criteria

As the branch crack expands the interaction of the crackresults in the continuous degradation of the macroscopicmechanical properties of the rock mass The wing crackinitiation from microcracks to the completely damage of therock mass is a process of evolution and accumulation of therock mass damage as well as a process of the fracture of theconnection between cracks [24] It has been proved by manyexperimental researches that the gradual damage process ofthe cracks generally has two forms [25 26] (1) the rock bridgeaxial transfixion failure and (2) the tension-shear compoundfailure According to the branch crack propagation evolutionmechanism this paper studies the gradual damage features ofthose damage models

31 The Axial Transfixion Failure Whenever the wing crackextends stably or unstably the tension wing crack connectsthe main crack in another row as shown in Figure 5 Whenthe wing crack reaches the critical length 119897

1119888= 119863 sin120595

1198711119862

= 1198971119888119886 the rock bridge starts to damage in the form

of axial transfixion failure Taking the stress intensity at thecrack tip as the criterion when the wing crack reaches the

Mathematical Problems in Engineering 5

critical length 1198971119888= 119863 sin120595 this paper establishes the failure

criteria

119870119868= 3120591119899119890radic119886119897119905119910

120587

sinminus1 ( 1

119897119905119910

) sin 120579 cos 1205792

minus

1

2

[(1205901+ 1205903) + (120590

1minus 1205903) cos2 (120579 + 120573)]radic120587119897

1119888

+ 119901radic1205871198971119888+

119886 (1205901minus 1205903) sin2120595 sin120595 + 2119901119897

1119888

119873minus12

119860minus 2 (1198971119888+ 119886 cos120595)

radic1205871198971119888

(13)

When 119870119868(1198711119862) ge 119870

119868119862 axial transfixion failure occurs in the

rock bridge

32 The Tension-Shear Compound Failure With the expan-sion of wing cracks under seepage pressure the cut-resistantcapacity of the rock bridge is increasingly weakened Whenthe wing crack expands to a certain degree the rock bridgeat the crack tip between adjacent wing cracks is cut off bythe shear stress consequently the crack is connected in theshear direction [27] The mechanical analyzing diagram forthe element of the rock bridge composite failure is shown inFigure 6 where 119860119861 is 12 the length of the bottom crack 119864119865is 12 the length of the upper crack 119862119863 is the rock bridge119861119862 and 119864119863 are the wing cracks produced by effective sheardriving force of the main crack 119860119861 and 119864119865 120579 is the anglebetween rock bridge and the maximum principal stress and120590119862119863

and 120591119862119863

are the normal stress and shear stress acting onthe rock bridge respectively

According to the element shown in Figure 6 and theprinciple of mechanics balance it could be deduced asfollows

sum119865119909= 0 sum119865

119910= 0

211990331205903minus 2119886 (120591

119899119890sin120595 + 120590

119899119890cos120595)

minus 21199032(120591119862119863

sin 120579 + 120590119862119863119899

cos 120579) minus 2119901119897 = 0

211990311205901+ 2119886 (120591

119899119890cos120595 minus 120590

119899119890sin120595)

+ 21199032(120591119862119863

cos 120579 minus 120590119862119863119899

sin 120579) = 0

(14)

where

1199031= 119886 sin120595 + 1

2

radic1198892+ ℎ2 sin 120579

1199032=

radic1198892+ ℎ2 cos 120579 minus 2119897

2 cos 120579

1199033= 119886 cos120595 + 1

2

radic1198892+ ℎ2 cos 120579

120579 = 120595 minus arctan 119889ℎ

(15)

Then the following equation can be obtained

120591119862119863

=

21198601tan 120579 minus 2119861

1

(radic1198892+ ℎ2 cos 120579 minus 2119897) (1 + tan2120579)

120590119862119863

=

21198601+ 21198611tan 120579

(radic1198892+ ℎ2 cos 120579 minus 2119897) (1 + tan2120579)

(16)

1198601= 11990331205903minus 119886 (120591

119899119890sin120595 + 120590

119899119890cos120595) minus 119901119897

1198611= 11990311205901+ 119886 (120591

119899119890cos120595 minus 120590

119899119890sin120595)

(17)

Assuming that the rock bridge shear damage follows theMohr-Coulomb strength criterion conditions for the damageare

120591119862119863

minus 119888 minus 120590119862119863

tan120593 ⩾ 0 (18)

Substituting (16) into (18) the fellow equation can beeasily obtained as follows

(

1

2

radic1198892+ ℎ2 cos 120579 minus 119897) 119888 tan2 120579 + (119861

1tan120593 minus 119860

1) tan 120579

+ 1198601tan120593 + 119861 + 1

2

119888radic1198892+ ℎ2 cos 120579 minus 119888119897 ⩽ 0

(19)

When angle 120579 meets the rock bridge shear failure condi-tion and the wing crack propagation length reaches criticalvalue 119897

2119888 the angle 120579 between the rock bridge and 120590

1also

reaches their critical value and then

120579119888= arctan

1198601minus 1198611tan120593 + radicΔ

(radic1198892+ ℎ2 cos 120579 minus 2119897

2119888) 119888

(20)

As the geometric relations show in Figure 6 it is easy tosee that

tan 120579119888=

radic1198892+ ℎ2 sin 120579

(radic1198892+ ℎ2 cos 120579 minus 2119897

2119888)

(21)

Combined (20) and (21) the critical length of the wingcrack can be defined as

1198972119888=

1198602minus radic119860

2

2minus 4 (119888 + 119901 tan120593) 119861

2

2 (119888 + 119901 tan120593)

(22)

where

1198602= 11990331205903tan120593 minus 119886 (120591

119899119890sin120595 + 120590

119899119890cos120595) tan120593 + 119861

1

+

1

2

radic1198892+ ℎ2[(2119888 minus 119901 tan120593) cos 120579 + 119901 sin 120579]

1198612=

1

4

(1198892+ ℎ2) 119888 +

1

2

radic1198892+ ℎ2

times[(11990331205903minus 119886120591119899119890sin120595 minus 119886120590

119899119890cos120595) (cos 120579 tan120593 minus sin 120579)

+1198611(sin 120579 tan120593 + cos 120579)]

(23)

6 Mathematical Problems in Engineering

0 12 24 36 48 6 72 84 960

015

03

045

06

075

09

105

Equivalent crack length 119871(119897119886)

119901 = 0119901 = 6MPa119901 = 15MPa

Dim

ensio

nles

s stre

ss in

tens

ity fa

ctor

119870119868120590 1(120587119886)

12

Figure 4 The relationship between intensity factor at the crack tipwing stress and crack propagation length

Meanwhile this paper gets the stress intensity factor at thecrack tip when the wing reaches the critical length as follows

119870119868(1198972119888) = 119870

119868+ 1198701015840

119868= 3120591eff

radic119886119897119905119910

120587

sinminus1 ( 1

119897119905119910

) sin (120579) cos (120579)

minus

1

2

[(1205901+ 1205903) + (120590

1minus 1205903) cos2 (120579 + 120573)]radic120587119897

2119888

+ 119901radic1205871198972119888+

119886 (1205901minus 1205903) sin2120595 sin120595 + 2119901119897

2119888

2 (1198972119888+ 119886 cos120595) minus 119873minus12

119860

radic1205871198972119888

(24)

4 The Damage Evolution Mechanism ofMulticrack Rock Masses

The macroscopic mechanical effect of fractured rock wouldbe reflected by the change of its flexibility and then thedamage tensor 119863 can be determined by the elastic flexibilitytensor1198620 and the equivalent damage flexibility tensor119862119889 [28]

119863 = 119868 minus

(119862119889)

minus1

1198620

(25)

where 119868 is the fourth-order unit tensor and119863 is a tensor of thefourth order for the 3D anisotropy mode of fractured rock

In terms of the complete rock the elastic flexibility tensor1198620 can be expressed as [29]

1198620

119894119895119896119897 =

1 + V0

1198640

120575119894119896120575119895119897minus

V0

1198640

120575119894119895120575119896119897 (26)

where1198640and V0represented the elasticmodulus andPoissonrsquos

ratio of the rock respectivelyThe equivalent damage flexibility tensor119862119889 can be drawn

from Betti energy reciprocal theorem Hence based on self-consistent method and strain energy equivalence in solid

mechanics the equivalent damage flexibility tensor 119862119889119894119895119896119897 isacquired through calculating the equivalent elastic strainenergy

41The Equivalent Damage Flexibility Tensor Based on Equiv-alent Elastic Strain Energy The flexibility matrix of the crackin compression shear state is [30]

[1198620] = [

[

119862011 119862011 0

119862021 1198620

220

0 0 1198620

33

]

]

(27)

Based onBettirsquos theorem that isMaxwell-Betti reciprocalwork theorem the work done by one set of forces throughthe displacements produced by another set of forces is equalto the work done by the latter set of forces through thedisplacements produced by the former set of forces

(1)When 120590119909= 120591119909119910= 0 120590

119910= 0

11988212= (2119887120590

119910minus 2119886119901) (119862

221205901015840

1199102119889) = 4 (119887120590

119910minus 119886119901) 120590

101584011988911986222

11988221= (2119887120590

1015840

119910) (120590119910minus 119901)119862

0

22

2119889 + Δ119882119888

(28)

where Δ119882119888= (2119886120590

1015840

119910)(120590119910minus 119901)119862

119899119870119899

Then we get

11988221= 41205901015840

1199101198891198871198620

22

(120590119910minus 119901) +

21198861205901015840

119910(120590119910minus 119901)119862

119899

119870119899

(29)

According to 11988221

= 11988212 whilst the crack size can be

neglected compared to the size of the rock mass

11986222= (1 minus 119877) (119862

0

22

+

119886119862119899

2119870119899119887119889

) (30)

where 119877 = 119901120590119910

(2)When 120590119909= 120590119910= 0 120591119909119910

= 0

11988212= (1205912119889) (119862

3312059110158402119887) = 4119862

331205911015840119887119889

11988221= (12059110158402119889) (119862

331205912119887) + Δ119882

119888= 4120591119862

331205911015840119887119889 + Δ119882

119888

Δ119882119888=

12059110158402119886120591119862119904

119870119904

(31)

Then we obtain the following equation

11988221= 412059110158401205911198891198871198620

33+

21198861205911015840120591119862119904

119870119904

(32)

According to11988221= 11988212 we get

11986233= 1198620

33

+

119886119862119904

2119870119904119887119889

(33)

Thus the parameters of the equivalent damage flexibilitymatrix [119862119889] can be expressed as follows

[119862119889] =

[[[[[

[

0 0 0

0

119886119862119899(1 minus 119877)

2119870119899119887119889

minus 1198771198620

220

0 0

119886119862119904

1198701199042119887119889

]]]]]

]

(34)

Mathematical Problems in Engineering 7

42TheEquivalent Damage Flexibility Tensor Based onCrackrsquosElastic Strain Energy The strain energy of single crack isassumed as [29]

119880(119894)

119889= 2int

119886

0

119866119889Γ = 2

1 minus ]20

1198640

int

119886

0

(1198702

119868+ 1198702

119868119868) 119889119886 (35)

where Γ is the line length of cracksIntegrating over the line length the stress intensity factors

of type 119868 or type 119868119868 for the crack under seepage pressure are

119870(119894)

119868= 120590(119894)

119891

radic120587119886(119894) 119870

(119894)

119868119868= 120591(119894)

119902radic120587119886(119894) (36)

where 120590(119894)119891

is the normal stress of the crack and 120591(119894)

119902is the

tangential shear stress Combining the former equation thenwe get

119880(119894)

119889=

1 minus ]20

1198640

120587119886(119894)2

[120590(119894)2

119891+ 120591(119894)2

119902] (37)

By assuming there are 119899 groups of cracks in the fracturedrock mass the strain energy of the fractured rock mass willbe

119880119889=

119899

sum

119894=1

119880(119894)

119889=

1 minus ]20

1198640

120587

119899

sum

119894=1

119886(119894)2120588(119894)[120590(119894)2

119891+ 120591(119894)2

119902] (38)

Set the equation 119867 = ((1 minus ]20)1198640)120587 119860(119894) = 119886

(119894)2120588(119894) in

which 120588(119894) is the density of cracks then

119880119889= 119867

119899

sum

119894=1

119860(119894)[120590(119894)2

119891+ 120591(119894)2

119902] (39)

Rewrite (39) into the tensor expression

119880119889=

1

2

120590119894119895119862119889

119894119895119896119897120590119896119897= 119867

119899

sum

119894=1

119860(119894)[120590(119894)

119891120590(119894)

119891+ 120591(119894)

119902120591(119894)

119902] (40)

According to the hydrostatic pressure principle andCauchy criterion the remote field stress is denoted as 120590

119894119895 and

the stress matrix on the surface can be defined as [31]

[

[

119879V119909

119879V119910

]

]

= [

120590119909119909minus 119901 120590

119909119910

120590119910119909

120590119909119909minus 119901

] [

]119909

]119910

] (41)

Defining 119899119894119899119894= 1 authors rewrite the equation into the

tensor expression

119879(119894)

119894= (120590(119894)

119894119895minus 120575119894119895119901) 119899(119894)

119895= 120590(119894)

119894119895119899(119894)

119895minus 119901119899(119894)

119894 (42)

Considering the compression transferring coefficient 119862119899

the equivalent normal stress on the surface of the crack canbe expressed as

120590(119894)

119891= 119879(119894)

119894119899(119894)

119895= (1 minus 119862

(119894)

119899) 120590(119894)

119894119895119899(119894)

119895119899(119894)

119894minus 120573(119894)119901

(120590(119894)

119891)119894= (1 minus 119862

119899) 120590(119894)

119895119896119899(119894)

119895119899(119894)

119896119899(119894)

119894minus 120573(119894)119901119899(119894)

119894

(43)

Considering the shear transferring coefficient 119862119904again

the shear stress on the crack surface can be expressed as

(120591(119894)

119891)119894= 119879(119894)

119894minus (120590(119894)

119891)119894= (1 minus 119862

(119894)

119899) 120590(119894)

119895119896119899(119894)

119895(120575119896119897minus 119899(119894)

119896119899(119894)

119894)

(44)

Then we get

(120590(119894)

119891)119894(120590(119894)

119891)119894= ((1 minus 119862

(119894)

119899) 120590(119894)

119895119896119899(119894)

119895119899(119894)

119896119899(119894)

119894minus 120573(119894)119901119899(119894)

119894)

times ((1 minus 119862(119894)

119899) 120590(119894)

119904119905119899(119894)

119904119899(119894)

119905119899(119894)

119894minus 120573(119894)119901119899(119894)

119894)

(45)

Considering the symmetry of 120590119894119895 the following equations

can be deduced

(120590(119894)

119891)119894(120590(119894)

119891)119894= (1 minus 119862

(119894)

119899)

2

120590(119894)

119895119896119899(119894)

119895119899(119894)

119896119899(119894)

119904119899(119894)

119905120590(119894)

119904119905

minus (1 minus 119862(119894)

119899) 120590(119894)

119895119896119899(119894)

119895119899(119894)

119896119901

minus (1 minus 119862(119894)

119899) 120590(119894)

119904119905119899(119894)

119904119899(119894)

119905119901 + 120573(119894)2

1199012

(120591(119894)

119891)119894(120591(119894)

119891)119894= (1 minus 119862

(119894)

119904)

2

120590(119894)

119895119896119899(119894)

119895(120575119896119897minus 119899(119894)

119896119899(119894)

119894) 120590(119894)

119904119905119899(119894)

119904

times (120575119905119894minus 119899(119894)

119894119899(119894)

119894)

= (1 minus 119862(119894)

119904)

2

120590(119894)

119895119896

times (120575119896119897119899(119894)

119895119899(119894)

119904minus 119899(119894)

119895119899(119894)

119896119899(119894)

119904119899(119894)

119905) 120590(119894)

119904119905

(46)

Supposing the proportion coefficient 119877 = 119901120590 120590 is theaverage stress 120590 = (12)120590

119894119894 120590119894119894is the first invariant stress

Then the seepage pressure can be transformed to

119901 = 119901

120590119904119904

120590119904119904

= 120590119904119905120575119904119905

119901

2120590

=

1

2

120590119904119905120575119904119905119877

1199012= 1199012 120590119896119896

120590119896119896

120590119904119904

120590119904119904

= 120590119896119895120575119896119895120590119904119905120575119905119904

119901

2120590

119901

2120590

=

1

4

1205901198961198951205751198961198951205901199041199051205751199051199041198772

(47)

Then we get

(120590(119894)

119891)119894(120590(119894)

119891)119894

= (1 minus 119862(119894)

119899)

2

120590(119894)

119895119896119899(119894)

119895119899(119894)

119896119899(119894)

119904119899(119894)

119905120590(119894)

119904119905

minus

1

2

(1 minus 119862(119894)

119899) 119877120590(119894)

119895119896

times (119899(119894)

119895119899(119894)

119896120590(119894)

119904119905+ 120575119895119896119899(119894)

119904119899(119894)

119905)

+

1

4

(120573(119894)119877)

2

120590(119894)

119895119896120590(119894)

119904119905120590119905119904

8 Mathematical Problems in Engineering

Tension crack

1205901

1205901

12059031205903

ℎ

119863

2119886

120595

2119886

Figure 5 Failure characteristic of crag bridge shearing

119880119889=

1

2

120590119895119896119862119889

119895119896119904119905120590119904119894

= 119867

119899

sum

119894=1

119860(119894)120590(119894)

119895119896[(1 minus 119862

(119894)

119899)

2

119899(119894)

119895119899(119894)

119896119899(119894)

119904119899(119894)

119905

minus

1

2

(1 minus 119862(119894)

119899) 120573(119894)119877120590(119894)

119895119896

times (119899(119894)

119895119899(119894)

119896120590(119894)

119904119905+ 120575119895119896119899(119894)

119904119899(119894)

119905)

+

1

4

(119877120573(119894))

2

120590(119894)

119895119896120590(119894)

119904119905120590119905119904+ (1 minus 119862

(119894)

119904)

2

times (120575119896119905119899(119894)

119895119899(119894)

119904minus 119899(119894)

119895119899(119894)

119896119899(119894)

119904119899(119894)

119905) ] 120590(119894)

119904119905

(48)

Finally it is clear to conclude the additional flexibilitytensor of fractured rock mass under seepage pressure asfollows

119862119889

119894119895119896119897= (1 minus 119862

(119894)

119899)

2

119899(119894)

119895119899(119894)

119896119899(119894)

119904119899(119894)

119905

minus

1

2

(1 minus 119862(119894)

119899) 120573(119894)119877120590(119894)

119895119896(119899(119894)

119895119899(119894)

119896120590(119894)

119904119905+ 120575119895119896119899(119894)

119904119899(119894)

119905)

+

1

4

(119877120573(119894))

2

120590(119894)

119895119896120590(119894)

119904119905120590119905119904

+ (1 minus 119862(119894)

119904)

2

(120575119896119905119899(119894)

119895119899(119894)

119904minus 119899(119894)

119895119899(119894)

119896119899(119894)

119904119899(119894)

119905)

(49)

43The Fracture Damage Evolution Equation The additionalelastic strain energy density 119906119894

119896119911

produced by the wing crackis [32]

119906(119894)

119896119911=

4

1198640

120588(119894)

V 119886(119894)2(

5

2

120590(119894)

eff119871(119894)+

2

radic3

120591(119894)

eff)2

(50)

where 119871(119894) is the equivalent length for the crack

Assuming there are 119899 groups of cracks in the fracturedrock mass the additional elastic strain energy density of thefractured rock 119880

119896119911is

119880119896119911=

119899

sum

119894=1

119906(119894)

119896119911=

1

2

120590119894119895119862119896119911

119894119895119896119897120590119896119897 (51)

Crack extension reduces the stiffness of rock masses andincreases its flexibility Through the derivative of 119880

119896119911with

respect to the stress tensor we can get the additional flexibilitytensor of fractured rock masses in the damage evolutionprocess

120597119880119896119911

120597120590119894119895

= 119862119896119911

119894119895119896119897120590119896119897 (52)

At the same time by calculating the partial derivativeof 119871(119894) 120590(119894)eff and 120591

(119894)

eff with respect to 120590119894119895 respectively and

considering the symmetry of 119862119896119911119894119895119896119897

we can get the followingresults

120597119880119896119911

120597120590119894119895

=

1

1198640

119899

sum

119894=1

119886(119894)2120588(119894)

V [119872

(119894)

1119899(119894)

119895119899(119894)

119896119899(119894)

119896119899(119894)

119905minus119872(119894)

2

times (119899(119894)

119895119899(119894)

119896120575119896119895+ 120575119896119894119899(119894)

119895119899(119894)

119905

+119899(119894)

119895119899(119894)

119896120575119894119895+ 120575119894119905119899(119894)

119895119899(119894)

119896)

minus

119872(119894)

3

3

120573119877120575119894119895120575119896119897]120590119896119897

(53)

Mathematical Problems in Engineering 9

Shear crack

Tension crack

1199031

1199033

120595

119862

119864

119865

119861

119863119897

120591CD

120590ne120591ne

120579

119903 2

119901

119901

1205901

1205903

119897

119886119860

2119886

120590CD119899

Figure 6 Failure characteristic of crag bridge shearing

where

119872(119894)

1= (20120590

(119894)

eff119871(119894)+

16

radic3

120591(119894)

eff)

times [

1

120590(119894)

119899

(

5

2

119871(119894)+

2

radic3

119891)

minus

1

120591(119894)

119899

(

5

2

cot 120579(119894)119871(119894) + 2

radic3

119891) +

5

2

120590(119894)

eff119873(119894)

1]

119872(119894)

2= (5120590

(119894)

eff119871(119894)+

4

radic3

120591(119894)

eff)

times [

1

120591(119894)

119899

(

5

2

cot 120579(119894)119871(119894) + 2

radic3

119891) +

5

2

120590(119894)

eff119873(119894)

2]

119872(119894)

3= (20120590

(119894)

eff119871(119894)+

16

radic3

120591(119894)

eff)

times [

2

radic3

119891 +

5

2

(120590(119894)

eff119873(119894)

3+ 119871(119894))]

119873(119894)

1=radic119871(119894)[minus4119886(119894) cos 120579(119894)120590(119894)2eff (119891120591

(119894)

119899minus 120590(119894)

119899)

minus (119870(119894)2

119868119862minus 4119886(119894) cos 120579(119894)120590(119894)eff

minus 2radic120587119886(119894)119860(119894)119870(119894)

119868119862120590(119894)

eff )

times (120591(119894)

119899minus 120590(119894)

119899cot120579(119894))]

times (2120587119886(119894)119860(119894)120590(119894)

eff120590(119894)

119899120591(119894)

119899)

minus1

119873(119894)

2=radic119871(119894)[minus4119886(119894) cos 120579(119894)120590(119894)2eff

minus (119870(119894)

119868119862

2

minus 4119886(119894) cos 120579(119894)120590(119894)eff120591

(119894)

eff

minus2radic120587119886(119894)119860(119894)119870(119894)

119868119862120590(119894)

eff) cot 120579(119894)]

times (2120587119886(119894)119860(119894)120590(119894)

eff120590(119894)

119899120591(119894)

119899)

minus1

119873(119894)

3

=[

[

minus(minus2(

119870(119894)

119868119862

radic120587119886(119894)

)

2

120590(119894)3

eff minus2

120587

cos 120579(119894)(119891

120590(119894)

eff

minus

120591(119894)

eff

120590(119894)2

eff

))]

]

times

radic119871(119894)

119860(119894)

minus

2119870(119894)

119868119862radic119871(119894)

120590(119894)2

effradic2120587119886

(119894)

119860(119894)= radic

119870(119894)

119868119862

2120590(119894)2

effradic2120587119886

(119894)

minus

2

120587

cos 120579(119894)120591(119894)

eff

120590(119894)2

eff

(54)

Then we can get the flexibility tensor 119862119896119911119894119895119896119897

due to thedamage evolution of fractured rock masses as follows

119862119896119911

119894119895119896119897=

1

1198640

119899

sum

119894=1

119886(119894)2120588(119894)

V [119872(119894)

1119899(119894)

119895119899(119894)

119896119899(119894)

119896119899(119894)

119905minus119872(119894)

2

times (119899(119894)

119895119899(119894)

119896120575119896119895+ 120575119896119894119899(119894)

119895119899(119894)

119905

+119899(119894)

119895119899(119894)

119896120575119894119895+ 120575119894119905119899(119894)

119895119899(119894)

119896)

minus

119872(119894)

3

3

120573119877120575119894119895120575119896119897]

(55)

44 The Constitutive Equation of the Multicrack Rock MassTo sum up combining the initial damage and damage

10 Mathematical Problems in Engineering

evolution tensor we can get the flexibility tensor of thefractured rock mass under seepage pressureas follows

119862119894119895119896119897

= 1198620

119894119895119896119897+ 119862119889

119894119895119896119897+ 119862119896119911

119894119895119896119897 (56)

where1198620119894119895119896119897

is the elastic flexibility tensor of the complete rock119862119889

119894119895119896119897is the initial equivalent damage flexibility tensor of the

fractured rock mass under seepage pressure and 119862119896119911119894119895119896119897

is theadditional damage flexibility tensor with crack propagationof the fractured rock mass under seepage pressure

According to generalized Hookersquos law [33]

120576119894119895= 119862119894119895119896119897120590119896119897

120590119894119895= 119864119894119895119896119897120576119896119897= (119862119894119895119896119897)

minus1

120576119896119897

(57)

and this paper sets up the constitutive relation of the multi-crack rock mass under seepage pressure

5 Conclusions

After discussing the damage and fracture evolution mecha-nism of the multicrack rock mass under compression-shearstress and seepage pressure the following main conclusionsare drawn

(1) Through proposing the fractured damage model ofthe multicrack rock mass under the combined actionof compression-shear stress and seepage pressurethis paper discusses the law of crack initiation andthe evolution law of stress intensity factor at the tipof a wing crack The interaction of multiple cracksmakes the stress intensity factor at the crack tiplarger than that of a single wing crack Regarding theseepage pressure the existence of that reinforces thewing crackrsquos propagation Moreover the high seepagepressure converts wing crack growth from stable formto unstable form

(2) The wing crack initiation from microcracks to thecompletely damage of the rock mass is a processof evolution and accumulation of the rock massdamage as well as a process of the fracture of theconnection between cracks Based on the criterionand mechanism for crack initiation and propagationthis paper puts forward the mechanical model fordifferent fracture transfixion failuremodes of the cragbridge under the combined action of seepage pressureand compression-shear stress

(3) According to the strain energy equivalent principlethis paper applies Bettirsquos reciprocal work theorem ofthe fractured rock mass to study the flexibility tensorof the rock massrsquos initial damage and its damage evo-lution and deduces the damage constitutive equationsfor the elastic-plastic fracture and damage evolutionThe theory provides a reliable theoretical principlefor quantitative research of the fractured rock massfailure under seepage pressure

Acknowledgments

This paper gets its funding from Projects (51174228 5127424941002090) supported by the National Natural ScienceFoundation of China Project supported by the HunanProvincial Innovation Foundation for Postgraduate Project(20120162120014) supported by the Specialized ResearchFund for the Doctoral Program of Higher Education ofChina Project (12KF01) supported by the Open Projects ofState Key Laboratory of Coal Resources and Safe MiningCUMTThe authors wish to acknowledge these supports

References

[1] V Kuksenko N Tomilin and A Chmel ldquoThe rock fractureexperiment with a drive control a spatial aspectrdquo Tectono-physics vol 431 no 1ndash4 pp 123ndash129 2007

[2] S Marfia and E Sacco ldquoA fracture evolution procedure forcohesive materialsrdquo International Journal of Fracture vol 110no 3 pp 241ndash261 2001

[3] L N Lens E Bittencourt and V M R drsquoAvila ldquoConstitutivemodels for cohesive zones in mixed-mode fracture of plainconcreterdquo Engineering Fracture Mechanics vol 76 no 14 pp2281ndash2297 2009

[4] M K Rahman Y A Suarez Z Chen and S S RahmanldquoUnsuccessful hydraulic fracturing cases in Australia investi-gation into causes of failures and their remediesrdquo Journal ofPetroleum Science and Engineering vol 57 no 1-2 pp 70ndash812007

[5] R P Orense S Shimoma K Maeda and I Towhata ldquoInstru-mented model slope failure due to water seepagerdquo Journal ofNatural Disaster Science vol 26 pp 15ndash26 2004

[6] M M Hossain and M K Rahman ldquoNumerical simulationof complex fracture growth during tight reservoir stimulationby hydraulic fracturingrdquo Journal of Petroleum Science andEngineering vol 60 no 2 pp 86ndash104 2008

[7] D Hoxha M Lespinasse J Sausse and F Homand ldquoAmicrostructural study of natural and experimentally inducedcracks in a granodioriterdquo Tectonophysics vol 395 no 1-2 pp99ndash112 2005

[8] M Sagong and A Bobet ldquoCoalescence of multiple flaws ina rock-model material in uniaxial compressionrdquo InternationalJournal of Rock Mechanics and Mining Sciences vol 39 no 2pp 229ndash241 2002

[9] R H C Wong K T Chau C A Tang and P Lin ldquoAnalysisof crack coalescence in rock-like materials containing threeflawsmdashpart I experimental approachrdquo International Journal ofRockMechanics andMining Sciences vol 38 no 7 pp 909ndash9242001

[10] S YWang SW SloanD C Sheng andCA Tang ldquoNumericalanalysis of the failure process around a circular opening in rockrdquoComputers and Geotechnics vol 39 pp 8ndash16 2012

[11] O Mutlu and A Bobet ldquoSlip initiation on frictional fracturesrdquoEngineering FractureMechanics vol 72 no 5 pp 729ndash747 2005

[12] Y Nara and K Kaneko ldquoStudy of subcritical crack growth inandesite using the Double Torsion testrdquo International Journal ofRock Mechanics and Mining Sciences vol 42 no 4 pp 521ndash5302005

[13] M Haggerty Q Lin and J F Labuz ldquoObserving deformationand fracture of rock with speckle patternsrdquo Rock Mechanics andRock Engineering vol 43 pp 417ndash426 2010

Mathematical Problems in Engineering 11

[14] C Dascalu B Francois and O Keita ldquoA two-scale model forsubcritical damage propagationrdquo International Journal of Solidsand Structures vol 47 no 3-4 pp 493ndash502 2010

[15] SMisra ldquoDeformation localization at the tips of shear fracturesan analytical approachrdquo Tectonophysics vol 503 no 1-2 pp182ndash187 2011

[16] M F Ashby and S D Hallam ldquoThe failure of brittle solidscontaining small cracks under compressive stress statesrdquo ActaMetallurgica vol 34 no 3 pp 497ndash510 1986

[17] C Fairhurst and N G W Cood ldquoThe phenomenon of rocksplitting parallel to the direction of maximum compression inthe neighbourhood of a surfacerdquo in Proceedings of the Congressof the International Society of RockMechanics L Muller Ed pp687ndash692 Balkema Rotterdam The Netherlands 1966

[18] H P Yuan P Cao and W Z Xu ldquoMechanism study onsubcritical crack growth of flabby and intricate ore rockrdquoTransactions of Nonferrous Metals Society of China vol 16 no3 pp 723ndash727 2006

[19] E Sahouryeh A V Dyskin and L N Germanovich ldquoCrackgrowth under biaxial compressionrdquo Engineering FractureMechanics vol 69 no 18 pp 2187ndash2198 2002

[20] S-H Zheng Research on Coupling Theory Between Seepageand Damage of Fractured Rock Mass and Its Application toEngineering Wuhan Institute of Rock and Soil MechanicsChinese Academy of Scinces Wuhan China 2000

[21] T Liu P Cao and H Lin ldquoAnalytical solutions for the seepageinduced fracture failure propagation in rock massrdquo Journal ofDisaster Advances vol 3 supplement 1 pp 307ndash314 2013

[22] L N Y Wong and H H Einstein ldquoUsing high speed videoimaging in the study of cracking processes in rockrdquoGeotechnicalTesting Journal vol 32 no 2 pp 164ndash180 2009

[23] W Zhu and Q Zhang ldquoBrittle elastic fracture damage constitu-tive model of jointed rockmass and its application to engineer-ingrdquoChinese Journal of RockMechanics and Engineering vol 18no 3 pp 245ndash249 1999

[24] K Y Lam and S P Phua ldquoMultiple crack interaction and itseffect on stress intensity factorrdquoEngineering FractureMechanicsvol 40 no 3 pp 585ndash592 1991

[25] M Kachanov ldquoOn the problems of crack interactions and crackcoalescencerdquo International Journal of Fracture vol 120 no 3pp 537ndash543 2003

[26] H Qing and W Yang ldquoCharacterization of strongly interactedmultiple cracks in an infinite platerdquo Theoretical and AppliedFracture Mechanics vol 46 no 3 pp 209ndash216 2006

[27] A P S Selvadurai and Q Yu ldquoMechanics of a discontinuity in ageomaterialrdquo Computers and Geotechnics vol 32 no 2 pp 92ndash106 2005

[28] X P Yuan H Y Liu and Z Q Wang ldquoAn interacting crack-mechanics basedmodel for elastoplastic damagemodel of rock-likematerials under compressionrdquo International Journal of RockMechanics and Mining Sciences vol 58 pp 92ndash102 2013

[29] B Franois and C Dascalu ldquoA two-scale time-dependentdamage model based on non-planar growth of micro-cracksrdquoJournal of the Mechanics and Physics of Solids vol 58 no 11 pp1928ndash1946 2010

[30] V Palchik andYHHatzor ldquoCrack damage stress as a compositefunction of porosity and elasticmatrix stiffness in dolomites andlimestonesrdquo Engineering Geology vol 63 no 3-4 pp 233ndash2452002

[31] G Alfano S Marfia and E Sacco ldquoA cohesive damage-friction interface model accounting for water pressure on crack

propagationrdquo Computer Methods in Applied Mechanics andEngineering vol 196 no 1ndash3 pp 192ndash209 2006

[32] V Monchiet C Gruescu O Cazacu and D Kondo ldquoAmicromechanical approach of crack-induced damage inorthotropic media application to a brittle matrix compositerdquoEngineering Fracture Mechanics vol 83 pp 40ndash53 2012

[33] A Golshani Y Okui M Oda and T Takemura ldquoA microme-chanical model for brittle failure of rock and its relation tocrack growth observed in triaxial compression tests of graniterdquoMechanics of Materials vol 38 no 4 pp 287ndash303 2006

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Mathematical Problems in Engineering

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Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

1205901

1205901

2119886

1198791198901205903 1205903

119897

119897

Figure 1 Sketch of wing cracks seeding and propagation

as studies of the damage and fracture evolutionmechanism ofthe fractured rock mass with multiple cracks under seepagepressure

As for the multicrack rock mass the mechanical stateon the crack surface could be changed by the action ofseepage pressure As the branch crack expands the inter-action of cracks leads to the continuous degradation of themacroscopic mechanical properties of the rock mass [8] Theinteraction theory of multiple cracks in rock masses is a keyfactor in the analysis of microdamage mechanismThereforebased on previous studies and the rock fracture mechanicscriterion this paper proposes the mechanical model of themulticrack rock mass under the action of seepage pressureand investigates traits of gradual fracture and damage evo-lution of the multicrack rock under seepage pressure on thebasis of exploring the law of the compression-shear crackinitiation branch crack growth and rock bridge connectionThen this paper applies the self-consistent theory to settingup the constitutive and damage evolution equation for themulticrack rock mass under seepage pressure

2 Analysis of Compression-Shear MulticrackDamage Fracture Models

21 Crack Initiation Theunderground rock usually exists in acompression stress state and a lot of testing results and theo-retical calculation prove that cracks expand approximately inthe direction perpendicular to the maximum principal stress[16 17] as shown in Figure 1

Firstly this paper assumes that the rock mass is catego-rized as crisp flexible meeting the theory of linear elasticfracture mechanics and the seepage pressure is equal inevery directions along the crack The fractured rock massis under the remote field stress 120590

1and 120590

3 where 120590

1is the

maximum principal stress 1205901ge 1205903 The angle between

the crack and vertical stress 1205901is 120595 and there is a seepage

pressure119901 in the crackThenormal stress on the crack surfaceis compressive stress The shear stress forces the crack toslide which generates a friction 120583120590

119899119890+ 119862 owing to the part

closure of the crack where 120583 is the friction coefficient onthe crack surface and 119862 is the cohesion on the crack surfaceMeanwhile introducing the coefficient 120573 which presents theratio of the connected area to the total area the seepagepressure contributes120573119901 to the surfaceTherefore the effectiveshear driving force 120591eff and effective normal stress 120590

119899119890appear

in following formulas [18] (here the compressive stress ispositive)

120590119899119890= 120590119899minus 120573119901 = (1 minus 119862

119899) (1205901sin2120595 + 120590

3cos2120595) minus 120573119901 (1a)

120591eff = (1 minus 119862V)1205901minus 1205903

2

sin2120595 minus 120583120590119899119890minus 119862 (1b)

119862119899=

120587119886

120587119886 + (1198640 (1 minus V2

0)119870119899)

119862V =120587119886

120587119886 + (1198640 (1 minus V2

0)119870119904)

(1c)

where119862119899and119862V are the compression transmitting factor and

shearing transmitting factor respectively which are resultedfrom the part closure of the crack

According to the maximum circumferential stress cri-terion the initial crack extends along the direction of themaximum normal stress Thus the cracking angle 120579

3can be

obtained (1205793= 705

∘) [19] Then the stress intensity factor atthe wing crack initiation can be concluded as [20]

119870119868=

2

radic3

120591effradic120587119886 (2)

Making 119870119868= 119870119868119862

in (2) 119870119868119862

is the fracture toughnessthe biggest crack stress intensity factor It is easy to get theultimate strength 120590

11of the fracture rock under seepage

pressure

12059011=

radic3119870119868119862radic120587119886 + 2119862 minus 2120583120573119901 + 119861120590

3

119860

(3a)

119860 = minus120583 (1 minus 119862119899) (1 minus cos2120595) + (1 minus 119862V) sin

2120595

119861 = 120583 (1 minus 119862119899) (1 + cos2120595) + (1 minus 119862V) sin

2120595

(3b)

Mathematical Problems in Engineering 3

1

1205901

1205901

1205903 12059031radic119873119860 1205909984003119905

119901 119901

119901 119901

1

119897

119865 = 119879119890 sin 120595

119865 = 119879119890 sin 120595

119886cos120595

Figure 2 Schematic drawing ofmultiple interacting cracks119865 is the horizontal stress in the crack 12059010158403is the horizontal stress in the rock bridge

and 119905 is the length of the rock bridge

From (3a) it is clear to see that the ultimate strength 12059011

decreases linearly with the increase of the seepage pressure119901

As for multicrack rock masses when at the initial stage ofcrack expansion or the crack space is relatively large it canbe treated as the wing crack propagation of a single crackUnder such circumstance the stress intensity factor119870

119868at the

crack tip could be defined through the revised wing crackcalculation model [21] Considering the additional seepagepressure in the crack the stress intensity factor 119870

119868consists

of the stress intensity factor produced by the effective sheardriving force 119879

119890and the remote field stress 120590

1and 120590

3 and it

can be shown as in (4)

119870119868= 3120591119899119890radic119886119897119905119910

120587

sinminus1 ( 1

119897119905119910

) sin 120579 cos 1205792

minus

1

2

[(1205901+ 1205903) + (120590

1minus 1205903) cos2 (120579 + 120573)]radic120587119897 + 119901radic120587119897

(4)

where 119897119905119910is the impact factor which is the function of thewing

crack length 119897 wing crack azimuth 120579 and main crack length119886 as shown in (5)

119897119905119910= [1 +

9119897

4119886

cos2 (1205792

)] (1 minus 119890minus119897119886

) + 0667 sec2 (1205792

) 119890minus119897119886

(5)

The wing crack extends approximately in the directionperpendicular to the maximum principal stress until 119870

119868=

119870119868119862 and then the wing crack propagation length 119897 under

the combined action of compression-shear stress and seepagepressure can be calculated as shown in (6)

119897 =

6120591119899119890radic(119886119897119905119910120587)sinminus1 (1119897

119905119910) sin 120579 cos (1205792) minus 2119870

119868119862

[(1205901+ 1205903) + (120590

1minus 1205903) cos2 (120579 + 120573) minus 2119901]radic120587

(6)

22 The Multicrack Interaction Models As the crack expandsor when the crack space is relatively small the interactionbetween cracks leads to a damaged connection and unstablebreak of the rock bridge [22] The rock bridge interactionmechanical model of the multicrack rock mass with the wingcrack expansion is shown in Figure 2

Assuming that the number of the compression-shearcracks per unit area is 119873

119860 the length between main crack

centers and the rock bridge lengths between wing cracksrespectively are presented as follows

119878 =

1

radic119873119860

119879 = 119878 minus 2 (119897 + 119886 cos120595)

(7)

In Figure 2 119865 = 119879119890sin120595 is balanced by the tensile stress

1205901015840

3in the rock bridge together with seepage pressure 119901in the

wing crack

1205901015840

3=

119879119890sin120595 + 2119901119897

119878 minus 2 (119897 + 119886 cos120595) (8)

where 119879119890= 2119886120591

119899119890

Acting on the wing crack 12059010158403produces an additional

intensity factor at the crack tip [23]

1198701015840

119868= minus1205901015840

3radic120587119897 =

119886 (1205901minus 1205903) sin2120595 sin120595 + 2119901119897

119873minus12

119860minus 2 (119897 + 119886 cos120595)

radic120587119897 (9)

4 Mathematical Problems in Engineering

Considering the damage to the rock bridge caused by thewing crack interaction the stress intensity factor at the wingcrack tip is composed of (4) and (9) as follows

119870119868= 119870119868+ 1198701015840

119868= 3120591119899119890radic119886119897119905119910

120587

sinminus1 ( 1

119897119905119910

) sin 120579 cos 1205792

minus

1

2

[(1205901+ 1205903) + (120590

1minus 1205903) cos2 (120579 + 120573)]radic120587119897

+ 119901radic120587119897 +

119886 (1205901minus 1205903) sin2120595 sin120595 + 2119901119897

119873minus12

119860minus 2 (119897 + 119886 cos120595)

radic120587119897

(10)

From (10) it is clear to see that the interaction of multiplewing cracks in rock bridge damage makes the stress intensityfactor at the crack tip larger than that at a single wing crack

The process of rock fracture is the one of the increasingdamage and inherent fracture in rock and is the couplingdamage result of the microdamage and macrodamage Thereare many ways to define the damage variable such as thecracks quantity crack length crack area and crack densityHere the cracks quantity is adopted to define119863

0for the initial

damage of the rockmass and119863 for the damagewhen thewingcrack expands to 119897

1198630= 120587(119886 cos120595)2119873

119860

119863 = 120587(119897 + 119886 cos120595)2119873119860

(11)

Substituting (11) into (8) the fellow equation can be easilyobtained

1205901015840

3= minus

(120591119899119890tan120595 + 2119901119897) (119863

0120587)12

119886 (1 minus 2(119863120587)12)

(12)

Combining (10) (11) and (12) the relation curvesbetween the damage variables 119863 and dimensionless stressintensity factor (119870

1198681205901radic120587119886) at the wing crack tip with

different crack density are shown in Figure 3The graph reveals that with the increase of the damage

variables the dimensionless stress intensity factor1198701198681205901radic120587119886

at the crack tip under seepage pressure decreases at thevery beginning but rises gradually when the equivalent cracklength is 05 (119897119886) The graph also illustrates that the sparserof the cracks are the higher of the stress intensity at the wingcrack will be when the rock bridge is connected During thewing crack expansion process the damage variable 119863 variesfrom119863

0to 1 When119863 = 1 and the wing crack stress intensity

factor119870119868⩾ 119870119868119862

at the crack tip the wing crack connects therock bridge cracks and the rock mass loses bearing capacity

Figure 4 shows the relationship between the stress inten-sity factor at the crack tip and the equivalent crack prop-agation length under different seepage pressure It canbe concluded that when the seepage pressure is rela-tively low (119901 = 0MPa) the wing crack expands stably

0 015 03 045 06 075 09 105

036

038

04

042

044

Damage variables (119863)

119873119860 = 100119873119860 = 225

119873119860 = 400

Dim

ensio

nles

s stre

ss in

tens

ity119870119868120590 1(120587119886)

12

Figure 3 The relationship between damage variables and intensityfactor at the crack tip of the branch stress

(120597119870119868120597119871 lt 0) While when the seepage pressure is relatively

high (119901 = 15MPa) the wing crack tends to expand unstablyBesides there is the expansion stage that 120597119870

119868120597119871 gt 0 under

such condition and the growth rate of wing crack stressintensity factor becomes greater with the increase of seepagepressure 119901 It indicates that under high seepage pressure thecompressive-shear rock crack expands at a high speed as longas it cracks namely and the high seepage pressure contributesto unstable expansion

3 Wing Crack Connection Model andDamage Criteria

As the branch crack expands the interaction of the crackresults in the continuous degradation of the macroscopicmechanical properties of the rock mass The wing crackinitiation from microcracks to the completely damage of therock mass is a process of evolution and accumulation of therock mass damage as well as a process of the fracture of theconnection between cracks [24] It has been proved by manyexperimental researches that the gradual damage process ofthe cracks generally has two forms [25 26] (1) the rock bridgeaxial transfixion failure and (2) the tension-shear compoundfailure According to the branch crack propagation evolutionmechanism this paper studies the gradual damage features ofthose damage models

31 The Axial Transfixion Failure Whenever the wing crackextends stably or unstably the tension wing crack connectsthe main crack in another row as shown in Figure 5 Whenthe wing crack reaches the critical length 119897

1119888= 119863 sin120595

1198711119862

= 1198971119888119886 the rock bridge starts to damage in the form

of axial transfixion failure Taking the stress intensity at thecrack tip as the criterion when the wing crack reaches the

Mathematical Problems in Engineering 5

critical length 1198971119888= 119863 sin120595 this paper establishes the failure

criteria

119870119868= 3120591119899119890radic119886119897119905119910

120587

sinminus1 ( 1

119897119905119910

) sin 120579 cos 1205792

minus

1

2

[(1205901+ 1205903) + (120590

1minus 1205903) cos2 (120579 + 120573)]radic120587119897

1119888

+ 119901radic1205871198971119888+

119886 (1205901minus 1205903) sin2120595 sin120595 + 2119901119897

1119888

119873minus12

119860minus 2 (1198971119888+ 119886 cos120595)

radic1205871198971119888

(13)

When 119870119868(1198711119862) ge 119870

119868119862 axial transfixion failure occurs in the

rock bridge

32 The Tension-Shear Compound Failure With the expan-sion of wing cracks under seepage pressure the cut-resistantcapacity of the rock bridge is increasingly weakened Whenthe wing crack expands to a certain degree the rock bridgeat the crack tip between adjacent wing cracks is cut off bythe shear stress consequently the crack is connected in theshear direction [27] The mechanical analyzing diagram forthe element of the rock bridge composite failure is shown inFigure 6 where 119860119861 is 12 the length of the bottom crack 119864119865is 12 the length of the upper crack 119862119863 is the rock bridge119861119862 and 119864119863 are the wing cracks produced by effective sheardriving force of the main crack 119860119861 and 119864119865 120579 is the anglebetween rock bridge and the maximum principal stress and120590119862119863

and 120591119862119863

are the normal stress and shear stress acting onthe rock bridge respectively

According to the element shown in Figure 6 and theprinciple of mechanics balance it could be deduced asfollows

sum119865119909= 0 sum119865

119910= 0

211990331205903minus 2119886 (120591

119899119890sin120595 + 120590

119899119890cos120595)

minus 21199032(120591119862119863

sin 120579 + 120590119862119863119899

cos 120579) minus 2119901119897 = 0

211990311205901+ 2119886 (120591

119899119890cos120595 minus 120590

119899119890sin120595)

+ 21199032(120591119862119863

cos 120579 minus 120590119862119863119899

sin 120579) = 0

(14)

where

1199031= 119886 sin120595 + 1

2

radic1198892+ ℎ2 sin 120579

1199032=

radic1198892+ ℎ2 cos 120579 minus 2119897

2 cos 120579

1199033= 119886 cos120595 + 1

2

radic1198892+ ℎ2 cos 120579

120579 = 120595 minus arctan 119889ℎ

(15)

Then the following equation can be obtained

120591119862119863

=

21198601tan 120579 minus 2119861

1

(radic1198892+ ℎ2 cos 120579 minus 2119897) (1 + tan2120579)

120590119862119863

=

21198601+ 21198611tan 120579

(radic1198892+ ℎ2 cos 120579 minus 2119897) (1 + tan2120579)

(16)

1198601= 11990331205903minus 119886 (120591

119899119890sin120595 + 120590

119899119890cos120595) minus 119901119897

1198611= 11990311205901+ 119886 (120591

119899119890cos120595 minus 120590

119899119890sin120595)

(17)

Assuming that the rock bridge shear damage follows theMohr-Coulomb strength criterion conditions for the damageare

120591119862119863

minus 119888 minus 120590119862119863

tan120593 ⩾ 0 (18)

Substituting (16) into (18) the fellow equation can beeasily obtained as follows

(

1

2

radic1198892+ ℎ2 cos 120579 minus 119897) 119888 tan2 120579 + (119861

1tan120593 minus 119860

1) tan 120579

+ 1198601tan120593 + 119861 + 1

2

119888radic1198892+ ℎ2 cos 120579 minus 119888119897 ⩽ 0

(19)

When angle 120579 meets the rock bridge shear failure condi-tion and the wing crack propagation length reaches criticalvalue 119897

2119888 the angle 120579 between the rock bridge and 120590

1also

reaches their critical value and then

120579119888= arctan

1198601minus 1198611tan120593 + radicΔ

(radic1198892+ ℎ2 cos 120579 minus 2119897

2119888) 119888

(20)

As the geometric relations show in Figure 6 it is easy tosee that

tan 120579119888=

radic1198892+ ℎ2 sin 120579

(radic1198892+ ℎ2 cos 120579 minus 2119897

2119888)

(21)

Combined (20) and (21) the critical length of the wingcrack can be defined as

1198972119888=

1198602minus radic119860

2

2minus 4 (119888 + 119901 tan120593) 119861

2

2 (119888 + 119901 tan120593)

(22)

where

1198602= 11990331205903tan120593 minus 119886 (120591

119899119890sin120595 + 120590

119899119890cos120595) tan120593 + 119861

1

+

1

2

radic1198892+ ℎ2[(2119888 minus 119901 tan120593) cos 120579 + 119901 sin 120579]

1198612=

1

4

(1198892+ ℎ2) 119888 +

1

2

radic1198892+ ℎ2

times[(11990331205903minus 119886120591119899119890sin120595 minus 119886120590

119899119890cos120595) (cos 120579 tan120593 minus sin 120579)

+1198611(sin 120579 tan120593 + cos 120579)]

(23)

6 Mathematical Problems in Engineering

0 12 24 36 48 6 72 84 960

015

03

045

06

075

09

105

Equivalent crack length 119871(119897119886)

119901 = 0119901 = 6MPa119901 = 15MPa

Dim

ensio

nles

s stre

ss in

tens

ity fa

ctor

119870119868120590 1(120587119886)

12

Figure 4 The relationship between intensity factor at the crack tipwing stress and crack propagation length

Meanwhile this paper gets the stress intensity factor at thecrack tip when the wing reaches the critical length as follows

119870119868(1198972119888) = 119870

119868+ 1198701015840

119868= 3120591eff

radic119886119897119905119910

120587

sinminus1 ( 1

119897119905119910

) sin (120579) cos (120579)

minus

1

2

[(1205901+ 1205903) + (120590

1minus 1205903) cos2 (120579 + 120573)]radic120587119897

2119888

+ 119901radic1205871198972119888+

119886 (1205901minus 1205903) sin2120595 sin120595 + 2119901119897

2119888

2 (1198972119888+ 119886 cos120595) minus 119873minus12

119860

radic1205871198972119888

(24)

4 The Damage Evolution Mechanism ofMulticrack Rock Masses

The macroscopic mechanical effect of fractured rock wouldbe reflected by the change of its flexibility and then thedamage tensor 119863 can be determined by the elastic flexibilitytensor1198620 and the equivalent damage flexibility tensor119862119889 [28]

119863 = 119868 minus

(119862119889)

minus1

1198620

(25)

where 119868 is the fourth-order unit tensor and119863 is a tensor of thefourth order for the 3D anisotropy mode of fractured rock

In terms of the complete rock the elastic flexibility tensor1198620 can be expressed as [29]

1198620

119894119895119896119897 =

1 + V0

1198640

120575119894119896120575119895119897minus

V0

1198640

120575119894119895120575119896119897 (26)

where1198640and V0represented the elasticmodulus andPoissonrsquos

ratio of the rock respectivelyThe equivalent damage flexibility tensor119862119889 can be drawn

from Betti energy reciprocal theorem Hence based on self-consistent method and strain energy equivalence in solid

mechanics the equivalent damage flexibility tensor 119862119889119894119895119896119897 isacquired through calculating the equivalent elastic strainenergy

41The Equivalent Damage Flexibility Tensor Based on Equiv-alent Elastic Strain Energy The flexibility matrix of the crackin compression shear state is [30]

[1198620] = [

[

119862011 119862011 0

119862021 1198620

220

0 0 1198620

33

]

]

(27)

Based onBettirsquos theorem that isMaxwell-Betti reciprocalwork theorem the work done by one set of forces throughthe displacements produced by another set of forces is equalto the work done by the latter set of forces through thedisplacements produced by the former set of forces

(1)When 120590119909= 120591119909119910= 0 120590

119910= 0

11988212= (2119887120590

119910minus 2119886119901) (119862

221205901015840

1199102119889) = 4 (119887120590

119910minus 119886119901) 120590

101584011988911986222

11988221= (2119887120590

1015840

119910) (120590119910minus 119901)119862

0

22

2119889 + Δ119882119888

(28)

where Δ119882119888= (2119886120590

1015840

119910)(120590119910minus 119901)119862

119899119870119899

Then we get

11988221= 41205901015840

1199101198891198871198620

22

(120590119910minus 119901) +

21198861205901015840

119910(120590119910minus 119901)119862

119899

119870119899

(29)

According to 11988221

= 11988212 whilst the crack size can be

neglected compared to the size of the rock mass

11986222= (1 minus 119877) (119862

0

22

+

119886119862119899

2119870119899119887119889

) (30)

where 119877 = 119901120590119910

(2)When 120590119909= 120590119910= 0 120591119909119910

= 0

11988212= (1205912119889) (119862

3312059110158402119887) = 4119862

331205911015840119887119889

11988221= (12059110158402119889) (119862

331205912119887) + Δ119882

119888= 4120591119862

331205911015840119887119889 + Δ119882

119888

Δ119882119888=

12059110158402119886120591119862119904

119870119904

(31)

Then we obtain the following equation

11988221= 412059110158401205911198891198871198620

33+

21198861205911015840120591119862119904

119870119904

(32)

According to11988221= 11988212 we get

11986233= 1198620

33

+

119886119862119904

2119870119904119887119889

(33)

Thus the parameters of the equivalent damage flexibilitymatrix [119862119889] can be expressed as follows

[119862119889] =

[[[[[

[

0 0 0

0

119886119862119899(1 minus 119877)

2119870119899119887119889

minus 1198771198620

220

0 0

119886119862119904

1198701199042119887119889

]]]]]

]

(34)

Mathematical Problems in Engineering 7

42TheEquivalent Damage Flexibility Tensor Based onCrackrsquosElastic Strain Energy The strain energy of single crack isassumed as [29]

119880(119894)

119889= 2int

119886

0

119866119889Γ = 2

1 minus ]20

1198640

int

119886

0

(1198702

119868+ 1198702

119868119868) 119889119886 (35)

where Γ is the line length of cracksIntegrating over the line length the stress intensity factors

of type 119868 or type 119868119868 for the crack under seepage pressure are

119870(119894)

119868= 120590(119894)

119891

radic120587119886(119894) 119870

(119894)

119868119868= 120591(119894)

119902radic120587119886(119894) (36)

where 120590(119894)119891

is the normal stress of the crack and 120591(119894)

119902is the

tangential shear stress Combining the former equation thenwe get

119880(119894)

119889=

1 minus ]20

1198640

120587119886(119894)2

[120590(119894)2

119891+ 120591(119894)2

119902] (37)

By assuming there are 119899 groups of cracks in the fracturedrock mass the strain energy of the fractured rock mass willbe

119880119889=

119899

sum

119894=1

119880(119894)

119889=

1 minus ]20

1198640

120587

119899

sum

119894=1

119886(119894)2120588(119894)[120590(119894)2

119891+ 120591(119894)2

119902] (38)

Set the equation 119867 = ((1 minus ]20)1198640)120587 119860(119894) = 119886

(119894)2120588(119894) in

which 120588(119894) is the density of cracks then

119880119889= 119867

119899

sum

119894=1

119860(119894)[120590(119894)2

119891+ 120591(119894)2

119902] (39)

Rewrite (39) into the tensor expression

119880119889=

1

2

120590119894119895119862119889

119894119895119896119897120590119896119897= 119867

119899

sum

119894=1

119860(119894)[120590(119894)

119891120590(119894)

119891+ 120591(119894)

119902120591(119894)

119902] (40)

According to the hydrostatic pressure principle andCauchy criterion the remote field stress is denoted as 120590

119894119895 and

the stress matrix on the surface can be defined as [31]

[

[

119879V119909

119879V119910

]

]

= [

120590119909119909minus 119901 120590

119909119910

120590119910119909

120590119909119909minus 119901

] [

]119909

]119910

] (41)

Defining 119899119894119899119894= 1 authors rewrite the equation into the

tensor expression

119879(119894)

119894= (120590(119894)

119894119895minus 120575119894119895119901) 119899(119894)

119895= 120590(119894)

119894119895119899(119894)

119895minus 119901119899(119894)

119894 (42)

Considering the compression transferring coefficient 119862119899

the equivalent normal stress on the surface of the crack canbe expressed as

120590(119894)

119891= 119879(119894)

119894119899(119894)

119895= (1 minus 119862

(119894)

119899) 120590(119894)

119894119895119899(119894)

119895119899(119894)

119894minus 120573(119894)119901

(120590(119894)

119891)119894= (1 minus 119862

119899) 120590(119894)

119895119896119899(119894)

119895119899(119894)

119896119899(119894)

119894minus 120573(119894)119901119899(119894)

119894

(43)

Considering the shear transferring coefficient 119862119904again

the shear stress on the crack surface can be expressed as

(120591(119894)

119891)119894= 119879(119894)

119894minus (120590(119894)

119891)119894= (1 minus 119862

(119894)

119899) 120590(119894)

119895119896119899(119894)

119895(120575119896119897minus 119899(119894)

119896119899(119894)

119894)

(44)

Then we get

(120590(119894)

119891)119894(120590(119894)

119891)119894= ((1 minus 119862

(119894)

119899) 120590(119894)

119895119896119899(119894)

119895119899(119894)

119896119899(119894)

119894minus 120573(119894)119901119899(119894)

119894)

times ((1 minus 119862(119894)

119899) 120590(119894)

119904119905119899(119894)

119904119899(119894)

119905119899(119894)

119894minus 120573(119894)119901119899(119894)

119894)

(45)

Considering the symmetry of 120590119894119895 the following equations

can be deduced

(120590(119894)

119891)119894(120590(119894)

119891)119894= (1 minus 119862

(119894)

119899)

2

120590(119894)

119895119896119899(119894)

119895119899(119894)

119896119899(119894)

119904119899(119894)

119905120590(119894)

119904119905

minus (1 minus 119862(119894)

119899) 120590(119894)

119895119896119899(119894)

119895119899(119894)

119896119901

minus (1 minus 119862(119894)

119899) 120590(119894)

119904119905119899(119894)

119904119899(119894)

119905119901 + 120573(119894)2

1199012

(120591(119894)

119891)119894(120591(119894)

119891)119894= (1 minus 119862

(119894)

119904)

2

120590(119894)

119895119896119899(119894)

119895(120575119896119897minus 119899(119894)

119896119899(119894)

119894) 120590(119894)

119904119905119899(119894)

119904

times (120575119905119894minus 119899(119894)

119894119899(119894)

119894)

= (1 minus 119862(119894)

119904)

2

120590(119894)

119895119896

times (120575119896119897119899(119894)

119895119899(119894)

119904minus 119899(119894)

119895119899(119894)

119896119899(119894)

119904119899(119894)

119905) 120590(119894)

119904119905

(46)

Supposing the proportion coefficient 119877 = 119901120590 120590 is theaverage stress 120590 = (12)120590

119894119894 120590119894119894is the first invariant stress

Then the seepage pressure can be transformed to

119901 = 119901

120590119904119904

120590119904119904

= 120590119904119905120575119904119905

119901

2120590

=

1

2

120590119904119905120575119904119905119877

1199012= 1199012 120590119896119896

120590119896119896

120590119904119904

120590119904119904

= 120590119896119895120575119896119895120590119904119905120575119905119904

119901

2120590

119901

2120590

=

1

4

1205901198961198951205751198961198951205901199041199051205751199051199041198772

(47)

Then we get

(120590(119894)

119891)119894(120590(119894)

119891)119894

= (1 minus 119862(119894)

119899)

2

120590(119894)

119895119896119899(119894)

119895119899(119894)

119896119899(119894)

119904119899(119894)

119905120590(119894)

119904119905

minus

1

2

(1 minus 119862(119894)

119899) 119877120590(119894)

119895119896

times (119899(119894)

119895119899(119894)

119896120590(119894)

119904119905+ 120575119895119896119899(119894)

119904119899(119894)

119905)

+

1

4

(120573(119894)119877)

2

120590(119894)

119895119896120590(119894)

119904119905120590119905119904

8 Mathematical Problems in Engineering

Tension crack

1205901

1205901

12059031205903

ℎ

119863

2119886

120595

2119886

Figure 5 Failure characteristic of crag bridge shearing

119880119889=

1

2

120590119895119896119862119889

119895119896119904119905120590119904119894

= 119867

119899

sum

119894=1

119860(119894)120590(119894)

119895119896[(1 minus 119862

(119894)

119899)

2

119899(119894)

119895119899(119894)

119896119899(119894)

119904119899(119894)

119905

minus

1

2

(1 minus 119862(119894)

119899) 120573(119894)119877120590(119894)

119895119896

times (119899(119894)

119895119899(119894)

119896120590(119894)

119904119905+ 120575119895119896119899(119894)

119904119899(119894)

119905)

+

1

4

(119877120573(119894))

2

120590(119894)

119895119896120590(119894)

119904119905120590119905119904+ (1 minus 119862

(119894)

119904)

2

times (120575119896119905119899(119894)

119895119899(119894)

119904minus 119899(119894)

119895119899(119894)

119896119899(119894)

119904119899(119894)

119905) ] 120590(119894)

119904119905

(48)

Finally it is clear to conclude the additional flexibilitytensor of fractured rock mass under seepage pressure asfollows

119862119889

119894119895119896119897= (1 minus 119862

(119894)

119899)

2

119899(119894)

119895119899(119894)

119896119899(119894)

119904119899(119894)

119905

minus

1

2

(1 minus 119862(119894)

119899) 120573(119894)119877120590(119894)

119895119896(119899(119894)

119895119899(119894)

119896120590(119894)

119904119905+ 120575119895119896119899(119894)

119904119899(119894)

119905)

+

1

4

(119877120573(119894))

2

120590(119894)

119895119896120590(119894)

119904119905120590119905119904

+ (1 minus 119862(119894)

119904)

2

(120575119896119905119899(119894)

119895119899(119894)

119904minus 119899(119894)

119895119899(119894)

119896119899(119894)

119904119899(119894)

119905)

(49)

43The Fracture Damage Evolution Equation The additionalelastic strain energy density 119906119894

119896119911

produced by the wing crackis [32]

119906(119894)

119896119911=

4

1198640

120588(119894)

V 119886(119894)2(

5

2

120590(119894)

eff119871(119894)+

2

radic3

120591(119894)

eff)2

(50)

where 119871(119894) is the equivalent length for the crack

Assuming there are 119899 groups of cracks in the fracturedrock mass the additional elastic strain energy density of thefractured rock 119880

119896119911is

119880119896119911=

119899

sum

119894=1

119906(119894)

119896119911=

1

2

120590119894119895119862119896119911

119894119895119896119897120590119896119897 (51)

Crack extension reduces the stiffness of rock masses andincreases its flexibility Through the derivative of 119880

119896119911with

respect to the stress tensor we can get the additional flexibilitytensor of fractured rock masses in the damage evolutionprocess

120597119880119896119911

120597120590119894119895

= 119862119896119911

119894119895119896119897120590119896119897 (52)

At the same time by calculating the partial derivativeof 119871(119894) 120590(119894)eff and 120591

(119894)

eff with respect to 120590119894119895 respectively and

considering the symmetry of 119862119896119911119894119895119896119897

we can get the followingresults

120597119880119896119911

120597120590119894119895

=

1

1198640

119899

sum

119894=1

119886(119894)2120588(119894)

V [119872

(119894)

1119899(119894)

119895119899(119894)

119896119899(119894)

119896119899(119894)

119905minus119872(119894)

2

times (119899(119894)

119895119899(119894)

119896120575119896119895+ 120575119896119894119899(119894)

119895119899(119894)

119905

+119899(119894)

119895119899(119894)

119896120575119894119895+ 120575119894119905119899(119894)

119895119899(119894)

119896)

minus

119872(119894)

3

3

120573119877120575119894119895120575119896119897]120590119896119897

(53)

Mathematical Problems in Engineering 9

Shear crack

Tension crack

1199031

1199033

120595

119862

119864

119865

119861

119863119897

120591CD

120590ne120591ne

120579

119903 2

119901

119901

1205901

1205903

119897

119886119860

2119886

120590CD119899

Figure 6 Failure characteristic of crag bridge shearing

where

119872(119894)

1= (20120590

(119894)

eff119871(119894)+

16

radic3

120591(119894)

eff)

times [

1

120590(119894)

119899

(

5

2

119871(119894)+

2

radic3

119891)

minus

1

120591(119894)

119899

(

5

2

cot 120579(119894)119871(119894) + 2

radic3

119891) +

5

2

120590(119894)

eff119873(119894)

1]

119872(119894)

2= (5120590

(119894)

eff119871(119894)+

4

radic3

120591(119894)

eff)

times [

1

120591(119894)

119899

(

5

2

cot 120579(119894)119871(119894) + 2

radic3

119891) +

5

2

120590(119894)

eff119873(119894)

2]

119872(119894)

3= (20120590

(119894)

eff119871(119894)+

16

radic3

120591(119894)

eff)

times [

2

radic3

119891 +

5

2

(120590(119894)

eff119873(119894)

3+ 119871(119894))]

119873(119894)

1=radic119871(119894)[minus4119886(119894) cos 120579(119894)120590(119894)2eff (119891120591

(119894)

119899minus 120590(119894)

119899)

minus (119870(119894)2

119868119862minus 4119886(119894) cos 120579(119894)120590(119894)eff

minus 2radic120587119886(119894)119860(119894)119870(119894)

119868119862120590(119894)

eff )

times (120591(119894)

119899minus 120590(119894)

119899cot120579(119894))]

times (2120587119886(119894)119860(119894)120590(119894)

eff120590(119894)

119899120591(119894)

119899)

minus1

119873(119894)

2=radic119871(119894)[minus4119886(119894) cos 120579(119894)120590(119894)2eff

minus (119870(119894)

119868119862

2

minus 4119886(119894) cos 120579(119894)120590(119894)eff120591

(119894)

eff

minus2radic120587119886(119894)119860(119894)119870(119894)

119868119862120590(119894)

eff) cot 120579(119894)]

times (2120587119886(119894)119860(119894)120590(119894)

eff120590(119894)

119899120591(119894)

119899)

minus1

119873(119894)

3

=[

[

minus(minus2(

119870(119894)

119868119862

radic120587119886(119894)

)

2

120590(119894)3

eff minus2

120587

cos 120579(119894)(119891

120590(119894)

eff

minus

120591(119894)

eff

120590(119894)2

eff

))]

]

times

radic119871(119894)

119860(119894)

minus

2119870(119894)

119868119862radic119871(119894)

120590(119894)2

effradic2120587119886

(119894)

119860(119894)= radic

119870(119894)

119868119862

2120590(119894)2

effradic2120587119886

(119894)

minus

2

120587

cos 120579(119894)120591(119894)

eff

120590(119894)2

eff

(54)

Then we can get the flexibility tensor 119862119896119911119894119895119896119897

due to thedamage evolution of fractured rock masses as follows

119862119896119911

119894119895119896119897=

1

1198640

119899

sum

119894=1

119886(119894)2120588(119894)

V [119872(119894)

1119899(119894)

119895119899(119894)

119896119899(119894)

119896119899(119894)

119905minus119872(119894)

2

times (119899(119894)

119895119899(119894)

119896120575119896119895+ 120575119896119894119899(119894)

119895119899(119894)

119905

+119899(119894)

119895119899(119894)

119896120575119894119895+ 120575119894119905119899(119894)

119895119899(119894)

119896)

minus

119872(119894)

3

3

120573119877120575119894119895120575119896119897]

(55)

44 The Constitutive Equation of the Multicrack Rock MassTo sum up combining the initial damage and damage

10 Mathematical Problems in Engineering

evolution tensor we can get the flexibility tensor of thefractured rock mass under seepage pressureas follows

119862119894119895119896119897

= 1198620

119894119895119896119897+ 119862119889

119894119895119896119897+ 119862119896119911

119894119895119896119897 (56)

where1198620119894119895119896119897

is the elastic flexibility tensor of the complete rock119862119889

119894119895119896119897is the initial equivalent damage flexibility tensor of the

fractured rock mass under seepage pressure and 119862119896119911119894119895119896119897

is theadditional damage flexibility tensor with crack propagationof the fractured rock mass under seepage pressure

According to generalized Hookersquos law [33]

120576119894119895= 119862119894119895119896119897120590119896119897

120590119894119895= 119864119894119895119896119897120576119896119897= (119862119894119895119896119897)

minus1

120576119896119897

(57)

and this paper sets up the constitutive relation of the multi-crack rock mass under seepage pressure

5 Conclusions

After discussing the damage and fracture evolution mecha-nism of the multicrack rock mass under compression-shearstress and seepage pressure the following main conclusionsare drawn

(1) Through proposing the fractured damage model ofthe multicrack rock mass under the combined actionof compression-shear stress and seepage pressurethis paper discusses the law of crack initiation andthe evolution law of stress intensity factor at the tipof a wing crack The interaction of multiple cracksmakes the stress intensity factor at the crack tiplarger than that of a single wing crack Regarding theseepage pressure the existence of that reinforces thewing crackrsquos propagation Moreover the high seepagepressure converts wing crack growth from stable formto unstable form

(2) The wing crack initiation from microcracks to thecompletely damage of the rock mass is a processof evolution and accumulation of the rock massdamage as well as a process of the fracture of theconnection between cracks Based on the criterionand mechanism for crack initiation and propagationthis paper puts forward the mechanical model fordifferent fracture transfixion failuremodes of the cragbridge under the combined action of seepage pressureand compression-shear stress

(3) According to the strain energy equivalent principlethis paper applies Bettirsquos reciprocal work theorem ofthe fractured rock mass to study the flexibility tensorof the rock massrsquos initial damage and its damage evo-lution and deduces the damage constitutive equationsfor the elastic-plastic fracture and damage evolutionThe theory provides a reliable theoretical principlefor quantitative research of the fractured rock massfailure under seepage pressure

Acknowledgments

This paper gets its funding from Projects (51174228 5127424941002090) supported by the National Natural ScienceFoundation of China Project supported by the HunanProvincial Innovation Foundation for Postgraduate Project(20120162120014) supported by the Specialized ResearchFund for the Doctoral Program of Higher Education ofChina Project (12KF01) supported by the Open Projects ofState Key Laboratory of Coal Resources and Safe MiningCUMTThe authors wish to acknowledge these supports

References

[1] V Kuksenko N Tomilin and A Chmel ldquoThe rock fractureexperiment with a drive control a spatial aspectrdquo Tectono-physics vol 431 no 1ndash4 pp 123ndash129 2007

[2] S Marfia and E Sacco ldquoA fracture evolution procedure forcohesive materialsrdquo International Journal of Fracture vol 110no 3 pp 241ndash261 2001

[3] L N Lens E Bittencourt and V M R drsquoAvila ldquoConstitutivemodels for cohesive zones in mixed-mode fracture of plainconcreterdquo Engineering Fracture Mechanics vol 76 no 14 pp2281ndash2297 2009

[4] M K Rahman Y A Suarez Z Chen and S S RahmanldquoUnsuccessful hydraulic fracturing cases in Australia investi-gation into causes of failures and their remediesrdquo Journal ofPetroleum Science and Engineering vol 57 no 1-2 pp 70ndash812007

[5] R P Orense S Shimoma K Maeda and I Towhata ldquoInstru-mented model slope failure due to water seepagerdquo Journal ofNatural Disaster Science vol 26 pp 15ndash26 2004

[6] M M Hossain and M K Rahman ldquoNumerical simulationof complex fracture growth during tight reservoir stimulationby hydraulic fracturingrdquo Journal of Petroleum Science andEngineering vol 60 no 2 pp 86ndash104 2008

[7] D Hoxha M Lespinasse J Sausse and F Homand ldquoAmicrostructural study of natural and experimentally inducedcracks in a granodioriterdquo Tectonophysics vol 395 no 1-2 pp99ndash112 2005

[8] M Sagong and A Bobet ldquoCoalescence of multiple flaws ina rock-model material in uniaxial compressionrdquo InternationalJournal of Rock Mechanics and Mining Sciences vol 39 no 2pp 229ndash241 2002

[9] R H C Wong K T Chau C A Tang and P Lin ldquoAnalysisof crack coalescence in rock-like materials containing threeflawsmdashpart I experimental approachrdquo International Journal ofRockMechanics andMining Sciences vol 38 no 7 pp 909ndash9242001

[10] S YWang SW SloanD C Sheng andCA Tang ldquoNumericalanalysis of the failure process around a circular opening in rockrdquoComputers and Geotechnics vol 39 pp 8ndash16 2012

[11] O Mutlu and A Bobet ldquoSlip initiation on frictional fracturesrdquoEngineering FractureMechanics vol 72 no 5 pp 729ndash747 2005

[12] Y Nara and K Kaneko ldquoStudy of subcritical crack growth inandesite using the Double Torsion testrdquo International Journal ofRock Mechanics and Mining Sciences vol 42 no 4 pp 521ndash5302005

[13] M Haggerty Q Lin and J F Labuz ldquoObserving deformationand fracture of rock with speckle patternsrdquo Rock Mechanics andRock Engineering vol 43 pp 417ndash426 2010

Mathematical Problems in Engineering 11

[14] C Dascalu B Francois and O Keita ldquoA two-scale model forsubcritical damage propagationrdquo International Journal of Solidsand Structures vol 47 no 3-4 pp 493ndash502 2010

[15] SMisra ldquoDeformation localization at the tips of shear fracturesan analytical approachrdquo Tectonophysics vol 503 no 1-2 pp182ndash187 2011

[16] M F Ashby and S D Hallam ldquoThe failure of brittle solidscontaining small cracks under compressive stress statesrdquo ActaMetallurgica vol 34 no 3 pp 497ndash510 1986

[17] C Fairhurst and N G W Cood ldquoThe phenomenon of rocksplitting parallel to the direction of maximum compression inthe neighbourhood of a surfacerdquo in Proceedings of the Congressof the International Society of RockMechanics L Muller Ed pp687ndash692 Balkema Rotterdam The Netherlands 1966

[18] H P Yuan P Cao and W Z Xu ldquoMechanism study onsubcritical crack growth of flabby and intricate ore rockrdquoTransactions of Nonferrous Metals Society of China vol 16 no3 pp 723ndash727 2006

[19] E Sahouryeh A V Dyskin and L N Germanovich ldquoCrackgrowth under biaxial compressionrdquo Engineering FractureMechanics vol 69 no 18 pp 2187ndash2198 2002

[20] S-H Zheng Research on Coupling Theory Between Seepageand Damage of Fractured Rock Mass and Its Application toEngineering Wuhan Institute of Rock and Soil MechanicsChinese Academy of Scinces Wuhan China 2000

[21] T Liu P Cao and H Lin ldquoAnalytical solutions for the seepageinduced fracture failure propagation in rock massrdquo Journal ofDisaster Advances vol 3 supplement 1 pp 307ndash314 2013

[22] L N Y Wong and H H Einstein ldquoUsing high speed videoimaging in the study of cracking processes in rockrdquoGeotechnicalTesting Journal vol 32 no 2 pp 164ndash180 2009

[23] W Zhu and Q Zhang ldquoBrittle elastic fracture damage constitu-tive model of jointed rockmass and its application to engineer-ingrdquoChinese Journal of RockMechanics and Engineering vol 18no 3 pp 245ndash249 1999

[24] K Y Lam and S P Phua ldquoMultiple crack interaction and itseffect on stress intensity factorrdquoEngineering FractureMechanicsvol 40 no 3 pp 585ndash592 1991

[25] M Kachanov ldquoOn the problems of crack interactions and crackcoalescencerdquo International Journal of Fracture vol 120 no 3pp 537ndash543 2003

[26] H Qing and W Yang ldquoCharacterization of strongly interactedmultiple cracks in an infinite platerdquo Theoretical and AppliedFracture Mechanics vol 46 no 3 pp 209ndash216 2006

[27] A P S Selvadurai and Q Yu ldquoMechanics of a discontinuity in ageomaterialrdquo Computers and Geotechnics vol 32 no 2 pp 92ndash106 2005

[28] X P Yuan H Y Liu and Z Q Wang ldquoAn interacting crack-mechanics basedmodel for elastoplastic damagemodel of rock-likematerials under compressionrdquo International Journal of RockMechanics and Mining Sciences vol 58 pp 92ndash102 2013

[29] B Franois and C Dascalu ldquoA two-scale time-dependentdamage model based on non-planar growth of micro-cracksrdquoJournal of the Mechanics and Physics of Solids vol 58 no 11 pp1928ndash1946 2010

[30] V Palchik andYHHatzor ldquoCrack damage stress as a compositefunction of porosity and elasticmatrix stiffness in dolomites andlimestonesrdquo Engineering Geology vol 63 no 3-4 pp 233ndash2452002

[31] G Alfano S Marfia and E Sacco ldquoA cohesive damage-friction interface model accounting for water pressure on crack

propagationrdquo Computer Methods in Applied Mechanics andEngineering vol 196 no 1ndash3 pp 192ndash209 2006

[32] V Monchiet C Gruescu O Cazacu and D Kondo ldquoAmicromechanical approach of crack-induced damage inorthotropic media application to a brittle matrix compositerdquoEngineering Fracture Mechanics vol 83 pp 40ndash53 2012

[33] A Golshani Y Okui M Oda and T Takemura ldquoA microme-chanical model for brittle failure of rock and its relation tocrack growth observed in triaxial compression tests of graniterdquoMechanics of Materials vol 38 no 4 pp 287ndash303 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

1

1205901

1205901

1205903 12059031radic119873119860 1205909984003119905

119901 119901

119901 119901

1

119897

119865 = 119879119890 sin 120595

119865 = 119879119890 sin 120595

119886cos120595

Figure 2 Schematic drawing ofmultiple interacting cracks119865 is the horizontal stress in the crack 12059010158403is the horizontal stress in the rock bridge

and 119905 is the length of the rock bridge

From (3a) it is clear to see that the ultimate strength 12059011

decreases linearly with the increase of the seepage pressure119901

As for multicrack rock masses when at the initial stage ofcrack expansion or the crack space is relatively large it canbe treated as the wing crack propagation of a single crackUnder such circumstance the stress intensity factor119870

119868at the

crack tip could be defined through the revised wing crackcalculation model [21] Considering the additional seepagepressure in the crack the stress intensity factor 119870

119868consists

of the stress intensity factor produced by the effective sheardriving force 119879

119890and the remote field stress 120590

1and 120590

3 and it

can be shown as in (4)

119870119868= 3120591119899119890radic119886119897119905119910

120587

sinminus1 ( 1

119897119905119910

) sin 120579 cos 1205792

minus

1

2

[(1205901+ 1205903) + (120590

1minus 1205903) cos2 (120579 + 120573)]radic120587119897 + 119901radic120587119897

(4)

where 119897119905119910is the impact factor which is the function of thewing

crack length 119897 wing crack azimuth 120579 and main crack length119886 as shown in (5)

119897119905119910= [1 +

9119897

4119886

cos2 (1205792

)] (1 minus 119890minus119897119886

) + 0667 sec2 (1205792

) 119890minus119897119886

(5)

The wing crack extends approximately in the directionperpendicular to the maximum principal stress until 119870

119868=

119870119868119862 and then the wing crack propagation length 119897 under

the combined action of compression-shear stress and seepagepressure can be calculated as shown in (6)

119897 =

6120591119899119890radic(119886119897119905119910120587)sinminus1 (1119897

119905119910) sin 120579 cos (1205792) minus 2119870

119868119862

[(1205901+ 1205903) + (120590

1minus 1205903) cos2 (120579 + 120573) minus 2119901]radic120587

(6)

22 The Multicrack Interaction Models As the crack expandsor when the crack space is relatively small the interactionbetween cracks leads to a damaged connection and unstablebreak of the rock bridge [22] The rock bridge interactionmechanical model of the multicrack rock mass with the wingcrack expansion is shown in Figure 2

Assuming that the number of the compression-shearcracks per unit area is 119873

119860 the length between main crack

centers and the rock bridge lengths between wing cracksrespectively are presented as follows

119878 =

1

radic119873119860

119879 = 119878 minus 2 (119897 + 119886 cos120595)

(7)

In Figure 2 119865 = 119879119890sin120595 is balanced by the tensile stress

1205901015840

3in the rock bridge together with seepage pressure 119901in the

wing crack

1205901015840

3=

119879119890sin120595 + 2119901119897

119878 minus 2 (119897 + 119886 cos120595) (8)

where 119879119890= 2119886120591

119899119890

Acting on the wing crack 12059010158403produces an additional

intensity factor at the crack tip [23]

1198701015840

119868= minus1205901015840

3radic120587119897 =

119886 (1205901minus 1205903) sin2120595 sin120595 + 2119901119897

119873minus12

119860minus 2 (119897 + 119886 cos120595)

radic120587119897 (9)

4 Mathematical Problems in Engineering

Considering the damage to the rock bridge caused by thewing crack interaction the stress intensity factor at the wingcrack tip is composed of (4) and (9) as follows

119870119868= 119870119868+ 1198701015840

119868= 3120591119899119890radic119886119897119905119910

120587

sinminus1 ( 1

119897119905119910

) sin 120579 cos 1205792

minus

1

2

[(1205901+ 1205903) + (120590

1minus 1205903) cos2 (120579 + 120573)]radic120587119897

+ 119901radic120587119897 +

119886 (1205901minus 1205903) sin2120595 sin120595 + 2119901119897

119873minus12

119860minus 2 (119897 + 119886 cos120595)

radic120587119897

(10)

From (10) it is clear to see that the interaction of multiplewing cracks in rock bridge damage makes the stress intensityfactor at the crack tip larger than that at a single wing crack

The process of rock fracture is the one of the increasingdamage and inherent fracture in rock and is the couplingdamage result of the microdamage and macrodamage Thereare many ways to define the damage variable such as thecracks quantity crack length crack area and crack densityHere the cracks quantity is adopted to define119863

0for the initial

damage of the rockmass and119863 for the damagewhen thewingcrack expands to 119897

1198630= 120587(119886 cos120595)2119873

119860

119863 = 120587(119897 + 119886 cos120595)2119873119860

(11)

Substituting (11) into (8) the fellow equation can be easilyobtained

1205901015840

3= minus

(120591119899119890tan120595 + 2119901119897) (119863

0120587)12

119886 (1 minus 2(119863120587)12)

(12)

Combining (10) (11) and (12) the relation curvesbetween the damage variables 119863 and dimensionless stressintensity factor (119870

1198681205901radic120587119886) at the wing crack tip with

different crack density are shown in Figure 3The graph reveals that with the increase of the damage

variables the dimensionless stress intensity factor1198701198681205901radic120587119886

at the crack tip under seepage pressure decreases at thevery beginning but rises gradually when the equivalent cracklength is 05 (119897119886) The graph also illustrates that the sparserof the cracks are the higher of the stress intensity at the wingcrack will be when the rock bridge is connected During thewing crack expansion process the damage variable 119863 variesfrom119863

0to 1 When119863 = 1 and the wing crack stress intensity

factor119870119868⩾ 119870119868119862

at the crack tip the wing crack connects therock bridge cracks and the rock mass loses bearing capacity

Figure 4 shows the relationship between the stress inten-sity factor at the crack tip and the equivalent crack prop-agation length under different seepage pressure It canbe concluded that when the seepage pressure is rela-tively low (119901 = 0MPa) the wing crack expands stably

0 015 03 045 06 075 09 105

036

038

04

042

044

Damage variables (119863)

119873119860 = 100119873119860 = 225

119873119860 = 400

Dim

ensio

nles

s stre

ss in

tens

ity119870119868120590 1(120587119886)

12

Figure 3 The relationship between damage variables and intensityfactor at the crack tip of the branch stress

(120597119870119868120597119871 lt 0) While when the seepage pressure is relatively

high (119901 = 15MPa) the wing crack tends to expand unstablyBesides there is the expansion stage that 120597119870

119868120597119871 gt 0 under

such condition and the growth rate of wing crack stressintensity factor becomes greater with the increase of seepagepressure 119901 It indicates that under high seepage pressure thecompressive-shear rock crack expands at a high speed as longas it cracks namely and the high seepage pressure contributesto unstable expansion

3 Wing Crack Connection Model andDamage Criteria

As the branch crack expands the interaction of the crackresults in the continuous degradation of the macroscopicmechanical properties of the rock mass The wing crackinitiation from microcracks to the completely damage of therock mass is a process of evolution and accumulation of therock mass damage as well as a process of the fracture of theconnection between cracks [24] It has been proved by manyexperimental researches that the gradual damage process ofthe cracks generally has two forms [25 26] (1) the rock bridgeaxial transfixion failure and (2) the tension-shear compoundfailure According to the branch crack propagation evolutionmechanism this paper studies the gradual damage features ofthose damage models

31 The Axial Transfixion Failure Whenever the wing crackextends stably or unstably the tension wing crack connectsthe main crack in another row as shown in Figure 5 Whenthe wing crack reaches the critical length 119897

1119888= 119863 sin120595

1198711119862

= 1198971119888119886 the rock bridge starts to damage in the form

of axial transfixion failure Taking the stress intensity at thecrack tip as the criterion when the wing crack reaches the

Mathematical Problems in Engineering 5

critical length 1198971119888= 119863 sin120595 this paper establishes the failure

criteria

119870119868= 3120591119899119890radic119886119897119905119910

120587

sinminus1 ( 1

119897119905119910

) sin 120579 cos 1205792

minus

1

2

[(1205901+ 1205903) + (120590

1minus 1205903) cos2 (120579 + 120573)]radic120587119897

1119888

+ 119901radic1205871198971119888+

119886 (1205901minus 1205903) sin2120595 sin120595 + 2119901119897

1119888

119873minus12

119860minus 2 (1198971119888+ 119886 cos120595)

radic1205871198971119888

(13)

When 119870119868(1198711119862) ge 119870

119868119862 axial transfixion failure occurs in the

rock bridge

32 The Tension-Shear Compound Failure With the expan-sion of wing cracks under seepage pressure the cut-resistantcapacity of the rock bridge is increasingly weakened Whenthe wing crack expands to a certain degree the rock bridgeat the crack tip between adjacent wing cracks is cut off bythe shear stress consequently the crack is connected in theshear direction [27] The mechanical analyzing diagram forthe element of the rock bridge composite failure is shown inFigure 6 where 119860119861 is 12 the length of the bottom crack 119864119865is 12 the length of the upper crack 119862119863 is the rock bridge119861119862 and 119864119863 are the wing cracks produced by effective sheardriving force of the main crack 119860119861 and 119864119865 120579 is the anglebetween rock bridge and the maximum principal stress and120590119862119863

and 120591119862119863

are the normal stress and shear stress acting onthe rock bridge respectively

According to the element shown in Figure 6 and theprinciple of mechanics balance it could be deduced asfollows

sum119865119909= 0 sum119865

119910= 0

211990331205903minus 2119886 (120591

119899119890sin120595 + 120590

119899119890cos120595)

minus 21199032(120591119862119863

sin 120579 + 120590119862119863119899

cos 120579) minus 2119901119897 = 0

211990311205901+ 2119886 (120591

119899119890cos120595 minus 120590

119899119890sin120595)

+ 21199032(120591119862119863

cos 120579 minus 120590119862119863119899

sin 120579) = 0

(14)

where

1199031= 119886 sin120595 + 1

2

radic1198892+ ℎ2 sin 120579

1199032=

radic1198892+ ℎ2 cos 120579 minus 2119897

2 cos 120579

1199033= 119886 cos120595 + 1

2

radic1198892+ ℎ2 cos 120579

120579 = 120595 minus arctan 119889ℎ

(15)

Then the following equation can be obtained

120591119862119863

=

21198601tan 120579 minus 2119861

1

(radic1198892+ ℎ2 cos 120579 minus 2119897) (1 + tan2120579)

120590119862119863

=

21198601+ 21198611tan 120579

(radic1198892+ ℎ2 cos 120579 minus 2119897) (1 + tan2120579)

(16)

1198601= 11990331205903minus 119886 (120591

119899119890sin120595 + 120590

119899119890cos120595) minus 119901119897

1198611= 11990311205901+ 119886 (120591

119899119890cos120595 minus 120590

119899119890sin120595)

(17)

Assuming that the rock bridge shear damage follows theMohr-Coulomb strength criterion conditions for the damageare

120591119862119863

minus 119888 minus 120590119862119863

tan120593 ⩾ 0 (18)

Substituting (16) into (18) the fellow equation can beeasily obtained as follows

(

1

2

radic1198892+ ℎ2 cos 120579 minus 119897) 119888 tan2 120579 + (119861

1tan120593 minus 119860

1) tan 120579

+ 1198601tan120593 + 119861 + 1

2

119888radic1198892+ ℎ2 cos 120579 minus 119888119897 ⩽ 0

(19)

When angle 120579 meets the rock bridge shear failure condi-tion and the wing crack propagation length reaches criticalvalue 119897

2119888 the angle 120579 between the rock bridge and 120590

1also

reaches their critical value and then

120579119888= arctan

1198601minus 1198611tan120593 + radicΔ

(radic1198892+ ℎ2 cos 120579 minus 2119897

2119888) 119888

(20)

As the geometric relations show in Figure 6 it is easy tosee that

tan 120579119888=

radic1198892+ ℎ2 sin 120579

(radic1198892+ ℎ2 cos 120579 minus 2119897

2119888)

(21)

Combined (20) and (21) the critical length of the wingcrack can be defined as

1198972119888=

1198602minus radic119860

2

2minus 4 (119888 + 119901 tan120593) 119861

2

2 (119888 + 119901 tan120593)

(22)

where

1198602= 11990331205903tan120593 minus 119886 (120591

119899119890sin120595 + 120590

119899119890cos120595) tan120593 + 119861

1

+

1

2

radic1198892+ ℎ2[(2119888 minus 119901 tan120593) cos 120579 + 119901 sin 120579]

1198612=

1

4

(1198892+ ℎ2) 119888 +

1

2

radic1198892+ ℎ2

times[(11990331205903minus 119886120591119899119890sin120595 minus 119886120590

119899119890cos120595) (cos 120579 tan120593 minus sin 120579)

+1198611(sin 120579 tan120593 + cos 120579)]

(23)

6 Mathematical Problems in Engineering

0 12 24 36 48 6 72 84 960

015

03

045

06

075

09

105

Equivalent crack length 119871(119897119886)

119901 = 0119901 = 6MPa119901 = 15MPa

Dim

ensio

nles

s stre

ss in

tens

ity fa

ctor

119870119868120590 1(120587119886)

12

Figure 4 The relationship between intensity factor at the crack tipwing stress and crack propagation length

Meanwhile this paper gets the stress intensity factor at thecrack tip when the wing reaches the critical length as follows

119870119868(1198972119888) = 119870

119868+ 1198701015840

119868= 3120591eff

radic119886119897119905119910

120587

sinminus1 ( 1

119897119905119910

) sin (120579) cos (120579)

minus

1

2

[(1205901+ 1205903) + (120590

1minus 1205903) cos2 (120579 + 120573)]radic120587119897

2119888

+ 119901radic1205871198972119888+

119886 (1205901minus 1205903) sin2120595 sin120595 + 2119901119897

2119888

2 (1198972119888+ 119886 cos120595) minus 119873minus12

119860

radic1205871198972119888

(24)

4 The Damage Evolution Mechanism ofMulticrack Rock Masses

The macroscopic mechanical effect of fractured rock wouldbe reflected by the change of its flexibility and then thedamage tensor 119863 can be determined by the elastic flexibilitytensor1198620 and the equivalent damage flexibility tensor119862119889 [28]

119863 = 119868 minus

(119862119889)

minus1

1198620

(25)

where 119868 is the fourth-order unit tensor and119863 is a tensor of thefourth order for the 3D anisotropy mode of fractured rock

In terms of the complete rock the elastic flexibility tensor1198620 can be expressed as [29]

1198620

119894119895119896119897 =

1 + V0

1198640

120575119894119896120575119895119897minus

V0

1198640

120575119894119895120575119896119897 (26)

where1198640and V0represented the elasticmodulus andPoissonrsquos

ratio of the rock respectivelyThe equivalent damage flexibility tensor119862119889 can be drawn

from Betti energy reciprocal theorem Hence based on self-consistent method and strain energy equivalence in solid

mechanics the equivalent damage flexibility tensor 119862119889119894119895119896119897 isacquired through calculating the equivalent elastic strainenergy

41The Equivalent Damage Flexibility Tensor Based on Equiv-alent Elastic Strain Energy The flexibility matrix of the crackin compression shear state is [30]

[1198620] = [

[

119862011 119862011 0

119862021 1198620

220

0 0 1198620

33

]

]

(27)

Based onBettirsquos theorem that isMaxwell-Betti reciprocalwork theorem the work done by one set of forces throughthe displacements produced by another set of forces is equalto the work done by the latter set of forces through thedisplacements produced by the former set of forces

(1)When 120590119909= 120591119909119910= 0 120590

119910= 0

11988212= (2119887120590

119910minus 2119886119901) (119862

221205901015840

1199102119889) = 4 (119887120590

119910minus 119886119901) 120590

101584011988911986222

11988221= (2119887120590

1015840

119910) (120590119910minus 119901)119862

0

22

2119889 + Δ119882119888

(28)

where Δ119882119888= (2119886120590

1015840

119910)(120590119910minus 119901)119862

119899119870119899

Then we get

11988221= 41205901015840

1199101198891198871198620

22

(120590119910minus 119901) +

21198861205901015840

119910(120590119910minus 119901)119862

119899

119870119899

(29)

According to 11988221

= 11988212 whilst the crack size can be

neglected compared to the size of the rock mass

11986222= (1 minus 119877) (119862

0

22

+

119886119862119899

2119870119899119887119889

) (30)

where 119877 = 119901120590119910

(2)When 120590119909= 120590119910= 0 120591119909119910

= 0

11988212= (1205912119889) (119862

3312059110158402119887) = 4119862

331205911015840119887119889

11988221= (12059110158402119889) (119862

331205912119887) + Δ119882

119888= 4120591119862

331205911015840119887119889 + Δ119882

119888

Δ119882119888=

12059110158402119886120591119862119904

119870119904

(31)

Then we obtain the following equation

11988221= 412059110158401205911198891198871198620

33+

21198861205911015840120591119862119904

119870119904

(32)

According to11988221= 11988212 we get

11986233= 1198620

33

+

119886119862119904

2119870119904119887119889

(33)

Thus the parameters of the equivalent damage flexibilitymatrix [119862119889] can be expressed as follows

[119862119889] =

[[[[[

[

0 0 0

0

119886119862119899(1 minus 119877)

2119870119899119887119889

minus 1198771198620

220

0 0

119886119862119904

1198701199042119887119889

]]]]]

]

(34)

Mathematical Problems in Engineering 7

42TheEquivalent Damage Flexibility Tensor Based onCrackrsquosElastic Strain Energy The strain energy of single crack isassumed as [29]

119880(119894)

119889= 2int

119886

0

119866119889Γ = 2

1 minus ]20

1198640

int

119886

0

(1198702

119868+ 1198702

119868119868) 119889119886 (35)

where Γ is the line length of cracksIntegrating over the line length the stress intensity factors

of type 119868 or type 119868119868 for the crack under seepage pressure are

119870(119894)

119868= 120590(119894)

119891

radic120587119886(119894) 119870

(119894)

119868119868= 120591(119894)

119902radic120587119886(119894) (36)

where 120590(119894)119891

is the normal stress of the crack and 120591(119894)

119902is the

tangential shear stress Combining the former equation thenwe get

119880(119894)

119889=

1 minus ]20

1198640

120587119886(119894)2

[120590(119894)2

119891+ 120591(119894)2

119902] (37)

By assuming there are 119899 groups of cracks in the fracturedrock mass the strain energy of the fractured rock mass willbe

119880119889=

119899

sum

119894=1

119880(119894)

119889=

1 minus ]20

1198640

120587

119899

sum

119894=1

119886(119894)2120588(119894)[120590(119894)2

119891+ 120591(119894)2

119902] (38)

Set the equation 119867 = ((1 minus ]20)1198640)120587 119860(119894) = 119886

(119894)2120588(119894) in

which 120588(119894) is the density of cracks then

119880119889= 119867

119899

sum

119894=1

119860(119894)[120590(119894)2

119891+ 120591(119894)2

119902] (39)

Rewrite (39) into the tensor expression

119880119889=

1

2

120590119894119895119862119889

119894119895119896119897120590119896119897= 119867

119899

sum

119894=1

119860(119894)[120590(119894)

119891120590(119894)

119891+ 120591(119894)

119902120591(119894)

119902] (40)

According to the hydrostatic pressure principle andCauchy criterion the remote field stress is denoted as 120590

119894119895 and

the stress matrix on the surface can be defined as [31]

[

[

119879V119909

119879V119910

]

]

= [

120590119909119909minus 119901 120590

119909119910

120590119910119909

120590119909119909minus 119901

] [

]119909

]119910

] (41)

Defining 119899119894119899119894= 1 authors rewrite the equation into the

tensor expression

119879(119894)

119894= (120590(119894)

119894119895minus 120575119894119895119901) 119899(119894)

119895= 120590(119894)

119894119895119899(119894)

119895minus 119901119899(119894)

119894 (42)

Considering the compression transferring coefficient 119862119899

the equivalent normal stress on the surface of the crack canbe expressed as

120590(119894)

119891= 119879(119894)

119894119899(119894)

119895= (1 minus 119862

(119894)

119899) 120590(119894)

119894119895119899(119894)

119895119899(119894)

119894minus 120573(119894)119901

(120590(119894)

119891)119894= (1 minus 119862

119899) 120590(119894)

119895119896119899(119894)

119895119899(119894)

119896119899(119894)

119894minus 120573(119894)119901119899(119894)

119894

(43)

Considering the shear transferring coefficient 119862119904again

the shear stress on the crack surface can be expressed as

(120591(119894)

119891)119894= 119879(119894)

119894minus (120590(119894)

119891)119894= (1 minus 119862

(119894)

119899) 120590(119894)

119895119896119899(119894)

119895(120575119896119897minus 119899(119894)

119896119899(119894)

119894)

(44)

Then we get

(120590(119894)

119891)119894(120590(119894)

119891)119894= ((1 minus 119862

(119894)

119899) 120590(119894)

119895119896119899(119894)

119895119899(119894)

119896119899(119894)

119894minus 120573(119894)119901119899(119894)

119894)

times ((1 minus 119862(119894)

119899) 120590(119894)

119904119905119899(119894)

119904119899(119894)

119905119899(119894)

119894minus 120573(119894)119901119899(119894)

119894)

(45)

Considering the symmetry of 120590119894119895 the following equations

can be deduced

(120590(119894)

119891)119894(120590(119894)

119891)119894= (1 minus 119862

(119894)

119899)

2

120590(119894)

119895119896119899(119894)

119895119899(119894)

119896119899(119894)

119904119899(119894)

119905120590(119894)

119904119905

minus (1 minus 119862(119894)

119899) 120590(119894)

119895119896119899(119894)

119895119899(119894)

119896119901

minus (1 minus 119862(119894)

119899) 120590(119894)

119904119905119899(119894)

119904119899(119894)

119905119901 + 120573(119894)2

1199012

(120591(119894)

119891)119894(120591(119894)

119891)119894= (1 minus 119862

(119894)

119904)

2

120590(119894)

119895119896119899(119894)

119895(120575119896119897minus 119899(119894)

119896119899(119894)

119894) 120590(119894)

119904119905119899(119894)

119904

times (120575119905119894minus 119899(119894)

119894119899(119894)

119894)

= (1 minus 119862(119894)

119904)

2

120590(119894)

119895119896

times (120575119896119897119899(119894)

119895119899(119894)

119904minus 119899(119894)

119895119899(119894)

119896119899(119894)

119904119899(119894)

119905) 120590(119894)

119904119905

(46)

Supposing the proportion coefficient 119877 = 119901120590 120590 is theaverage stress 120590 = (12)120590

119894119894 120590119894119894is the first invariant stress

Then the seepage pressure can be transformed to

119901 = 119901

120590119904119904

120590119904119904

= 120590119904119905120575119904119905

119901

2120590

=

1

2

120590119904119905120575119904119905119877

1199012= 1199012 120590119896119896

120590119896119896

120590119904119904

120590119904119904

= 120590119896119895120575119896119895120590119904119905120575119905119904

119901

2120590

119901

2120590

=

1

4

1205901198961198951205751198961198951205901199041199051205751199051199041198772

(47)

Then we get

(120590(119894)

119891)119894(120590(119894)

119891)119894

= (1 minus 119862(119894)

119899)

2

120590(119894)

119895119896119899(119894)

119895119899(119894)

119896119899(119894)

119904119899(119894)

119905120590(119894)

119904119905

minus

1

2

(1 minus 119862(119894)

119899) 119877120590(119894)

119895119896

times (119899(119894)

119895119899(119894)

119896120590(119894)

119904119905+ 120575119895119896119899(119894)

119904119899(119894)

119905)

+

1

4

(120573(119894)119877)

2

120590(119894)

119895119896120590(119894)

119904119905120590119905119904

8 Mathematical Problems in Engineering

Tension crack

1205901

1205901

12059031205903

ℎ

119863

2119886

120595

2119886

Figure 5 Failure characteristic of crag bridge shearing

119880119889=

1

2

120590119895119896119862119889

119895119896119904119905120590119904119894

= 119867

119899

sum

119894=1

119860(119894)120590(119894)

119895119896[(1 minus 119862

(119894)

119899)

2

119899(119894)

119895119899(119894)

119896119899(119894)

119904119899(119894)

119905

minus

1

2

(1 minus 119862(119894)

119899) 120573(119894)119877120590(119894)

119895119896

times (119899(119894)

119895119899(119894)

119896120590(119894)

119904119905+ 120575119895119896119899(119894)

119904119899(119894)

119905)

+

1

4

(119877120573(119894))

2

120590(119894)

119895119896120590(119894)

119904119905120590119905119904+ (1 minus 119862

(119894)

119904)

2

times (120575119896119905119899(119894)

119895119899(119894)

119904minus 119899(119894)

119895119899(119894)

119896119899(119894)

119904119899(119894)

119905) ] 120590(119894)

119904119905

(48)

Finally it is clear to conclude the additional flexibilitytensor of fractured rock mass under seepage pressure asfollows

119862119889

119894119895119896119897= (1 minus 119862

(119894)

119899)

2

119899(119894)

119895119899(119894)

119896119899(119894)

119904119899(119894)

119905

minus

1

2

(1 minus 119862(119894)

119899) 120573(119894)119877120590(119894)

119895119896(119899(119894)

119895119899(119894)

119896120590(119894)

119904119905+ 120575119895119896119899(119894)

119904119899(119894)

119905)

+

1

4

(119877120573(119894))

2

120590(119894)

119895119896120590(119894)

119904119905120590119905119904

+ (1 minus 119862(119894)

119904)

2

(120575119896119905119899(119894)

119895119899(119894)

119904minus 119899(119894)

119895119899(119894)

119896119899(119894)

119904119899(119894)

119905)

(49)

43The Fracture Damage Evolution Equation The additionalelastic strain energy density 119906119894

119896119911

produced by the wing crackis [32]

119906(119894)

119896119911=

4

1198640

120588(119894)

V 119886(119894)2(

5

2

120590(119894)

eff119871(119894)+

2

radic3

120591(119894)

eff)2

(50)

where 119871(119894) is the equivalent length for the crack

Assuming there are 119899 groups of cracks in the fracturedrock mass the additional elastic strain energy density of thefractured rock 119880

119896119911is

119880119896119911=

119899

sum

119894=1

119906(119894)

119896119911=

1

2

120590119894119895119862119896119911

119894119895119896119897120590119896119897 (51)

Crack extension reduces the stiffness of rock masses andincreases its flexibility Through the derivative of 119880

119896119911with

respect to the stress tensor we can get the additional flexibilitytensor of fractured rock masses in the damage evolutionprocess

120597119880119896119911

120597120590119894119895

= 119862119896119911

119894119895119896119897120590119896119897 (52)

At the same time by calculating the partial derivativeof 119871(119894) 120590(119894)eff and 120591

(119894)

eff with respect to 120590119894119895 respectively and

considering the symmetry of 119862119896119911119894119895119896119897

we can get the followingresults

120597119880119896119911

120597120590119894119895

=

1

1198640

119899

sum

119894=1

119886(119894)2120588(119894)

V [119872

(119894)

1119899(119894)

119895119899(119894)

119896119899(119894)

119896119899(119894)

119905minus119872(119894)

2

times (119899(119894)

119895119899(119894)

119896120575119896119895+ 120575119896119894119899(119894)

119895119899(119894)

119905

+119899(119894)

119895119899(119894)

119896120575119894119895+ 120575119894119905119899(119894)

119895119899(119894)

119896)

minus

119872(119894)

3

3

120573119877120575119894119895120575119896119897]120590119896119897

(53)

Mathematical Problems in Engineering 9

Shear crack

Tension crack

1199031

1199033

120595

119862

119864

119865

119861

119863119897

120591CD

120590ne120591ne

120579

119903 2

119901

119901

1205901

1205903

119897

119886119860

2119886

120590CD119899

Figure 6 Failure characteristic of crag bridge shearing

where

119872(119894)

1= (20120590

(119894)

eff119871(119894)+

16

radic3

120591(119894)

eff)

times [

1

120590(119894)

119899

(

5

2

119871(119894)+

2

radic3

119891)

minus

1

120591(119894)

119899

(

5

2

cot 120579(119894)119871(119894) + 2

radic3

119891) +

5

2

120590(119894)

eff119873(119894)

1]

119872(119894)

2= (5120590

(119894)

eff119871(119894)+

4

radic3

120591(119894)

eff)

times [

1

120591(119894)

119899

(

5

2

cot 120579(119894)119871(119894) + 2

radic3

119891) +

5

2

120590(119894)

eff119873(119894)

2]

119872(119894)

3= (20120590

(119894)

eff119871(119894)+

16

radic3

120591(119894)

eff)

times [

2

radic3

119891 +

5

2

(120590(119894)

eff119873(119894)

3+ 119871(119894))]

119873(119894)

1=radic119871(119894)[minus4119886(119894) cos 120579(119894)120590(119894)2eff (119891120591

(119894)

119899minus 120590(119894)

119899)

minus (119870(119894)2

119868119862minus 4119886(119894) cos 120579(119894)120590(119894)eff

minus 2radic120587119886(119894)119860(119894)119870(119894)

119868119862120590(119894)

eff )

times (120591(119894)

119899minus 120590(119894)

119899cot120579(119894))]

times (2120587119886(119894)119860(119894)120590(119894)

eff120590(119894)

119899120591(119894)

119899)

minus1

119873(119894)

2=radic119871(119894)[minus4119886(119894) cos 120579(119894)120590(119894)2eff

minus (119870(119894)

119868119862

2

minus 4119886(119894) cos 120579(119894)120590(119894)eff120591

(119894)

eff

minus2radic120587119886(119894)119860(119894)119870(119894)

119868119862120590(119894)

eff) cot 120579(119894)]

times (2120587119886(119894)119860(119894)120590(119894)

eff120590(119894)

119899120591(119894)

119899)

minus1

119873(119894)

3

=[

[

minus(minus2(

119870(119894)

119868119862

radic120587119886(119894)

)

2

120590(119894)3

eff minus2

120587

cos 120579(119894)(119891

120590(119894)

eff

minus

120591(119894)

eff

120590(119894)2

eff

))]

]

times

radic119871(119894)

119860(119894)

minus

2119870(119894)

119868119862radic119871(119894)

120590(119894)2

effradic2120587119886

(119894)

119860(119894)= radic

119870(119894)

119868119862

2120590(119894)2

effradic2120587119886

(119894)

minus

2

120587

cos 120579(119894)120591(119894)

eff

120590(119894)2

eff

(54)

Then we can get the flexibility tensor 119862119896119911119894119895119896119897

due to thedamage evolution of fractured rock masses as follows

119862119896119911

119894119895119896119897=

1

1198640

119899

sum

119894=1

119886(119894)2120588(119894)

V [119872(119894)

1119899(119894)

119895119899(119894)

119896119899(119894)

119896119899(119894)

119905minus119872(119894)

2

times (119899(119894)

119895119899(119894)

119896120575119896119895+ 120575119896119894119899(119894)

119895119899(119894)

119905

+119899(119894)

119895119899(119894)

119896120575119894119895+ 120575119894119905119899(119894)

119895119899(119894)

119896)

minus

119872(119894)

3

3

120573119877120575119894119895120575119896119897]

(55)

44 The Constitutive Equation of the Multicrack Rock MassTo sum up combining the initial damage and damage

10 Mathematical Problems in Engineering

evolution tensor we can get the flexibility tensor of thefractured rock mass under seepage pressureas follows

119862119894119895119896119897

= 1198620

119894119895119896119897+ 119862119889

119894119895119896119897+ 119862119896119911

119894119895119896119897 (56)

where1198620119894119895119896119897

is the elastic flexibility tensor of the complete rock119862119889

119894119895119896119897is the initial equivalent damage flexibility tensor of the

fractured rock mass under seepage pressure and 119862119896119911119894119895119896119897

is theadditional damage flexibility tensor with crack propagationof the fractured rock mass under seepage pressure

According to generalized Hookersquos law [33]

120576119894119895= 119862119894119895119896119897120590119896119897

120590119894119895= 119864119894119895119896119897120576119896119897= (119862119894119895119896119897)

minus1

120576119896119897

(57)

and this paper sets up the constitutive relation of the multi-crack rock mass under seepage pressure

5 Conclusions

After discussing the damage and fracture evolution mecha-nism of the multicrack rock mass under compression-shearstress and seepage pressure the following main conclusionsare drawn

(1) Through proposing the fractured damage model ofthe multicrack rock mass under the combined actionof compression-shear stress and seepage pressurethis paper discusses the law of crack initiation andthe evolution law of stress intensity factor at the tipof a wing crack The interaction of multiple cracksmakes the stress intensity factor at the crack tiplarger than that of a single wing crack Regarding theseepage pressure the existence of that reinforces thewing crackrsquos propagation Moreover the high seepagepressure converts wing crack growth from stable formto unstable form

(2) The wing crack initiation from microcracks to thecompletely damage of the rock mass is a processof evolution and accumulation of the rock massdamage as well as a process of the fracture of theconnection between cracks Based on the criterionand mechanism for crack initiation and propagationthis paper puts forward the mechanical model fordifferent fracture transfixion failuremodes of the cragbridge under the combined action of seepage pressureand compression-shear stress

(3) According to the strain energy equivalent principlethis paper applies Bettirsquos reciprocal work theorem ofthe fractured rock mass to study the flexibility tensorof the rock massrsquos initial damage and its damage evo-lution and deduces the damage constitutive equationsfor the elastic-plastic fracture and damage evolutionThe theory provides a reliable theoretical principlefor quantitative research of the fractured rock massfailure under seepage pressure

Acknowledgments

This paper gets its funding from Projects (51174228 5127424941002090) supported by the National Natural ScienceFoundation of China Project supported by the HunanProvincial Innovation Foundation for Postgraduate Project(20120162120014) supported by the Specialized ResearchFund for the Doctoral Program of Higher Education ofChina Project (12KF01) supported by the Open Projects ofState Key Laboratory of Coal Resources and Safe MiningCUMTThe authors wish to acknowledge these supports

References

[1] V Kuksenko N Tomilin and A Chmel ldquoThe rock fractureexperiment with a drive control a spatial aspectrdquo Tectono-physics vol 431 no 1ndash4 pp 123ndash129 2007

[2] S Marfia and E Sacco ldquoA fracture evolution procedure forcohesive materialsrdquo International Journal of Fracture vol 110no 3 pp 241ndash261 2001

[3] L N Lens E Bittencourt and V M R drsquoAvila ldquoConstitutivemodels for cohesive zones in mixed-mode fracture of plainconcreterdquo Engineering Fracture Mechanics vol 76 no 14 pp2281ndash2297 2009

[4] M K Rahman Y A Suarez Z Chen and S S RahmanldquoUnsuccessful hydraulic fracturing cases in Australia investi-gation into causes of failures and their remediesrdquo Journal ofPetroleum Science and Engineering vol 57 no 1-2 pp 70ndash812007

[5] R P Orense S Shimoma K Maeda and I Towhata ldquoInstru-mented model slope failure due to water seepagerdquo Journal ofNatural Disaster Science vol 26 pp 15ndash26 2004

[6] M M Hossain and M K Rahman ldquoNumerical simulationof complex fracture growth during tight reservoir stimulationby hydraulic fracturingrdquo Journal of Petroleum Science andEngineering vol 60 no 2 pp 86ndash104 2008

[7] D Hoxha M Lespinasse J Sausse and F Homand ldquoAmicrostructural study of natural and experimentally inducedcracks in a granodioriterdquo Tectonophysics vol 395 no 1-2 pp99ndash112 2005

[8] M Sagong and A Bobet ldquoCoalescence of multiple flaws ina rock-model material in uniaxial compressionrdquo InternationalJournal of Rock Mechanics and Mining Sciences vol 39 no 2pp 229ndash241 2002

[9] R H C Wong K T Chau C A Tang and P Lin ldquoAnalysisof crack coalescence in rock-like materials containing threeflawsmdashpart I experimental approachrdquo International Journal ofRockMechanics andMining Sciences vol 38 no 7 pp 909ndash9242001

[10] S YWang SW SloanD C Sheng andCA Tang ldquoNumericalanalysis of the failure process around a circular opening in rockrdquoComputers and Geotechnics vol 39 pp 8ndash16 2012

[11] O Mutlu and A Bobet ldquoSlip initiation on frictional fracturesrdquoEngineering FractureMechanics vol 72 no 5 pp 729ndash747 2005

[12] Y Nara and K Kaneko ldquoStudy of subcritical crack growth inandesite using the Double Torsion testrdquo International Journal ofRock Mechanics and Mining Sciences vol 42 no 4 pp 521ndash5302005

[13] M Haggerty Q Lin and J F Labuz ldquoObserving deformationand fracture of rock with speckle patternsrdquo Rock Mechanics andRock Engineering vol 43 pp 417ndash426 2010

Mathematical Problems in Engineering 11

[14] C Dascalu B Francois and O Keita ldquoA two-scale model forsubcritical damage propagationrdquo International Journal of Solidsand Structures vol 47 no 3-4 pp 493ndash502 2010

[15] SMisra ldquoDeformation localization at the tips of shear fracturesan analytical approachrdquo Tectonophysics vol 503 no 1-2 pp182ndash187 2011

[16] M F Ashby and S D Hallam ldquoThe failure of brittle solidscontaining small cracks under compressive stress statesrdquo ActaMetallurgica vol 34 no 3 pp 497ndash510 1986

[17] C Fairhurst and N G W Cood ldquoThe phenomenon of rocksplitting parallel to the direction of maximum compression inthe neighbourhood of a surfacerdquo in Proceedings of the Congressof the International Society of RockMechanics L Muller Ed pp687ndash692 Balkema Rotterdam The Netherlands 1966

[18] H P Yuan P Cao and W Z Xu ldquoMechanism study onsubcritical crack growth of flabby and intricate ore rockrdquoTransactions of Nonferrous Metals Society of China vol 16 no3 pp 723ndash727 2006

[19] E Sahouryeh A V Dyskin and L N Germanovich ldquoCrackgrowth under biaxial compressionrdquo Engineering FractureMechanics vol 69 no 18 pp 2187ndash2198 2002

[20] S-H Zheng Research on Coupling Theory Between Seepageand Damage of Fractured Rock Mass and Its Application toEngineering Wuhan Institute of Rock and Soil MechanicsChinese Academy of Scinces Wuhan China 2000

[21] T Liu P Cao and H Lin ldquoAnalytical solutions for the seepageinduced fracture failure propagation in rock massrdquo Journal ofDisaster Advances vol 3 supplement 1 pp 307ndash314 2013

[22] L N Y Wong and H H Einstein ldquoUsing high speed videoimaging in the study of cracking processes in rockrdquoGeotechnicalTesting Journal vol 32 no 2 pp 164ndash180 2009

[23] W Zhu and Q Zhang ldquoBrittle elastic fracture damage constitu-tive model of jointed rockmass and its application to engineer-ingrdquoChinese Journal of RockMechanics and Engineering vol 18no 3 pp 245ndash249 1999

[24] K Y Lam and S P Phua ldquoMultiple crack interaction and itseffect on stress intensity factorrdquoEngineering FractureMechanicsvol 40 no 3 pp 585ndash592 1991

[25] M Kachanov ldquoOn the problems of crack interactions and crackcoalescencerdquo International Journal of Fracture vol 120 no 3pp 537ndash543 2003

[26] H Qing and W Yang ldquoCharacterization of strongly interactedmultiple cracks in an infinite platerdquo Theoretical and AppliedFracture Mechanics vol 46 no 3 pp 209ndash216 2006

[27] A P S Selvadurai and Q Yu ldquoMechanics of a discontinuity in ageomaterialrdquo Computers and Geotechnics vol 32 no 2 pp 92ndash106 2005

[28] X P Yuan H Y Liu and Z Q Wang ldquoAn interacting crack-mechanics basedmodel for elastoplastic damagemodel of rock-likematerials under compressionrdquo International Journal of RockMechanics and Mining Sciences vol 58 pp 92ndash102 2013

[29] B Franois and C Dascalu ldquoA two-scale time-dependentdamage model based on non-planar growth of micro-cracksrdquoJournal of the Mechanics and Physics of Solids vol 58 no 11 pp1928ndash1946 2010

[30] V Palchik andYHHatzor ldquoCrack damage stress as a compositefunction of porosity and elasticmatrix stiffness in dolomites andlimestonesrdquo Engineering Geology vol 63 no 3-4 pp 233ndash2452002

[31] G Alfano S Marfia and E Sacco ldquoA cohesive damage-friction interface model accounting for water pressure on crack

propagationrdquo Computer Methods in Applied Mechanics andEngineering vol 196 no 1ndash3 pp 192ndash209 2006

[32] V Monchiet C Gruescu O Cazacu and D Kondo ldquoAmicromechanical approach of crack-induced damage inorthotropic media application to a brittle matrix compositerdquoEngineering Fracture Mechanics vol 83 pp 40ndash53 2012

[33] A Golshani Y Okui M Oda and T Takemura ldquoA microme-chanical model for brittle failure of rock and its relation tocrack growth observed in triaxial compression tests of graniterdquoMechanics of Materials vol 38 no 4 pp 287ndash303 2006

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

Considering the damage to the rock bridge caused by thewing crack interaction the stress intensity factor at the wingcrack tip is composed of (4) and (9) as follows

119870119868= 119870119868+ 1198701015840

119868= 3120591119899119890radic119886119897119905119910

120587

sinminus1 ( 1

119897119905119910

) sin 120579 cos 1205792

minus

1

2

[(1205901+ 1205903) + (120590

1minus 1205903) cos2 (120579 + 120573)]radic120587119897

+ 119901radic120587119897 +

119886 (1205901minus 1205903) sin2120595 sin120595 + 2119901119897

119873minus12

119860minus 2 (119897 + 119886 cos120595)

radic120587119897

(10)

From (10) it is clear to see that the interaction of multiplewing cracks in rock bridge damage makes the stress intensityfactor at the crack tip larger than that at a single wing crack

The process of rock fracture is the one of the increasingdamage and inherent fracture in rock and is the couplingdamage result of the microdamage and macrodamage Thereare many ways to define the damage variable such as thecracks quantity crack length crack area and crack densityHere the cracks quantity is adopted to define119863

0for the initial

damage of the rockmass and119863 for the damagewhen thewingcrack expands to 119897

1198630= 120587(119886 cos120595)2119873

119860

119863 = 120587(119897 + 119886 cos120595)2119873119860

(11)

Substituting (11) into (8) the fellow equation can be easilyobtained

1205901015840

3= minus

(120591119899119890tan120595 + 2119901119897) (119863

0120587)12

119886 (1 minus 2(119863120587)12)

(12)

Combining (10) (11) and (12) the relation curvesbetween the damage variables 119863 and dimensionless stressintensity factor (119870

1198681205901radic120587119886) at the wing crack tip with

different crack density are shown in Figure 3The graph reveals that with the increase of the damage

variables the dimensionless stress intensity factor1198701198681205901radic120587119886

at the crack tip under seepage pressure decreases at thevery beginning but rises gradually when the equivalent cracklength is 05 (119897119886) The graph also illustrates that the sparserof the cracks are the higher of the stress intensity at the wingcrack will be when the rock bridge is connected During thewing crack expansion process the damage variable 119863 variesfrom119863

0to 1 When119863 = 1 and the wing crack stress intensity

factor119870119868⩾ 119870119868119862

at the crack tip the wing crack connects therock bridge cracks and the rock mass loses bearing capacity

Figure 4 shows the relationship between the stress inten-sity factor at the crack tip and the equivalent crack prop-agation length under different seepage pressure It canbe concluded that when the seepage pressure is rela-tively low (119901 = 0MPa) the wing crack expands stably

0 015 03 045 06 075 09 105

036

038

04

042

044

Damage variables (119863)

119873119860 = 100119873119860 = 225

119873119860 = 400

Dim

ensio

nles

s stre

ss in

tens

ity119870119868120590 1(120587119886)

12

Figure 3 The relationship between damage variables and intensityfactor at the crack tip of the branch stress

(120597119870119868120597119871 lt 0) While when the seepage pressure is relatively

high (119901 = 15MPa) the wing crack tends to expand unstablyBesides there is the expansion stage that 120597119870

119868120597119871 gt 0 under

such condition and the growth rate of wing crack stressintensity factor becomes greater with the increase of seepagepressure 119901 It indicates that under high seepage pressure thecompressive-shear rock crack expands at a high speed as longas it cracks namely and the high seepage pressure contributesto unstable expansion

3 Wing Crack Connection Model andDamage Criteria

As the branch crack expands the interaction of the crackresults in the continuous degradation of the macroscopicmechanical properties of the rock mass The wing crackinitiation from microcracks to the completely damage of therock mass is a process of evolution and accumulation of therock mass damage as well as a process of the fracture of theconnection between cracks [24] It has been proved by manyexperimental researches that the gradual damage process ofthe cracks generally has two forms [25 26] (1) the rock bridgeaxial transfixion failure and (2) the tension-shear compoundfailure According to the branch crack propagation evolutionmechanism this paper studies the gradual damage features ofthose damage models

31 The Axial Transfixion Failure Whenever the wing crackextends stably or unstably the tension wing crack connectsthe main crack in another row as shown in Figure 5 Whenthe wing crack reaches the critical length 119897

1119888= 119863 sin120595

1198711119862

= 1198971119888119886 the rock bridge starts to damage in the form

of axial transfixion failure Taking the stress intensity at thecrack tip as the criterion when the wing crack reaches the

Mathematical Problems in Engineering 5

critical length 1198971119888= 119863 sin120595 this paper establishes the failure

criteria

119870119868= 3120591119899119890radic119886119897119905119910

120587

sinminus1 ( 1

119897119905119910

) sin 120579 cos 1205792

minus

1

2

[(1205901+ 1205903) + (120590

1minus 1205903) cos2 (120579 + 120573)]radic120587119897

1119888

+ 119901radic1205871198971119888+

119886 (1205901minus 1205903) sin2120595 sin120595 + 2119901119897

1119888

119873minus12

119860minus 2 (1198971119888+ 119886 cos120595)

radic1205871198971119888

(13)

When 119870119868(1198711119862) ge 119870

119868119862 axial transfixion failure occurs in the

rock bridge

32 The Tension-Shear Compound Failure With the expan-sion of wing cracks under seepage pressure the cut-resistantcapacity of the rock bridge is increasingly weakened Whenthe wing crack expands to a certain degree the rock bridgeat the crack tip between adjacent wing cracks is cut off bythe shear stress consequently the crack is connected in theshear direction [27] The mechanical analyzing diagram forthe element of the rock bridge composite failure is shown inFigure 6 where 119860119861 is 12 the length of the bottom crack 119864119865is 12 the length of the upper crack 119862119863 is the rock bridge119861119862 and 119864119863 are the wing cracks produced by effective sheardriving force of the main crack 119860119861 and 119864119865 120579 is the anglebetween rock bridge and the maximum principal stress and120590119862119863

and 120591119862119863

are the normal stress and shear stress acting onthe rock bridge respectively

According to the element shown in Figure 6 and theprinciple of mechanics balance it could be deduced asfollows

sum119865119909= 0 sum119865

119910= 0

211990331205903minus 2119886 (120591

119899119890sin120595 + 120590

119899119890cos120595)

minus 21199032(120591119862119863

sin 120579 + 120590119862119863119899

cos 120579) minus 2119901119897 = 0

211990311205901+ 2119886 (120591

119899119890cos120595 minus 120590

119899119890sin120595)

+ 21199032(120591119862119863

cos 120579 minus 120590119862119863119899

sin 120579) = 0

(14)

where

1199031= 119886 sin120595 + 1

2

radic1198892+ ℎ2 sin 120579

1199032=

radic1198892+ ℎ2 cos 120579 minus 2119897

2 cos 120579

1199033= 119886 cos120595 + 1

2

radic1198892+ ℎ2 cos 120579

120579 = 120595 minus arctan 119889ℎ

(15)

Then the following equation can be obtained

120591119862119863

=

21198601tan 120579 minus 2119861

1

(radic1198892+ ℎ2 cos 120579 minus 2119897) (1 + tan2120579)

120590119862119863

=

21198601+ 21198611tan 120579

(radic1198892+ ℎ2 cos 120579 minus 2119897) (1 + tan2120579)

(16)

1198601= 11990331205903minus 119886 (120591

119899119890sin120595 + 120590

119899119890cos120595) minus 119901119897

1198611= 11990311205901+ 119886 (120591

119899119890cos120595 minus 120590

119899119890sin120595)

(17)

Assuming that the rock bridge shear damage follows theMohr-Coulomb strength criterion conditions for the damageare

120591119862119863

minus 119888 minus 120590119862119863

tan120593 ⩾ 0 (18)

Substituting (16) into (18) the fellow equation can beeasily obtained as follows

(

1

2

radic1198892+ ℎ2 cos 120579 minus 119897) 119888 tan2 120579 + (119861

1tan120593 minus 119860

1) tan 120579

+ 1198601tan120593 + 119861 + 1

2

119888radic1198892+ ℎ2 cos 120579 minus 119888119897 ⩽ 0

(19)

When angle 120579 meets the rock bridge shear failure condi-tion and the wing crack propagation length reaches criticalvalue 119897

2119888 the angle 120579 between the rock bridge and 120590

1also

reaches their critical value and then

120579119888= arctan

1198601minus 1198611tan120593 + radicΔ

(radic1198892+ ℎ2 cos 120579 minus 2119897

2119888) 119888

(20)

As the geometric relations show in Figure 6 it is easy tosee that

tan 120579119888=

radic1198892+ ℎ2 sin 120579

(radic1198892+ ℎ2 cos 120579 minus 2119897

2119888)

(21)

Combined (20) and (21) the critical length of the wingcrack can be defined as

1198972119888=

1198602minus radic119860

2

2minus 4 (119888 + 119901 tan120593) 119861

2

2 (119888 + 119901 tan120593)

(22)

where

1198602= 11990331205903tan120593 minus 119886 (120591

119899119890sin120595 + 120590

119899119890cos120595) tan120593 + 119861

1

+

1

2

radic1198892+ ℎ2[(2119888 minus 119901 tan120593) cos 120579 + 119901 sin 120579]

1198612=

1

4

(1198892+ ℎ2) 119888 +

1

2

radic1198892+ ℎ2

times[(11990331205903minus 119886120591119899119890sin120595 minus 119886120590

119899119890cos120595) (cos 120579 tan120593 minus sin 120579)

+1198611(sin 120579 tan120593 + cos 120579)]

(23)

6 Mathematical Problems in Engineering

0 12 24 36 48 6 72 84 960

015

03

045

06

075

09

105

Equivalent crack length 119871(119897119886)

119901 = 0119901 = 6MPa119901 = 15MPa

Dim

ensio

nles

s stre

ss in

tens

ity fa

ctor

119870119868120590 1(120587119886)

12

Figure 4 The relationship between intensity factor at the crack tipwing stress and crack propagation length

Meanwhile this paper gets the stress intensity factor at thecrack tip when the wing reaches the critical length as follows

119870119868(1198972119888) = 119870

119868+ 1198701015840

119868= 3120591eff

radic119886119897119905119910

120587

sinminus1 ( 1

119897119905119910

) sin (120579) cos (120579)

minus

1

2

[(1205901+ 1205903) + (120590

1minus 1205903) cos2 (120579 + 120573)]radic120587119897

2119888

+ 119901radic1205871198972119888+

119886 (1205901minus 1205903) sin2120595 sin120595 + 2119901119897

2119888

2 (1198972119888+ 119886 cos120595) minus 119873minus12

119860

radic1205871198972119888

(24)

4 The Damage Evolution Mechanism ofMulticrack Rock Masses

The macroscopic mechanical effect of fractured rock wouldbe reflected by the change of its flexibility and then thedamage tensor 119863 can be determined by the elastic flexibilitytensor1198620 and the equivalent damage flexibility tensor119862119889 [28]

119863 = 119868 minus

(119862119889)

minus1

1198620

(25)

where 119868 is the fourth-order unit tensor and119863 is a tensor of thefourth order for the 3D anisotropy mode of fractured rock

In terms of the complete rock the elastic flexibility tensor1198620 can be expressed as [29]

1198620

119894119895119896119897 =

1 + V0

1198640

120575119894119896120575119895119897minus

V0

1198640

120575119894119895120575119896119897 (26)

where1198640and V0represented the elasticmodulus andPoissonrsquos

ratio of the rock respectivelyThe equivalent damage flexibility tensor119862119889 can be drawn

from Betti energy reciprocal theorem Hence based on self-consistent method and strain energy equivalence in solid

mechanics the equivalent damage flexibility tensor 119862119889119894119895119896119897 isacquired through calculating the equivalent elastic strainenergy

41The Equivalent Damage Flexibility Tensor Based on Equiv-alent Elastic Strain Energy The flexibility matrix of the crackin compression shear state is [30]

[1198620] = [

[

119862011 119862011 0

119862021 1198620

220

0 0 1198620

33

]

]

(27)

Based onBettirsquos theorem that isMaxwell-Betti reciprocalwork theorem the work done by one set of forces throughthe displacements produced by another set of forces is equalto the work done by the latter set of forces through thedisplacements produced by the former set of forces

(1)When 120590119909= 120591119909119910= 0 120590

119910= 0

11988212= (2119887120590

119910minus 2119886119901) (119862

221205901015840

1199102119889) = 4 (119887120590

119910minus 119886119901) 120590

101584011988911986222

11988221= (2119887120590

1015840

119910) (120590119910minus 119901)119862

0

22

2119889 + Δ119882119888

(28)

where Δ119882119888= (2119886120590

1015840

119910)(120590119910minus 119901)119862

119899119870119899

Then we get

11988221= 41205901015840

1199101198891198871198620

22

(120590119910minus 119901) +

21198861205901015840

119910(120590119910minus 119901)119862

119899

119870119899

(29)

According to 11988221

= 11988212 whilst the crack size can be

neglected compared to the size of the rock mass

11986222= (1 minus 119877) (119862

0

22

+

119886119862119899

2119870119899119887119889

) (30)

where 119877 = 119901120590119910

(2)When 120590119909= 120590119910= 0 120591119909119910

= 0

11988212= (1205912119889) (119862

3312059110158402119887) = 4119862

331205911015840119887119889

11988221= (12059110158402119889) (119862

331205912119887) + Δ119882

119888= 4120591119862

331205911015840119887119889 + Δ119882

119888

Δ119882119888=

12059110158402119886120591119862119904

119870119904

(31)

Then we obtain the following equation

11988221= 412059110158401205911198891198871198620

33+

21198861205911015840120591119862119904

119870119904

(32)

According to11988221= 11988212 we get

11986233= 1198620

33

+

119886119862119904

2119870119904119887119889

(33)

Thus the parameters of the equivalent damage flexibilitymatrix [119862119889] can be expressed as follows

[119862119889] =

[[[[[

[

0 0 0

0

119886119862119899(1 minus 119877)

2119870119899119887119889

minus 1198771198620

220

0 0

119886119862119904

1198701199042119887119889

]]]]]

]

(34)

Mathematical Problems in Engineering 7

42TheEquivalent Damage Flexibility Tensor Based onCrackrsquosElastic Strain Energy The strain energy of single crack isassumed as [29]

119880(119894)

119889= 2int

119886

0

119866119889Γ = 2

1 minus ]20

1198640

int

119886

0

(1198702

119868+ 1198702

119868119868) 119889119886 (35)

where Γ is the line length of cracksIntegrating over the line length the stress intensity factors

of type 119868 or type 119868119868 for the crack under seepage pressure are

119870(119894)

119868= 120590(119894)

119891

radic120587119886(119894) 119870

(119894)

119868119868= 120591(119894)

119902radic120587119886(119894) (36)

where 120590(119894)119891

is the normal stress of the crack and 120591(119894)

119902is the

tangential shear stress Combining the former equation thenwe get

119880(119894)

119889=

1 minus ]20

1198640

120587119886(119894)2

[120590(119894)2

119891+ 120591(119894)2

119902] (37)

By assuming there are 119899 groups of cracks in the fracturedrock mass the strain energy of the fractured rock mass willbe

119880119889=

119899

sum

119894=1

119880(119894)

119889=

1 minus ]20

1198640

120587

119899

sum

119894=1

119886(119894)2120588(119894)[120590(119894)2

119891+ 120591(119894)2

119902] (38)

Set the equation 119867 = ((1 minus ]20)1198640)120587 119860(119894) = 119886

(119894)2120588(119894) in

which 120588(119894) is the density of cracks then

119880119889= 119867

119899

sum

119894=1

119860(119894)[120590(119894)2

119891+ 120591(119894)2

119902] (39)

Rewrite (39) into the tensor expression

119880119889=

1

2

120590119894119895119862119889

119894119895119896119897120590119896119897= 119867

119899

sum

119894=1

119860(119894)[120590(119894)

119891120590(119894)

119891+ 120591(119894)

119902120591(119894)

119902] (40)

According to the hydrostatic pressure principle andCauchy criterion the remote field stress is denoted as 120590

119894119895 and

the stress matrix on the surface can be defined as [31]

[

[

119879V119909

119879V119910

]

]

= [

120590119909119909minus 119901 120590

119909119910

120590119910119909

120590119909119909minus 119901

] [

]119909

]119910

] (41)

Defining 119899119894119899119894= 1 authors rewrite the equation into the

tensor expression

119879(119894)

119894= (120590(119894)

119894119895minus 120575119894119895119901) 119899(119894)

119895= 120590(119894)

119894119895119899(119894)

119895minus 119901119899(119894)

119894 (42)

Considering the compression transferring coefficient 119862119899

the equivalent normal stress on the surface of the crack canbe expressed as

120590(119894)

119891= 119879(119894)

119894119899(119894)

119895= (1 minus 119862

(119894)

119899) 120590(119894)

119894119895119899(119894)

119895119899(119894)

119894minus 120573(119894)119901

(120590(119894)

119891)119894= (1 minus 119862

119899) 120590(119894)

119895119896119899(119894)

119895119899(119894)

119896119899(119894)

119894minus 120573(119894)119901119899(119894)

119894

(43)

Considering the shear transferring coefficient 119862119904again

the shear stress on the crack surface can be expressed as

(120591(119894)

119891)119894= 119879(119894)

119894minus (120590(119894)

119891)119894= (1 minus 119862

(119894)

119899) 120590(119894)

119895119896119899(119894)

119895(120575119896119897minus 119899(119894)

119896119899(119894)

119894)

(44)

Then we get

(120590(119894)

119891)119894(120590(119894)

119891)119894= ((1 minus 119862

(119894)

119899) 120590(119894)

119895119896119899(119894)

119895119899(119894)

119896119899(119894)

119894minus 120573(119894)119901119899(119894)

119894)

times ((1 minus 119862(119894)

119899) 120590(119894)

119904119905119899(119894)

119904119899(119894)

119905119899(119894)

119894minus 120573(119894)119901119899(119894)

119894)

(45)

Considering the symmetry of 120590119894119895 the following equations

can be deduced

(120590(119894)

119891)119894(120590(119894)

119891)119894= (1 minus 119862

(119894)

119899)

2

120590(119894)

119895119896119899(119894)

119895119899(119894)

119896119899(119894)

119904119899(119894)

119905120590(119894)

119904119905

minus (1 minus 119862(119894)

119899) 120590(119894)

119895119896119899(119894)

119895119899(119894)

119896119901

minus (1 minus 119862(119894)

119899) 120590(119894)

119904119905119899(119894)

119904119899(119894)

119905119901 + 120573(119894)2

1199012

(120591(119894)

119891)119894(120591(119894)

119891)119894= (1 minus 119862

(119894)

119904)

2

120590(119894)

119895119896119899(119894)

119895(120575119896119897minus 119899(119894)

119896119899(119894)

119894) 120590(119894)

119904119905119899(119894)

119904

times (120575119905119894minus 119899(119894)

119894119899(119894)

119894)

= (1 minus 119862(119894)

119904)

2

120590(119894)

119895119896

times (120575119896119897119899(119894)

119895119899(119894)

119904minus 119899(119894)

119895119899(119894)

119896119899(119894)

119904119899(119894)

119905) 120590(119894)

119904119905

(46)

Supposing the proportion coefficient 119877 = 119901120590 120590 is theaverage stress 120590 = (12)120590

119894119894 120590119894119894is the first invariant stress

Then the seepage pressure can be transformed to

119901 = 119901

120590119904119904

120590119904119904

= 120590119904119905120575119904119905

119901

2120590

=

1

2

120590119904119905120575119904119905119877

1199012= 1199012 120590119896119896

120590119896119896

120590119904119904

120590119904119904

= 120590119896119895120575119896119895120590119904119905120575119905119904

119901

2120590

119901

2120590

=

1

4

1205901198961198951205751198961198951205901199041199051205751199051199041198772

(47)

Then we get

(120590(119894)

119891)119894(120590(119894)

119891)119894

= (1 minus 119862(119894)

119899)

2

120590(119894)

119895119896119899(119894)

119895119899(119894)

119896119899(119894)

119904119899(119894)

119905120590(119894)

119904119905

minus

1

2

(1 minus 119862(119894)

119899) 119877120590(119894)

119895119896

times (119899(119894)

119895119899(119894)

119896120590(119894)

119904119905+ 120575119895119896119899(119894)

119904119899(119894)

119905)

+

1

4

(120573(119894)119877)

2

120590(119894)

119895119896120590(119894)

119904119905120590119905119904

8 Mathematical Problems in Engineering

Tension crack

1205901

1205901

12059031205903

ℎ

119863

2119886

120595

2119886

Figure 5 Failure characteristic of crag bridge shearing

119880119889=

1

2

120590119895119896119862119889

119895119896119904119905120590119904119894

= 119867

119899

sum

119894=1

119860(119894)120590(119894)

119895119896[(1 minus 119862

(119894)

119899)

2

119899(119894)

119895119899(119894)

119896119899(119894)

119904119899(119894)

119905

minus

1

2

(1 minus 119862(119894)

119899) 120573(119894)119877120590(119894)

119895119896

times (119899(119894)

119895119899(119894)

119896120590(119894)

119904119905+ 120575119895119896119899(119894)

119904119899(119894)

119905)

+

1

4

(119877120573(119894))

2

120590(119894)

119895119896120590(119894)

119904119905120590119905119904+ (1 minus 119862

(119894)

119904)

2

times (120575119896119905119899(119894)

119895119899(119894)

119904minus 119899(119894)

119895119899(119894)

119896119899(119894)

119904119899(119894)

119905) ] 120590(119894)

119904119905

(48)

Finally it is clear to conclude the additional flexibilitytensor of fractured rock mass under seepage pressure asfollows

119862119889

119894119895119896119897= (1 minus 119862

(119894)

119899)

2

119899(119894)

119895119899(119894)

119896119899(119894)

119904119899(119894)

119905

minus

1

2

(1 minus 119862(119894)

119899) 120573(119894)119877120590(119894)

119895119896(119899(119894)

119895119899(119894)

119896120590(119894)

119904119905+ 120575119895119896119899(119894)

119904119899(119894)

119905)

+

1

4

(119877120573(119894))

2

120590(119894)

119895119896120590(119894)

119904119905120590119905119904

+ (1 minus 119862(119894)

119904)

2

(120575119896119905119899(119894)

119895119899(119894)

119904minus 119899(119894)

119895119899(119894)

119896119899(119894)

119904119899(119894)

119905)

(49)

43The Fracture Damage Evolution Equation The additionalelastic strain energy density 119906119894

119896119911

produced by the wing crackis [32]

119906(119894)

119896119911=

4

1198640

120588(119894)

V 119886(119894)2(

5

2

120590(119894)

eff119871(119894)+

2

radic3

120591(119894)

eff)2

(50)

where 119871(119894) is the equivalent length for the crack

Assuming there are 119899 groups of cracks in the fracturedrock mass the additional elastic strain energy density of thefractured rock 119880

119896119911is

119880119896119911=

119899

sum

119894=1

119906(119894)

119896119911=

1

2

120590119894119895119862119896119911

119894119895119896119897120590119896119897 (51)

Crack extension reduces the stiffness of rock masses andincreases its flexibility Through the derivative of 119880

119896119911with

respect to the stress tensor we can get the additional flexibilitytensor of fractured rock masses in the damage evolutionprocess

120597119880119896119911

120597120590119894119895

= 119862119896119911

119894119895119896119897120590119896119897 (52)

At the same time by calculating the partial derivativeof 119871(119894) 120590(119894)eff and 120591

(119894)

eff with respect to 120590119894119895 respectively and

considering the symmetry of 119862119896119911119894119895119896119897

we can get the followingresults

120597119880119896119911

120597120590119894119895

=

1

1198640

119899

sum

119894=1

119886(119894)2120588(119894)

V [119872

(119894)

1119899(119894)

119895119899(119894)

119896119899(119894)

119896119899(119894)

119905minus119872(119894)

2

times (119899(119894)

119895119899(119894)

119896120575119896119895+ 120575119896119894119899(119894)

119895119899(119894)

119905

+119899(119894)

119895119899(119894)

119896120575119894119895+ 120575119894119905119899(119894)

119895119899(119894)

119896)

minus

119872(119894)

3

3

120573119877120575119894119895120575119896119897]120590119896119897

(53)

Mathematical Problems in Engineering 9

Shear crack

Tension crack

1199031

1199033

120595

119862

119864

119865

119861

119863119897

120591CD

120590ne120591ne

120579

119903 2

119901

119901

1205901

1205903

119897

119886119860

2119886

120590CD119899

Figure 6 Failure characteristic of crag bridge shearing

where

119872(119894)

1= (20120590

(119894)

eff119871(119894)+

16

radic3

120591(119894)

eff)

times [

1

120590(119894)

119899

(

5

2

119871(119894)+

2

radic3

119891)

minus

1

120591(119894)

119899

(

5

2

cot 120579(119894)119871(119894) + 2

radic3

119891) +

5

2

120590(119894)

eff119873(119894)

1]

119872(119894)

2= (5120590

(119894)

eff119871(119894)+

4

radic3

120591(119894)

eff)

times [

1

120591(119894)

119899

(

5

2

cot 120579(119894)119871(119894) + 2

radic3

119891) +

5

2

120590(119894)

eff119873(119894)

2]

119872(119894)

3= (20120590

(119894)

eff119871(119894)+

16

radic3

120591(119894)

eff)

times [

2

radic3

119891 +

5

2

(120590(119894)

eff119873(119894)

3+ 119871(119894))]

119873(119894)

1=radic119871(119894)[minus4119886(119894) cos 120579(119894)120590(119894)2eff (119891120591

(119894)

119899minus 120590(119894)

119899)

minus (119870(119894)2

119868119862minus 4119886(119894) cos 120579(119894)120590(119894)eff

minus 2radic120587119886(119894)119860(119894)119870(119894)

119868119862120590(119894)

eff )

times (120591(119894)

119899minus 120590(119894)

119899cot120579(119894))]

times (2120587119886(119894)119860(119894)120590(119894)

eff120590(119894)

119899120591(119894)

119899)

minus1

119873(119894)

2=radic119871(119894)[minus4119886(119894) cos 120579(119894)120590(119894)2eff

minus (119870(119894)

119868119862

2

minus 4119886(119894) cos 120579(119894)120590(119894)eff120591

(119894)

eff

minus2radic120587119886(119894)119860(119894)119870(119894)

119868119862120590(119894)

eff) cot 120579(119894)]

times (2120587119886(119894)119860(119894)120590(119894)

eff120590(119894)

119899120591(119894)

119899)

minus1

119873(119894)

3

=[

[

minus(minus2(

119870(119894)

119868119862

radic120587119886(119894)

)

2

120590(119894)3

eff minus2

120587

cos 120579(119894)(119891

120590(119894)

eff

minus

120591(119894)

eff

120590(119894)2

eff

))]

]

times

radic119871(119894)

119860(119894)

minus

2119870(119894)

119868119862radic119871(119894)

120590(119894)2

effradic2120587119886

(119894)

119860(119894)= radic

119870(119894)

119868119862

2120590(119894)2

effradic2120587119886

(119894)

minus

2

120587

cos 120579(119894)120591(119894)

eff

120590(119894)2

eff

(54)

Then we can get the flexibility tensor 119862119896119911119894119895119896119897

due to thedamage evolution of fractured rock masses as follows

119862119896119911

119894119895119896119897=

1

1198640

119899

sum

119894=1

119886(119894)2120588(119894)

V [119872(119894)

1119899(119894)

119895119899(119894)

119896119899(119894)

119896119899(119894)

119905minus119872(119894)

2

times (119899(119894)

119895119899(119894)

119896120575119896119895+ 120575119896119894119899(119894)

119895119899(119894)

119905

+119899(119894)

119895119899(119894)

119896120575119894119895+ 120575119894119905119899(119894)

119895119899(119894)

119896)

minus

119872(119894)

3

3

120573119877120575119894119895120575119896119897]

(55)

44 The Constitutive Equation of the Multicrack Rock MassTo sum up combining the initial damage and damage

10 Mathematical Problems in Engineering

evolution tensor we can get the flexibility tensor of thefractured rock mass under seepage pressureas follows

119862119894119895119896119897

= 1198620

119894119895119896119897+ 119862119889

119894119895119896119897+ 119862119896119911

119894119895119896119897 (56)

where1198620119894119895119896119897

is the elastic flexibility tensor of the complete rock119862119889

119894119895119896119897is the initial equivalent damage flexibility tensor of the

fractured rock mass under seepage pressure and 119862119896119911119894119895119896119897

is theadditional damage flexibility tensor with crack propagationof the fractured rock mass under seepage pressure

According to generalized Hookersquos law [33]

120576119894119895= 119862119894119895119896119897120590119896119897

120590119894119895= 119864119894119895119896119897120576119896119897= (119862119894119895119896119897)

minus1

120576119896119897

(57)

and this paper sets up the constitutive relation of the multi-crack rock mass under seepage pressure

5 Conclusions

After discussing the damage and fracture evolution mecha-nism of the multicrack rock mass under compression-shearstress and seepage pressure the following main conclusionsare drawn

(1) Through proposing the fractured damage model ofthe multicrack rock mass under the combined actionof compression-shear stress and seepage pressurethis paper discusses the law of crack initiation andthe evolution law of stress intensity factor at the tipof a wing crack The interaction of multiple cracksmakes the stress intensity factor at the crack tiplarger than that of a single wing crack Regarding theseepage pressure the existence of that reinforces thewing crackrsquos propagation Moreover the high seepagepressure converts wing crack growth from stable formto unstable form

(2) The wing crack initiation from microcracks to thecompletely damage of the rock mass is a processof evolution and accumulation of the rock massdamage as well as a process of the fracture of theconnection between cracks Based on the criterionand mechanism for crack initiation and propagationthis paper puts forward the mechanical model fordifferent fracture transfixion failuremodes of the cragbridge under the combined action of seepage pressureand compression-shear stress

(3) According to the strain energy equivalent principlethis paper applies Bettirsquos reciprocal work theorem ofthe fractured rock mass to study the flexibility tensorof the rock massrsquos initial damage and its damage evo-lution and deduces the damage constitutive equationsfor the elastic-plastic fracture and damage evolutionThe theory provides a reliable theoretical principlefor quantitative research of the fractured rock massfailure under seepage pressure

Acknowledgments

This paper gets its funding from Projects (51174228 5127424941002090) supported by the National Natural ScienceFoundation of China Project supported by the HunanProvincial Innovation Foundation for Postgraduate Project(20120162120014) supported by the Specialized ResearchFund for the Doctoral Program of Higher Education ofChina Project (12KF01) supported by the Open Projects ofState Key Laboratory of Coal Resources and Safe MiningCUMTThe authors wish to acknowledge these supports

References

[1] V Kuksenko N Tomilin and A Chmel ldquoThe rock fractureexperiment with a drive control a spatial aspectrdquo Tectono-physics vol 431 no 1ndash4 pp 123ndash129 2007

[2] S Marfia and E Sacco ldquoA fracture evolution procedure forcohesive materialsrdquo International Journal of Fracture vol 110no 3 pp 241ndash261 2001

[3] L N Lens E Bittencourt and V M R drsquoAvila ldquoConstitutivemodels for cohesive zones in mixed-mode fracture of plainconcreterdquo Engineering Fracture Mechanics vol 76 no 14 pp2281ndash2297 2009

[4] M K Rahman Y A Suarez Z Chen and S S RahmanldquoUnsuccessful hydraulic fracturing cases in Australia investi-gation into causes of failures and their remediesrdquo Journal ofPetroleum Science and Engineering vol 57 no 1-2 pp 70ndash812007

[5] R P Orense S Shimoma K Maeda and I Towhata ldquoInstru-mented model slope failure due to water seepagerdquo Journal ofNatural Disaster Science vol 26 pp 15ndash26 2004

[6] M M Hossain and M K Rahman ldquoNumerical simulationof complex fracture growth during tight reservoir stimulationby hydraulic fracturingrdquo Journal of Petroleum Science andEngineering vol 60 no 2 pp 86ndash104 2008

[7] D Hoxha M Lespinasse J Sausse and F Homand ldquoAmicrostructural study of natural and experimentally inducedcracks in a granodioriterdquo Tectonophysics vol 395 no 1-2 pp99ndash112 2005

[8] M Sagong and A Bobet ldquoCoalescence of multiple flaws ina rock-model material in uniaxial compressionrdquo InternationalJournal of Rock Mechanics and Mining Sciences vol 39 no 2pp 229ndash241 2002

[9] R H C Wong K T Chau C A Tang and P Lin ldquoAnalysisof crack coalescence in rock-like materials containing threeflawsmdashpart I experimental approachrdquo International Journal ofRockMechanics andMining Sciences vol 38 no 7 pp 909ndash9242001

[10] S YWang SW SloanD C Sheng andCA Tang ldquoNumericalanalysis of the failure process around a circular opening in rockrdquoComputers and Geotechnics vol 39 pp 8ndash16 2012

[11] O Mutlu and A Bobet ldquoSlip initiation on frictional fracturesrdquoEngineering FractureMechanics vol 72 no 5 pp 729ndash747 2005

[12] Y Nara and K Kaneko ldquoStudy of subcritical crack growth inandesite using the Double Torsion testrdquo International Journal ofRock Mechanics and Mining Sciences vol 42 no 4 pp 521ndash5302005

[13] M Haggerty Q Lin and J F Labuz ldquoObserving deformationand fracture of rock with speckle patternsrdquo Rock Mechanics andRock Engineering vol 43 pp 417ndash426 2010

Mathematical Problems in Engineering 11

[14] C Dascalu B Francois and O Keita ldquoA two-scale model forsubcritical damage propagationrdquo International Journal of Solidsand Structures vol 47 no 3-4 pp 493ndash502 2010

[15] SMisra ldquoDeformation localization at the tips of shear fracturesan analytical approachrdquo Tectonophysics vol 503 no 1-2 pp182ndash187 2011

[16] M F Ashby and S D Hallam ldquoThe failure of brittle solidscontaining small cracks under compressive stress statesrdquo ActaMetallurgica vol 34 no 3 pp 497ndash510 1986

[17] C Fairhurst and N G W Cood ldquoThe phenomenon of rocksplitting parallel to the direction of maximum compression inthe neighbourhood of a surfacerdquo in Proceedings of the Congressof the International Society of RockMechanics L Muller Ed pp687ndash692 Balkema Rotterdam The Netherlands 1966

[18] H P Yuan P Cao and W Z Xu ldquoMechanism study onsubcritical crack growth of flabby and intricate ore rockrdquoTransactions of Nonferrous Metals Society of China vol 16 no3 pp 723ndash727 2006

[19] E Sahouryeh A V Dyskin and L N Germanovich ldquoCrackgrowth under biaxial compressionrdquo Engineering FractureMechanics vol 69 no 18 pp 2187ndash2198 2002

[20] S-H Zheng Research on Coupling Theory Between Seepageand Damage of Fractured Rock Mass and Its Application toEngineering Wuhan Institute of Rock and Soil MechanicsChinese Academy of Scinces Wuhan China 2000

[21] T Liu P Cao and H Lin ldquoAnalytical solutions for the seepageinduced fracture failure propagation in rock massrdquo Journal ofDisaster Advances vol 3 supplement 1 pp 307ndash314 2013

[22] L N Y Wong and H H Einstein ldquoUsing high speed videoimaging in the study of cracking processes in rockrdquoGeotechnicalTesting Journal vol 32 no 2 pp 164ndash180 2009

[23] W Zhu and Q Zhang ldquoBrittle elastic fracture damage constitu-tive model of jointed rockmass and its application to engineer-ingrdquoChinese Journal of RockMechanics and Engineering vol 18no 3 pp 245ndash249 1999

[24] K Y Lam and S P Phua ldquoMultiple crack interaction and itseffect on stress intensity factorrdquoEngineering FractureMechanicsvol 40 no 3 pp 585ndash592 1991

[25] M Kachanov ldquoOn the problems of crack interactions and crackcoalescencerdquo International Journal of Fracture vol 120 no 3pp 537ndash543 2003

[26] H Qing and W Yang ldquoCharacterization of strongly interactedmultiple cracks in an infinite platerdquo Theoretical and AppliedFracture Mechanics vol 46 no 3 pp 209ndash216 2006

[27] A P S Selvadurai and Q Yu ldquoMechanics of a discontinuity in ageomaterialrdquo Computers and Geotechnics vol 32 no 2 pp 92ndash106 2005

[28] X P Yuan H Y Liu and Z Q Wang ldquoAn interacting crack-mechanics basedmodel for elastoplastic damagemodel of rock-likematerials under compressionrdquo International Journal of RockMechanics and Mining Sciences vol 58 pp 92ndash102 2013

[29] B Franois and C Dascalu ldquoA two-scale time-dependentdamage model based on non-planar growth of micro-cracksrdquoJournal of the Mechanics and Physics of Solids vol 58 no 11 pp1928ndash1946 2010

[30] V Palchik andYHHatzor ldquoCrack damage stress as a compositefunction of porosity and elasticmatrix stiffness in dolomites andlimestonesrdquo Engineering Geology vol 63 no 3-4 pp 233ndash2452002

[31] G Alfano S Marfia and E Sacco ldquoA cohesive damage-friction interface model accounting for water pressure on crack

propagationrdquo Computer Methods in Applied Mechanics andEngineering vol 196 no 1ndash3 pp 192ndash209 2006

[32] V Monchiet C Gruescu O Cazacu and D Kondo ldquoAmicromechanical approach of crack-induced damage inorthotropic media application to a brittle matrix compositerdquoEngineering Fracture Mechanics vol 83 pp 40ndash53 2012

[33] A Golshani Y Okui M Oda and T Takemura ldquoA microme-chanical model for brittle failure of rock and its relation tocrack growth observed in triaxial compression tests of graniterdquoMechanics of Materials vol 38 no 4 pp 287ndash303 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

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Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

critical length 1198971119888= 119863 sin120595 this paper establishes the failure

criteria

119870119868= 3120591119899119890radic119886119897119905119910

120587

sinminus1 ( 1

119897119905119910

) sin 120579 cos 1205792

minus

1

2

[(1205901+ 1205903) + (120590

1minus 1205903) cos2 (120579 + 120573)]radic120587119897

1119888

+ 119901radic1205871198971119888+

119886 (1205901minus 1205903) sin2120595 sin120595 + 2119901119897

1119888

119873minus12

119860minus 2 (1198971119888+ 119886 cos120595)

radic1205871198971119888

(13)

When 119870119868(1198711119862) ge 119870

119868119862 axial transfixion failure occurs in the

rock bridge

32 The Tension-Shear Compound Failure With the expan-sion of wing cracks under seepage pressure the cut-resistantcapacity of the rock bridge is increasingly weakened Whenthe wing crack expands to a certain degree the rock bridgeat the crack tip between adjacent wing cracks is cut off bythe shear stress consequently the crack is connected in theshear direction [27] The mechanical analyzing diagram forthe element of the rock bridge composite failure is shown inFigure 6 where 119860119861 is 12 the length of the bottom crack 119864119865is 12 the length of the upper crack 119862119863 is the rock bridge119861119862 and 119864119863 are the wing cracks produced by effective sheardriving force of the main crack 119860119861 and 119864119865 120579 is the anglebetween rock bridge and the maximum principal stress and120590119862119863

and 120591119862119863

are the normal stress and shear stress acting onthe rock bridge respectively

According to the element shown in Figure 6 and theprinciple of mechanics balance it could be deduced asfollows

sum119865119909= 0 sum119865

119910= 0

211990331205903minus 2119886 (120591

119899119890sin120595 + 120590

119899119890cos120595)

minus 21199032(120591119862119863

sin 120579 + 120590119862119863119899

cos 120579) minus 2119901119897 = 0

211990311205901+ 2119886 (120591

119899119890cos120595 minus 120590

119899119890sin120595)

+ 21199032(120591119862119863

cos 120579 minus 120590119862119863119899

sin 120579) = 0

(14)

where

1199031= 119886 sin120595 + 1

2

radic1198892+ ℎ2 sin 120579

1199032=

radic1198892+ ℎ2 cos 120579 minus 2119897

2 cos 120579

1199033= 119886 cos120595 + 1

2

radic1198892+ ℎ2 cos 120579

120579 = 120595 minus arctan 119889ℎ

(15)

Then the following equation can be obtained

120591119862119863

=

21198601tan 120579 minus 2119861

1

(radic1198892+ ℎ2 cos 120579 minus 2119897) (1 + tan2120579)

120590119862119863

=

21198601+ 21198611tan 120579

(radic1198892+ ℎ2 cos 120579 minus 2119897) (1 + tan2120579)

(16)

1198601= 11990331205903minus 119886 (120591

119899119890sin120595 + 120590

119899119890cos120595) minus 119901119897

1198611= 11990311205901+ 119886 (120591

119899119890cos120595 minus 120590

119899119890sin120595)

(17)

Assuming that the rock bridge shear damage follows theMohr-Coulomb strength criterion conditions for the damageare

120591119862119863

minus 119888 minus 120590119862119863

tan120593 ⩾ 0 (18)

Substituting (16) into (18) the fellow equation can beeasily obtained as follows

(

1

2

radic1198892+ ℎ2 cos 120579 minus 119897) 119888 tan2 120579 + (119861

1tan120593 minus 119860

1) tan 120579

+ 1198601tan120593 + 119861 + 1

2

119888radic1198892+ ℎ2 cos 120579 minus 119888119897 ⩽ 0

(19)

When angle 120579 meets the rock bridge shear failure condi-tion and the wing crack propagation length reaches criticalvalue 119897

2119888 the angle 120579 between the rock bridge and 120590

1also

reaches their critical value and then

120579119888= arctan

1198601minus 1198611tan120593 + radicΔ

(radic1198892+ ℎ2 cos 120579 minus 2119897

2119888) 119888

(20)

As the geometric relations show in Figure 6 it is easy tosee that

tan 120579119888=

radic1198892+ ℎ2 sin 120579

(radic1198892+ ℎ2 cos 120579 minus 2119897

2119888)

(21)

Combined (20) and (21) the critical length of the wingcrack can be defined as

1198972119888=

1198602minus radic119860

2

2minus 4 (119888 + 119901 tan120593) 119861

2

2 (119888 + 119901 tan120593)

(22)

where

1198602= 11990331205903tan120593 minus 119886 (120591

119899119890sin120595 + 120590

119899119890cos120595) tan120593 + 119861

1

+

1

2

radic1198892+ ℎ2[(2119888 minus 119901 tan120593) cos 120579 + 119901 sin 120579]

1198612=

1

4

(1198892+ ℎ2) 119888 +

1

2

radic1198892+ ℎ2

times[(11990331205903minus 119886120591119899119890sin120595 minus 119886120590

119899119890cos120595) (cos 120579 tan120593 minus sin 120579)

+1198611(sin 120579 tan120593 + cos 120579)]

(23)

6 Mathematical Problems in Engineering

0 12 24 36 48 6 72 84 960

015

03

045

06

075

09

105

Equivalent crack length 119871(119897119886)

119901 = 0119901 = 6MPa119901 = 15MPa

Dim

ensio

nles

s stre

ss in

tens

ity fa

ctor

119870119868120590 1(120587119886)

12

Figure 4 The relationship between intensity factor at the crack tipwing stress and crack propagation length

Meanwhile this paper gets the stress intensity factor at thecrack tip when the wing reaches the critical length as follows

119870119868(1198972119888) = 119870

119868+ 1198701015840

119868= 3120591eff

radic119886119897119905119910

120587

sinminus1 ( 1

119897119905119910

) sin (120579) cos (120579)

minus

1

2

[(1205901+ 1205903) + (120590

1minus 1205903) cos2 (120579 + 120573)]radic120587119897

2119888

+ 119901radic1205871198972119888+

119886 (1205901minus 1205903) sin2120595 sin120595 + 2119901119897

2119888

2 (1198972119888+ 119886 cos120595) minus 119873minus12

119860

radic1205871198972119888

(24)

4 The Damage Evolution Mechanism ofMulticrack Rock Masses

The macroscopic mechanical effect of fractured rock wouldbe reflected by the change of its flexibility and then thedamage tensor 119863 can be determined by the elastic flexibilitytensor1198620 and the equivalent damage flexibility tensor119862119889 [28]

119863 = 119868 minus

(119862119889)

minus1

1198620

(25)

where 119868 is the fourth-order unit tensor and119863 is a tensor of thefourth order for the 3D anisotropy mode of fractured rock

In terms of the complete rock the elastic flexibility tensor1198620 can be expressed as [29]

1198620

119894119895119896119897 =

1 + V0

1198640

120575119894119896120575119895119897minus

V0

1198640

120575119894119895120575119896119897 (26)

where1198640and V0represented the elasticmodulus andPoissonrsquos

ratio of the rock respectivelyThe equivalent damage flexibility tensor119862119889 can be drawn

from Betti energy reciprocal theorem Hence based on self-consistent method and strain energy equivalence in solid

mechanics the equivalent damage flexibility tensor 119862119889119894119895119896119897 isacquired through calculating the equivalent elastic strainenergy

41The Equivalent Damage Flexibility Tensor Based on Equiv-alent Elastic Strain Energy The flexibility matrix of the crackin compression shear state is [30]

[1198620] = [

[

119862011 119862011 0

119862021 1198620

220

0 0 1198620

33

]

]

(27)

Based onBettirsquos theorem that isMaxwell-Betti reciprocalwork theorem the work done by one set of forces throughthe displacements produced by another set of forces is equalto the work done by the latter set of forces through thedisplacements produced by the former set of forces

(1)When 120590119909= 120591119909119910= 0 120590

119910= 0

11988212= (2119887120590

119910minus 2119886119901) (119862

221205901015840

1199102119889) = 4 (119887120590

119910minus 119886119901) 120590

101584011988911986222

11988221= (2119887120590

1015840

119910) (120590119910minus 119901)119862

0

22

2119889 + Δ119882119888

(28)

where Δ119882119888= (2119886120590

1015840

119910)(120590119910minus 119901)119862

119899119870119899

Then we get

11988221= 41205901015840

1199101198891198871198620

22

(120590119910minus 119901) +

21198861205901015840

119910(120590119910minus 119901)119862

119899

119870119899

(29)

According to 11988221

= 11988212 whilst the crack size can be

neglected compared to the size of the rock mass

11986222= (1 minus 119877) (119862

0

22

+

119886119862119899

2119870119899119887119889

) (30)

where 119877 = 119901120590119910

(2)When 120590119909= 120590119910= 0 120591119909119910

= 0

11988212= (1205912119889) (119862

3312059110158402119887) = 4119862

331205911015840119887119889

11988221= (12059110158402119889) (119862

331205912119887) + Δ119882

119888= 4120591119862

331205911015840119887119889 + Δ119882

119888

Δ119882119888=

12059110158402119886120591119862119904

119870119904

(31)

Then we obtain the following equation

11988221= 412059110158401205911198891198871198620

33+

21198861205911015840120591119862119904

119870119904

(32)

According to11988221= 11988212 we get

11986233= 1198620

33

+

119886119862119904

2119870119904119887119889

(33)

Thus the parameters of the equivalent damage flexibilitymatrix [119862119889] can be expressed as follows

[119862119889] =

[[[[[

[

0 0 0

0

119886119862119899(1 minus 119877)

2119870119899119887119889

minus 1198771198620

220

0 0

119886119862119904

1198701199042119887119889

]]]]]

]

(34)

Mathematical Problems in Engineering 7

42TheEquivalent Damage Flexibility Tensor Based onCrackrsquosElastic Strain Energy The strain energy of single crack isassumed as [29]

119880(119894)

119889= 2int

119886

0

119866119889Γ = 2

1 minus ]20

1198640

int

119886

0

(1198702

119868+ 1198702

119868119868) 119889119886 (35)

where Γ is the line length of cracksIntegrating over the line length the stress intensity factors

of type 119868 or type 119868119868 for the crack under seepage pressure are

119870(119894)

119868= 120590(119894)

119891

radic120587119886(119894) 119870

(119894)

119868119868= 120591(119894)

119902radic120587119886(119894) (36)

where 120590(119894)119891

is the normal stress of the crack and 120591(119894)

119902is the

tangential shear stress Combining the former equation thenwe get

119880(119894)

119889=

1 minus ]20

1198640

120587119886(119894)2

[120590(119894)2

119891+ 120591(119894)2

119902] (37)

By assuming there are 119899 groups of cracks in the fracturedrock mass the strain energy of the fractured rock mass willbe

119880119889=

119899

sum

119894=1

119880(119894)

119889=

1 minus ]20

1198640

120587

119899

sum

119894=1

119886(119894)2120588(119894)[120590(119894)2

119891+ 120591(119894)2

119902] (38)

Set the equation 119867 = ((1 minus ]20)1198640)120587 119860(119894) = 119886

(119894)2120588(119894) in

which 120588(119894) is the density of cracks then

119880119889= 119867

119899

sum

119894=1

119860(119894)[120590(119894)2

119891+ 120591(119894)2

119902] (39)

Rewrite (39) into the tensor expression

119880119889=

1

2

120590119894119895119862119889

119894119895119896119897120590119896119897= 119867

119899

sum

119894=1

119860(119894)[120590(119894)

119891120590(119894)

119891+ 120591(119894)

119902120591(119894)

119902] (40)

According to the hydrostatic pressure principle andCauchy criterion the remote field stress is denoted as 120590

119894119895 and

the stress matrix on the surface can be defined as [31]

[

[

119879V119909

119879V119910

]

]

= [

120590119909119909minus 119901 120590

119909119910

120590119910119909

120590119909119909minus 119901

] [

]119909

]119910

] (41)

Defining 119899119894119899119894= 1 authors rewrite the equation into the

tensor expression

119879(119894)

119894= (120590(119894)

119894119895minus 120575119894119895119901) 119899(119894)

119895= 120590(119894)

119894119895119899(119894)

119895minus 119901119899(119894)

119894 (42)

Considering the compression transferring coefficient 119862119899

the equivalent normal stress on the surface of the crack canbe expressed as

120590(119894)

119891= 119879(119894)

119894119899(119894)

119895= (1 minus 119862

(119894)

119899) 120590(119894)

119894119895119899(119894)

119895119899(119894)

119894minus 120573(119894)119901

(120590(119894)

119891)119894= (1 minus 119862

119899) 120590(119894)

119895119896119899(119894)

119895119899(119894)

119896119899(119894)

119894minus 120573(119894)119901119899(119894)

119894

(43)

Considering the shear transferring coefficient 119862119904again

the shear stress on the crack surface can be expressed as

(120591(119894)

119891)119894= 119879(119894)

119894minus (120590(119894)

119891)119894= (1 minus 119862

(119894)

119899) 120590(119894)

119895119896119899(119894)

119895(120575119896119897minus 119899(119894)

119896119899(119894)

119894)

(44)

Then we get

(120590(119894)

119891)119894(120590(119894)

119891)119894= ((1 minus 119862

(119894)

119899) 120590(119894)

119895119896119899(119894)

119895119899(119894)

119896119899(119894)

119894minus 120573(119894)119901119899(119894)

119894)

times ((1 minus 119862(119894)

119899) 120590(119894)

119904119905119899(119894)

119904119899(119894)

119905119899(119894)

119894minus 120573(119894)119901119899(119894)

119894)

(45)

Considering the symmetry of 120590119894119895 the following equations

can be deduced

(120590(119894)

119891)119894(120590(119894)

119891)119894= (1 minus 119862

(119894)

119899)

2

120590(119894)

119895119896119899(119894)

119895119899(119894)

119896119899(119894)

119904119899(119894)

119905120590(119894)

119904119905

minus (1 minus 119862(119894)

119899) 120590(119894)

119895119896119899(119894)

119895119899(119894)

119896119901

minus (1 minus 119862(119894)

119899) 120590(119894)

119904119905119899(119894)

119904119899(119894)

119905119901 + 120573(119894)2

1199012

(120591(119894)

119891)119894(120591(119894)

119891)119894= (1 minus 119862

(119894)

119904)

2

120590(119894)

119895119896119899(119894)

119895(120575119896119897minus 119899(119894)

119896119899(119894)

119894) 120590(119894)

119904119905119899(119894)

119904

times (120575119905119894minus 119899(119894)

119894119899(119894)

119894)

= (1 minus 119862(119894)

119904)

2

120590(119894)

119895119896

times (120575119896119897119899(119894)

119895119899(119894)

119904minus 119899(119894)

119895119899(119894)

119896119899(119894)

119904119899(119894)

119905) 120590(119894)

119904119905

(46)

Supposing the proportion coefficient 119877 = 119901120590 120590 is theaverage stress 120590 = (12)120590

119894119894 120590119894119894is the first invariant stress

Then the seepage pressure can be transformed to

119901 = 119901

120590119904119904

120590119904119904

= 120590119904119905120575119904119905

119901

2120590

=

1

2

120590119904119905120575119904119905119877

1199012= 1199012 120590119896119896

120590119896119896

120590119904119904

120590119904119904

= 120590119896119895120575119896119895120590119904119905120575119905119904

119901

2120590

119901

2120590

=

1

4

1205901198961198951205751198961198951205901199041199051205751199051199041198772

(47)

Then we get

(120590(119894)

119891)119894(120590(119894)

119891)119894

= (1 minus 119862(119894)

119899)

2

120590(119894)

119895119896119899(119894)

119895119899(119894)

119896119899(119894)

119904119899(119894)

119905120590(119894)

119904119905

minus

1

2

(1 minus 119862(119894)

119899) 119877120590(119894)

119895119896

times (119899(119894)

119895119899(119894)

119896120590(119894)

119904119905+ 120575119895119896119899(119894)

119904119899(119894)

119905)

+

1

4

(120573(119894)119877)

2

120590(119894)

119895119896120590(119894)

119904119905120590119905119904

8 Mathematical Problems in Engineering

Tension crack

1205901

1205901

12059031205903

ℎ

119863

2119886

120595

2119886

Figure 5 Failure characteristic of crag bridge shearing

119880119889=

1

2

120590119895119896119862119889

119895119896119904119905120590119904119894

= 119867

119899

sum

119894=1

119860(119894)120590(119894)

119895119896[(1 minus 119862

(119894)

119899)

2

119899(119894)

119895119899(119894)

119896119899(119894)

119904119899(119894)

119905

minus

1

2

(1 minus 119862(119894)

119899) 120573(119894)119877120590(119894)

119895119896

times (119899(119894)

119895119899(119894)

119896120590(119894)

119904119905+ 120575119895119896119899(119894)

119904119899(119894)

119905)

+

1

4

(119877120573(119894))

2

120590(119894)

119895119896120590(119894)

119904119905120590119905119904+ (1 minus 119862

(119894)

119904)

2

times (120575119896119905119899(119894)

119895119899(119894)

119904minus 119899(119894)

119895119899(119894)

119896119899(119894)

119904119899(119894)

119905) ] 120590(119894)

119904119905

(48)

Finally it is clear to conclude the additional flexibilitytensor of fractured rock mass under seepage pressure asfollows

119862119889

119894119895119896119897= (1 minus 119862

(119894)

119899)

2

119899(119894)

119895119899(119894)

119896119899(119894)

119904119899(119894)

119905

minus

1

2

(1 minus 119862(119894)

119899) 120573(119894)119877120590(119894)

119895119896(119899(119894)

119895119899(119894)

119896120590(119894)

119904119905+ 120575119895119896119899(119894)

119904119899(119894)

119905)

+

1

4

(119877120573(119894))

2

120590(119894)

119895119896120590(119894)

119904119905120590119905119904

+ (1 minus 119862(119894)

119904)

2

(120575119896119905119899(119894)

119895119899(119894)

119904minus 119899(119894)

119895119899(119894)

119896119899(119894)

119904119899(119894)

119905)

(49)

43The Fracture Damage Evolution Equation The additionalelastic strain energy density 119906119894

119896119911

produced by the wing crackis [32]

119906(119894)

119896119911=

4

1198640

120588(119894)

V 119886(119894)2(

5

2

120590(119894)

eff119871(119894)+

2

radic3

120591(119894)

eff)2

(50)

where 119871(119894) is the equivalent length for the crack

Assuming there are 119899 groups of cracks in the fracturedrock mass the additional elastic strain energy density of thefractured rock 119880

119896119911is

119880119896119911=

119899

sum

119894=1

119906(119894)

119896119911=

1

2

120590119894119895119862119896119911

119894119895119896119897120590119896119897 (51)

Crack extension reduces the stiffness of rock masses andincreases its flexibility Through the derivative of 119880

119896119911with

respect to the stress tensor we can get the additional flexibilitytensor of fractured rock masses in the damage evolutionprocess

120597119880119896119911

120597120590119894119895

= 119862119896119911

119894119895119896119897120590119896119897 (52)

At the same time by calculating the partial derivativeof 119871(119894) 120590(119894)eff and 120591

(119894)

eff with respect to 120590119894119895 respectively and

considering the symmetry of 119862119896119911119894119895119896119897

we can get the followingresults

120597119880119896119911

120597120590119894119895

=

1

1198640

119899

sum

119894=1

119886(119894)2120588(119894)

V [119872

(119894)

1119899(119894)

119895119899(119894)

119896119899(119894)

119896119899(119894)

119905minus119872(119894)

2

times (119899(119894)

119895119899(119894)

119896120575119896119895+ 120575119896119894119899(119894)

119895119899(119894)

119905

+119899(119894)

119895119899(119894)

119896120575119894119895+ 120575119894119905119899(119894)

119895119899(119894)

119896)

minus

119872(119894)

3

3

120573119877120575119894119895120575119896119897]120590119896119897

(53)

Mathematical Problems in Engineering 9

Shear crack

Tension crack

1199031

1199033

120595

119862

119864

119865

119861

119863119897

120591CD

120590ne120591ne

120579

119903 2

119901

119901

1205901

1205903

119897

119886119860

2119886

120590CD119899

Figure 6 Failure characteristic of crag bridge shearing

where

119872(119894)

1= (20120590

(119894)

eff119871(119894)+

16

radic3

120591(119894)

eff)

times [

1

120590(119894)

119899

(

5

2

119871(119894)+

2

radic3

119891)

minus

1

120591(119894)

119899

(

5

2

cot 120579(119894)119871(119894) + 2

radic3

119891) +

5

2

120590(119894)

eff119873(119894)

1]

119872(119894)

2= (5120590

(119894)

eff119871(119894)+

4

radic3

120591(119894)

eff)

times [

1

120591(119894)

119899

(

5

2

cot 120579(119894)119871(119894) + 2

radic3

119891) +

5

2

120590(119894)

eff119873(119894)

2]

119872(119894)

3= (20120590

(119894)

eff119871(119894)+

16

radic3

120591(119894)

eff)

times [

2

radic3

119891 +

5

2

(120590(119894)

eff119873(119894)

3+ 119871(119894))]

119873(119894)

1=radic119871(119894)[minus4119886(119894) cos 120579(119894)120590(119894)2eff (119891120591

(119894)

119899minus 120590(119894)

119899)

minus (119870(119894)2

119868119862minus 4119886(119894) cos 120579(119894)120590(119894)eff

minus 2radic120587119886(119894)119860(119894)119870(119894)

119868119862120590(119894)

eff )

times (120591(119894)

119899minus 120590(119894)

119899cot120579(119894))]

times (2120587119886(119894)119860(119894)120590(119894)

eff120590(119894)

119899120591(119894)

119899)

minus1

119873(119894)

2=radic119871(119894)[minus4119886(119894) cos 120579(119894)120590(119894)2eff

minus (119870(119894)

119868119862

2

minus 4119886(119894) cos 120579(119894)120590(119894)eff120591

(119894)

eff

minus2radic120587119886(119894)119860(119894)119870(119894)

119868119862120590(119894)

eff) cot 120579(119894)]

times (2120587119886(119894)119860(119894)120590(119894)

eff120590(119894)

119899120591(119894)

119899)

minus1

119873(119894)

3

=[

[

minus(minus2(

119870(119894)

119868119862

radic120587119886(119894)

)

2

120590(119894)3

eff minus2

120587

cos 120579(119894)(119891

120590(119894)

eff

minus

120591(119894)

eff

120590(119894)2

eff

))]

]

times

radic119871(119894)

119860(119894)

minus

2119870(119894)

119868119862radic119871(119894)

120590(119894)2

effradic2120587119886

(119894)

119860(119894)= radic

119870(119894)

119868119862

2120590(119894)2

effradic2120587119886

(119894)

minus

2

120587

cos 120579(119894)120591(119894)

eff

120590(119894)2

eff

(54)

Then we can get the flexibility tensor 119862119896119911119894119895119896119897

due to thedamage evolution of fractured rock masses as follows

119862119896119911

119894119895119896119897=

1

1198640

119899

sum

119894=1

119886(119894)2120588(119894)

V [119872(119894)

1119899(119894)

119895119899(119894)

119896119899(119894)

119896119899(119894)

119905minus119872(119894)

2

times (119899(119894)

119895119899(119894)

119896120575119896119895+ 120575119896119894119899(119894)

119895119899(119894)

119905

+119899(119894)

119895119899(119894)

119896120575119894119895+ 120575119894119905119899(119894)

119895119899(119894)

119896)

minus

119872(119894)

3

3

120573119877120575119894119895120575119896119897]

(55)

44 The Constitutive Equation of the Multicrack Rock MassTo sum up combining the initial damage and damage

10 Mathematical Problems in Engineering

evolution tensor we can get the flexibility tensor of thefractured rock mass under seepage pressureas follows

119862119894119895119896119897

= 1198620

119894119895119896119897+ 119862119889

119894119895119896119897+ 119862119896119911

119894119895119896119897 (56)

where1198620119894119895119896119897

is the elastic flexibility tensor of the complete rock119862119889

119894119895119896119897is the initial equivalent damage flexibility tensor of the

fractured rock mass under seepage pressure and 119862119896119911119894119895119896119897

is theadditional damage flexibility tensor with crack propagationof the fractured rock mass under seepage pressure

According to generalized Hookersquos law [33]

120576119894119895= 119862119894119895119896119897120590119896119897

120590119894119895= 119864119894119895119896119897120576119896119897= (119862119894119895119896119897)

minus1

120576119896119897

(57)

and this paper sets up the constitutive relation of the multi-crack rock mass under seepage pressure

5 Conclusions

After discussing the damage and fracture evolution mecha-nism of the multicrack rock mass under compression-shearstress and seepage pressure the following main conclusionsare drawn

(1) Through proposing the fractured damage model ofthe multicrack rock mass under the combined actionof compression-shear stress and seepage pressurethis paper discusses the law of crack initiation andthe evolution law of stress intensity factor at the tipof a wing crack The interaction of multiple cracksmakes the stress intensity factor at the crack tiplarger than that of a single wing crack Regarding theseepage pressure the existence of that reinforces thewing crackrsquos propagation Moreover the high seepagepressure converts wing crack growth from stable formto unstable form

(2) The wing crack initiation from microcracks to thecompletely damage of the rock mass is a processof evolution and accumulation of the rock massdamage as well as a process of the fracture of theconnection between cracks Based on the criterionand mechanism for crack initiation and propagationthis paper puts forward the mechanical model fordifferent fracture transfixion failuremodes of the cragbridge under the combined action of seepage pressureand compression-shear stress

(3) According to the strain energy equivalent principlethis paper applies Bettirsquos reciprocal work theorem ofthe fractured rock mass to study the flexibility tensorof the rock massrsquos initial damage and its damage evo-lution and deduces the damage constitutive equationsfor the elastic-plastic fracture and damage evolutionThe theory provides a reliable theoretical principlefor quantitative research of the fractured rock massfailure under seepage pressure

Acknowledgments

This paper gets its funding from Projects (51174228 5127424941002090) supported by the National Natural ScienceFoundation of China Project supported by the HunanProvincial Innovation Foundation for Postgraduate Project(20120162120014) supported by the Specialized ResearchFund for the Doctoral Program of Higher Education ofChina Project (12KF01) supported by the Open Projects ofState Key Laboratory of Coal Resources and Safe MiningCUMTThe authors wish to acknowledge these supports

References

[1] V Kuksenko N Tomilin and A Chmel ldquoThe rock fractureexperiment with a drive control a spatial aspectrdquo Tectono-physics vol 431 no 1ndash4 pp 123ndash129 2007

[2] S Marfia and E Sacco ldquoA fracture evolution procedure forcohesive materialsrdquo International Journal of Fracture vol 110no 3 pp 241ndash261 2001

[3] L N Lens E Bittencourt and V M R drsquoAvila ldquoConstitutivemodels for cohesive zones in mixed-mode fracture of plainconcreterdquo Engineering Fracture Mechanics vol 76 no 14 pp2281ndash2297 2009

[4] M K Rahman Y A Suarez Z Chen and S S RahmanldquoUnsuccessful hydraulic fracturing cases in Australia investi-gation into causes of failures and their remediesrdquo Journal ofPetroleum Science and Engineering vol 57 no 1-2 pp 70ndash812007

[5] R P Orense S Shimoma K Maeda and I Towhata ldquoInstru-mented model slope failure due to water seepagerdquo Journal ofNatural Disaster Science vol 26 pp 15ndash26 2004

[6] M M Hossain and M K Rahman ldquoNumerical simulationof complex fracture growth during tight reservoir stimulationby hydraulic fracturingrdquo Journal of Petroleum Science andEngineering vol 60 no 2 pp 86ndash104 2008

[7] D Hoxha M Lespinasse J Sausse and F Homand ldquoAmicrostructural study of natural and experimentally inducedcracks in a granodioriterdquo Tectonophysics vol 395 no 1-2 pp99ndash112 2005

[8] M Sagong and A Bobet ldquoCoalescence of multiple flaws ina rock-model material in uniaxial compressionrdquo InternationalJournal of Rock Mechanics and Mining Sciences vol 39 no 2pp 229ndash241 2002

[9] R H C Wong K T Chau C A Tang and P Lin ldquoAnalysisof crack coalescence in rock-like materials containing threeflawsmdashpart I experimental approachrdquo International Journal ofRockMechanics andMining Sciences vol 38 no 7 pp 909ndash9242001

[10] S YWang SW SloanD C Sheng andCA Tang ldquoNumericalanalysis of the failure process around a circular opening in rockrdquoComputers and Geotechnics vol 39 pp 8ndash16 2012

[11] O Mutlu and A Bobet ldquoSlip initiation on frictional fracturesrdquoEngineering FractureMechanics vol 72 no 5 pp 729ndash747 2005

[12] Y Nara and K Kaneko ldquoStudy of subcritical crack growth inandesite using the Double Torsion testrdquo International Journal ofRock Mechanics and Mining Sciences vol 42 no 4 pp 521ndash5302005

[13] M Haggerty Q Lin and J F Labuz ldquoObserving deformationand fracture of rock with speckle patternsrdquo Rock Mechanics andRock Engineering vol 43 pp 417ndash426 2010

Mathematical Problems in Engineering 11

[14] C Dascalu B Francois and O Keita ldquoA two-scale model forsubcritical damage propagationrdquo International Journal of Solidsand Structures vol 47 no 3-4 pp 493ndash502 2010

[15] SMisra ldquoDeformation localization at the tips of shear fracturesan analytical approachrdquo Tectonophysics vol 503 no 1-2 pp182ndash187 2011

[16] M F Ashby and S D Hallam ldquoThe failure of brittle solidscontaining small cracks under compressive stress statesrdquo ActaMetallurgica vol 34 no 3 pp 497ndash510 1986

[17] C Fairhurst and N G W Cood ldquoThe phenomenon of rocksplitting parallel to the direction of maximum compression inthe neighbourhood of a surfacerdquo in Proceedings of the Congressof the International Society of RockMechanics L Muller Ed pp687ndash692 Balkema Rotterdam The Netherlands 1966

[18] H P Yuan P Cao and W Z Xu ldquoMechanism study onsubcritical crack growth of flabby and intricate ore rockrdquoTransactions of Nonferrous Metals Society of China vol 16 no3 pp 723ndash727 2006

[19] E Sahouryeh A V Dyskin and L N Germanovich ldquoCrackgrowth under biaxial compressionrdquo Engineering FractureMechanics vol 69 no 18 pp 2187ndash2198 2002

[20] S-H Zheng Research on Coupling Theory Between Seepageand Damage of Fractured Rock Mass and Its Application toEngineering Wuhan Institute of Rock and Soil MechanicsChinese Academy of Scinces Wuhan China 2000

[21] T Liu P Cao and H Lin ldquoAnalytical solutions for the seepageinduced fracture failure propagation in rock massrdquo Journal ofDisaster Advances vol 3 supplement 1 pp 307ndash314 2013

[22] L N Y Wong and H H Einstein ldquoUsing high speed videoimaging in the study of cracking processes in rockrdquoGeotechnicalTesting Journal vol 32 no 2 pp 164ndash180 2009

[23] W Zhu and Q Zhang ldquoBrittle elastic fracture damage constitu-tive model of jointed rockmass and its application to engineer-ingrdquoChinese Journal of RockMechanics and Engineering vol 18no 3 pp 245ndash249 1999

[24] K Y Lam and S P Phua ldquoMultiple crack interaction and itseffect on stress intensity factorrdquoEngineering FractureMechanicsvol 40 no 3 pp 585ndash592 1991

[25] M Kachanov ldquoOn the problems of crack interactions and crackcoalescencerdquo International Journal of Fracture vol 120 no 3pp 537ndash543 2003

[26] H Qing and W Yang ldquoCharacterization of strongly interactedmultiple cracks in an infinite platerdquo Theoretical and AppliedFracture Mechanics vol 46 no 3 pp 209ndash216 2006

[27] A P S Selvadurai and Q Yu ldquoMechanics of a discontinuity in ageomaterialrdquo Computers and Geotechnics vol 32 no 2 pp 92ndash106 2005

[28] X P Yuan H Y Liu and Z Q Wang ldquoAn interacting crack-mechanics basedmodel for elastoplastic damagemodel of rock-likematerials under compressionrdquo International Journal of RockMechanics and Mining Sciences vol 58 pp 92ndash102 2013

[29] B Franois and C Dascalu ldquoA two-scale time-dependentdamage model based on non-planar growth of micro-cracksrdquoJournal of the Mechanics and Physics of Solids vol 58 no 11 pp1928ndash1946 2010

[30] V Palchik andYHHatzor ldquoCrack damage stress as a compositefunction of porosity and elasticmatrix stiffness in dolomites andlimestonesrdquo Engineering Geology vol 63 no 3-4 pp 233ndash2452002

[31] G Alfano S Marfia and E Sacco ldquoA cohesive damage-friction interface model accounting for water pressure on crack

propagationrdquo Computer Methods in Applied Mechanics andEngineering vol 196 no 1ndash3 pp 192ndash209 2006

[32] V Monchiet C Gruescu O Cazacu and D Kondo ldquoAmicromechanical approach of crack-induced damage inorthotropic media application to a brittle matrix compositerdquoEngineering Fracture Mechanics vol 83 pp 40ndash53 2012

[33] A Golshani Y Okui M Oda and T Takemura ldquoA microme-chanical model for brittle failure of rock and its relation tocrack growth observed in triaxial compression tests of graniterdquoMechanics of Materials vol 38 no 4 pp 287ndash303 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

0 12 24 36 48 6 72 84 960

015

03

045

06

075

09

105

Equivalent crack length 119871(119897119886)

119901 = 0119901 = 6MPa119901 = 15MPa

Dim

ensio

nles

s stre

ss in

tens

ity fa

ctor

119870119868120590 1(120587119886)

12

Figure 4 The relationship between intensity factor at the crack tipwing stress and crack propagation length

Meanwhile this paper gets the stress intensity factor at thecrack tip when the wing reaches the critical length as follows

119870119868(1198972119888) = 119870

119868+ 1198701015840

119868= 3120591eff

radic119886119897119905119910

120587

sinminus1 ( 1

119897119905119910

) sin (120579) cos (120579)

minus

1

2

[(1205901+ 1205903) + (120590

1minus 1205903) cos2 (120579 + 120573)]radic120587119897

2119888

+ 119901radic1205871198972119888+

119886 (1205901minus 1205903) sin2120595 sin120595 + 2119901119897

2119888

2 (1198972119888+ 119886 cos120595) minus 119873minus12

119860

radic1205871198972119888

(24)

4 The Damage Evolution Mechanism ofMulticrack Rock Masses

The macroscopic mechanical effect of fractured rock wouldbe reflected by the change of its flexibility and then thedamage tensor 119863 can be determined by the elastic flexibilitytensor1198620 and the equivalent damage flexibility tensor119862119889 [28]

119863 = 119868 minus

(119862119889)

minus1

1198620

(25)

where 119868 is the fourth-order unit tensor and119863 is a tensor of thefourth order for the 3D anisotropy mode of fractured rock

In terms of the complete rock the elastic flexibility tensor1198620 can be expressed as [29]

1198620

119894119895119896119897 =

1 + V0

1198640

120575119894119896120575119895119897minus

V0

1198640

120575119894119895120575119896119897 (26)

where1198640and V0represented the elasticmodulus andPoissonrsquos

ratio of the rock respectivelyThe equivalent damage flexibility tensor119862119889 can be drawn

from Betti energy reciprocal theorem Hence based on self-consistent method and strain energy equivalence in solid

mechanics the equivalent damage flexibility tensor 119862119889119894119895119896119897 isacquired through calculating the equivalent elastic strainenergy

41The Equivalent Damage Flexibility Tensor Based on Equiv-alent Elastic Strain Energy The flexibility matrix of the crackin compression shear state is [30]

[1198620] = [

[

119862011 119862011 0

119862021 1198620

220

0 0 1198620

33

]

]

(27)

Based onBettirsquos theorem that isMaxwell-Betti reciprocalwork theorem the work done by one set of forces throughthe displacements produced by another set of forces is equalto the work done by the latter set of forces through thedisplacements produced by the former set of forces

(1)When 120590119909= 120591119909119910= 0 120590

119910= 0

11988212= (2119887120590

119910minus 2119886119901) (119862

221205901015840

1199102119889) = 4 (119887120590

119910minus 119886119901) 120590

101584011988911986222

11988221= (2119887120590

1015840

119910) (120590119910minus 119901)119862

0

22

2119889 + Δ119882119888

(28)

where Δ119882119888= (2119886120590

1015840

119910)(120590119910minus 119901)119862

119899119870119899

Then we get

11988221= 41205901015840

1199101198891198871198620

22

(120590119910minus 119901) +

21198861205901015840

119910(120590119910minus 119901)119862

119899

119870119899

(29)

According to 11988221

= 11988212 whilst the crack size can be

neglected compared to the size of the rock mass

11986222= (1 minus 119877) (119862

0

22

+

119886119862119899

2119870119899119887119889

) (30)

where 119877 = 119901120590119910

(2)When 120590119909= 120590119910= 0 120591119909119910

= 0

11988212= (1205912119889) (119862

3312059110158402119887) = 4119862

331205911015840119887119889

11988221= (12059110158402119889) (119862

331205912119887) + Δ119882

119888= 4120591119862

331205911015840119887119889 + Δ119882

119888

Δ119882119888=

12059110158402119886120591119862119904

119870119904

(31)

Then we obtain the following equation

11988221= 412059110158401205911198891198871198620

33+

21198861205911015840120591119862119904

119870119904

(32)

According to11988221= 11988212 we get

11986233= 1198620

33

+

119886119862119904

2119870119904119887119889

(33)

Thus the parameters of the equivalent damage flexibilitymatrix [119862119889] can be expressed as follows

[119862119889] =

[[[[[

[

0 0 0

0

119886119862119899(1 minus 119877)

2119870119899119887119889

minus 1198771198620

220

0 0

119886119862119904

1198701199042119887119889

]]]]]

]

(34)

Mathematical Problems in Engineering 7

42TheEquivalent Damage Flexibility Tensor Based onCrackrsquosElastic Strain Energy The strain energy of single crack isassumed as [29]

119880(119894)

119889= 2int

119886

0

119866119889Γ = 2

1 minus ]20

1198640

int

119886

0

(1198702

119868+ 1198702

119868119868) 119889119886 (35)

where Γ is the line length of cracksIntegrating over the line length the stress intensity factors

of type 119868 or type 119868119868 for the crack under seepage pressure are

119870(119894)

119868= 120590(119894)

119891

radic120587119886(119894) 119870

(119894)

119868119868= 120591(119894)

119902radic120587119886(119894) (36)

where 120590(119894)119891

is the normal stress of the crack and 120591(119894)

119902is the

tangential shear stress Combining the former equation thenwe get

119880(119894)

119889=

1 minus ]20

1198640

120587119886(119894)2

[120590(119894)2

119891+ 120591(119894)2

119902] (37)

By assuming there are 119899 groups of cracks in the fracturedrock mass the strain energy of the fractured rock mass willbe

119880119889=

119899

sum

119894=1

119880(119894)

119889=

1 minus ]20

1198640

120587

119899

sum

119894=1

119886(119894)2120588(119894)[120590(119894)2

119891+ 120591(119894)2

119902] (38)

Set the equation 119867 = ((1 minus ]20)1198640)120587 119860(119894) = 119886

(119894)2120588(119894) in

which 120588(119894) is the density of cracks then

119880119889= 119867

119899

sum

119894=1

119860(119894)[120590(119894)2

119891+ 120591(119894)2

119902] (39)

Rewrite (39) into the tensor expression

119880119889=

1

2

120590119894119895119862119889

119894119895119896119897120590119896119897= 119867

119899

sum

119894=1

119860(119894)[120590(119894)

119891120590(119894)

119891+ 120591(119894)

119902120591(119894)

119902] (40)

According to the hydrostatic pressure principle andCauchy criterion the remote field stress is denoted as 120590

119894119895 and

the stress matrix on the surface can be defined as [31]

[

[

119879V119909

119879V119910

]

]

= [

120590119909119909minus 119901 120590

119909119910

120590119910119909

120590119909119909minus 119901

] [

]119909

]119910

] (41)

Defining 119899119894119899119894= 1 authors rewrite the equation into the

tensor expression

119879(119894)

119894= (120590(119894)

119894119895minus 120575119894119895119901) 119899(119894)

119895= 120590(119894)

119894119895119899(119894)

119895minus 119901119899(119894)

119894 (42)

Considering the compression transferring coefficient 119862119899

the equivalent normal stress on the surface of the crack canbe expressed as

120590(119894)

119891= 119879(119894)

119894119899(119894)

119895= (1 minus 119862

(119894)

119899) 120590(119894)

119894119895119899(119894)

119895119899(119894)

119894minus 120573(119894)119901

(120590(119894)

119891)119894= (1 minus 119862

119899) 120590(119894)

119895119896119899(119894)

119895119899(119894)

119896119899(119894)

119894minus 120573(119894)119901119899(119894)

119894

(43)

Considering the shear transferring coefficient 119862119904again

the shear stress on the crack surface can be expressed as

(120591(119894)

119891)119894= 119879(119894)

119894minus (120590(119894)

119891)119894= (1 minus 119862

(119894)

119899) 120590(119894)

119895119896119899(119894)

119895(120575119896119897minus 119899(119894)

119896119899(119894)

119894)

(44)

Then we get

(120590(119894)

119891)119894(120590(119894)

119891)119894= ((1 minus 119862

(119894)

119899) 120590(119894)

119895119896119899(119894)

119895119899(119894)

119896119899(119894)

119894minus 120573(119894)119901119899(119894)

119894)

times ((1 minus 119862(119894)

119899) 120590(119894)

119904119905119899(119894)

119904119899(119894)

119905119899(119894)

119894minus 120573(119894)119901119899(119894)

119894)

(45)

Considering the symmetry of 120590119894119895 the following equations

can be deduced

(120590(119894)

119891)119894(120590(119894)

119891)119894= (1 minus 119862

(119894)

119899)

2

120590(119894)

119895119896119899(119894)

119895119899(119894)

119896119899(119894)

119904119899(119894)

119905120590(119894)

119904119905

minus (1 minus 119862(119894)

119899) 120590(119894)

119895119896119899(119894)

119895119899(119894)

119896119901

minus (1 minus 119862(119894)

119899) 120590(119894)

119904119905119899(119894)

119904119899(119894)

119905119901 + 120573(119894)2

1199012

(120591(119894)

119891)119894(120591(119894)

119891)119894= (1 minus 119862

(119894)

119904)

2

120590(119894)

119895119896119899(119894)

119895(120575119896119897minus 119899(119894)

119896119899(119894)

119894) 120590(119894)

119904119905119899(119894)

119904

times (120575119905119894minus 119899(119894)

119894119899(119894)

119894)

= (1 minus 119862(119894)

119904)

2

120590(119894)

119895119896

times (120575119896119897119899(119894)

119895119899(119894)

119904minus 119899(119894)

119895119899(119894)

119896119899(119894)

119904119899(119894)

119905) 120590(119894)

119904119905

(46)

Supposing the proportion coefficient 119877 = 119901120590 120590 is theaverage stress 120590 = (12)120590

119894119894 120590119894119894is the first invariant stress

Then the seepage pressure can be transformed to

119901 = 119901

120590119904119904

120590119904119904

= 120590119904119905120575119904119905

119901

2120590

=

1

2

120590119904119905120575119904119905119877

1199012= 1199012 120590119896119896

120590119896119896

120590119904119904

120590119904119904

= 120590119896119895120575119896119895120590119904119905120575119905119904

119901

2120590

119901

2120590

=

1

4

1205901198961198951205751198961198951205901199041199051205751199051199041198772

(47)

Then we get

(120590(119894)

119891)119894(120590(119894)

119891)119894

= (1 minus 119862(119894)

119899)

2

120590(119894)

119895119896119899(119894)

119895119899(119894)

119896119899(119894)

119904119899(119894)

119905120590(119894)

119904119905

minus

1

2

(1 minus 119862(119894)

119899) 119877120590(119894)

119895119896

times (119899(119894)

119895119899(119894)

119896120590(119894)

119904119905+ 120575119895119896119899(119894)

119904119899(119894)

119905)

+

1

4

(120573(119894)119877)

2

120590(119894)

119895119896120590(119894)

119904119905120590119905119904

8 Mathematical Problems in Engineering

Tension crack

1205901

1205901

12059031205903

ℎ

119863

2119886

120595

2119886

Figure 5 Failure characteristic of crag bridge shearing

119880119889=

1

2

120590119895119896119862119889

119895119896119904119905120590119904119894

= 119867

119899

sum

119894=1

119860(119894)120590(119894)

119895119896[(1 minus 119862

(119894)

119899)

2

119899(119894)

119895119899(119894)

119896119899(119894)

119904119899(119894)

119905

minus

1

2

(1 minus 119862(119894)

119899) 120573(119894)119877120590(119894)

119895119896

times (119899(119894)

119895119899(119894)

119896120590(119894)

119904119905+ 120575119895119896119899(119894)

119904119899(119894)

119905)

+

1

4

(119877120573(119894))

2

120590(119894)

119895119896120590(119894)

119904119905120590119905119904+ (1 minus 119862

(119894)

119904)

2

times (120575119896119905119899(119894)

119895119899(119894)

119904minus 119899(119894)

119895119899(119894)

119896119899(119894)

119904119899(119894)

119905) ] 120590(119894)

119904119905

(48)

Finally it is clear to conclude the additional flexibilitytensor of fractured rock mass under seepage pressure asfollows

119862119889

119894119895119896119897= (1 minus 119862

(119894)

119899)

2

119899(119894)

119895119899(119894)

119896119899(119894)

119904119899(119894)

119905

minus

1

2

(1 minus 119862(119894)

119899) 120573(119894)119877120590(119894)

119895119896(119899(119894)

119895119899(119894)

119896120590(119894)

119904119905+ 120575119895119896119899(119894)

119904119899(119894)

119905)

+

1

4

(119877120573(119894))

2

120590(119894)

119895119896120590(119894)

119904119905120590119905119904

+ (1 minus 119862(119894)

119904)

2

(120575119896119905119899(119894)

119895119899(119894)

119904minus 119899(119894)

119895119899(119894)

119896119899(119894)

119904119899(119894)

119905)

(49)

43The Fracture Damage Evolution Equation The additionalelastic strain energy density 119906119894

119896119911

produced by the wing crackis [32]

119906(119894)

119896119911=

4

1198640

120588(119894)

V 119886(119894)2(

5

2

120590(119894)

eff119871(119894)+

2

radic3

120591(119894)

eff)2

(50)

where 119871(119894) is the equivalent length for the crack

Assuming there are 119899 groups of cracks in the fracturedrock mass the additional elastic strain energy density of thefractured rock 119880

119896119911is

119880119896119911=

119899

sum

119894=1

119906(119894)

119896119911=

1

2

120590119894119895119862119896119911

119894119895119896119897120590119896119897 (51)

Crack extension reduces the stiffness of rock masses andincreases its flexibility Through the derivative of 119880

119896119911with

respect to the stress tensor we can get the additional flexibilitytensor of fractured rock masses in the damage evolutionprocess

120597119880119896119911

120597120590119894119895

= 119862119896119911

119894119895119896119897120590119896119897 (52)

At the same time by calculating the partial derivativeof 119871(119894) 120590(119894)eff and 120591

(119894)

eff with respect to 120590119894119895 respectively and

considering the symmetry of 119862119896119911119894119895119896119897

we can get the followingresults

120597119880119896119911

120597120590119894119895

=

1

1198640

119899

sum

119894=1

119886(119894)2120588(119894)

V [119872

(119894)

1119899(119894)

119895119899(119894)

119896119899(119894)

119896119899(119894)

119905minus119872(119894)

2

times (119899(119894)

119895119899(119894)

119896120575119896119895+ 120575119896119894119899(119894)

119895119899(119894)

119905

+119899(119894)

119895119899(119894)

119896120575119894119895+ 120575119894119905119899(119894)

119895119899(119894)

119896)

minus

119872(119894)

3

3

120573119877120575119894119895120575119896119897]120590119896119897

(53)

Mathematical Problems in Engineering 9

Shear crack

Tension crack

1199031

1199033

120595

119862

119864

119865

119861

119863119897

120591CD

120590ne120591ne

120579

119903 2

119901

119901

1205901

1205903

119897

119886119860

2119886

120590CD119899

Figure 6 Failure characteristic of crag bridge shearing

where

119872(119894)

1= (20120590

(119894)

eff119871(119894)+

16

radic3

120591(119894)

eff)

times [

1

120590(119894)

119899

(

5

2

119871(119894)+

2

radic3

119891)

minus

1

120591(119894)

119899

(

5

2

cot 120579(119894)119871(119894) + 2

radic3

119891) +

5

2

120590(119894)

eff119873(119894)

1]

119872(119894)

2= (5120590

(119894)

eff119871(119894)+

4

radic3

120591(119894)

eff)

times [

1

120591(119894)

119899

(

5

2

cot 120579(119894)119871(119894) + 2

radic3

119891) +

5

2

120590(119894)

eff119873(119894)

2]

119872(119894)

3= (20120590

(119894)

eff119871(119894)+

16

radic3

120591(119894)

eff)

times [

2

radic3

119891 +

5

2

(120590(119894)

eff119873(119894)

3+ 119871(119894))]

119873(119894)

1=radic119871(119894)[minus4119886(119894) cos 120579(119894)120590(119894)2eff (119891120591

(119894)

119899minus 120590(119894)

119899)

minus (119870(119894)2

119868119862minus 4119886(119894) cos 120579(119894)120590(119894)eff

minus 2radic120587119886(119894)119860(119894)119870(119894)

119868119862120590(119894)

eff )

times (120591(119894)

119899minus 120590(119894)

119899cot120579(119894))]

times (2120587119886(119894)119860(119894)120590(119894)

eff120590(119894)

119899120591(119894)

119899)

minus1

119873(119894)

2=radic119871(119894)[minus4119886(119894) cos 120579(119894)120590(119894)2eff

minus (119870(119894)

119868119862

2

minus 4119886(119894) cos 120579(119894)120590(119894)eff120591

(119894)

eff

minus2radic120587119886(119894)119860(119894)119870(119894)

119868119862120590(119894)

eff) cot 120579(119894)]

times (2120587119886(119894)119860(119894)120590(119894)

eff120590(119894)

119899120591(119894)

119899)

minus1

119873(119894)

3

=[

[

minus(minus2(

119870(119894)

119868119862

radic120587119886(119894)

)

2

120590(119894)3

eff minus2

120587

cos 120579(119894)(119891

120590(119894)

eff

minus

120591(119894)

eff

120590(119894)2

eff

))]

]

times

radic119871(119894)

119860(119894)

minus

2119870(119894)

119868119862radic119871(119894)

120590(119894)2

effradic2120587119886

(119894)

119860(119894)= radic

119870(119894)

119868119862

2120590(119894)2

effradic2120587119886

(119894)

minus

2

120587

cos 120579(119894)120591(119894)

eff

120590(119894)2

eff

(54)

Then we can get the flexibility tensor 119862119896119911119894119895119896119897

due to thedamage evolution of fractured rock masses as follows

119862119896119911

119894119895119896119897=

1

1198640

119899

sum

119894=1

119886(119894)2120588(119894)

V [119872(119894)

1119899(119894)

119895119899(119894)

119896119899(119894)

119896119899(119894)

119905minus119872(119894)

2

times (119899(119894)

119895119899(119894)

119896120575119896119895+ 120575119896119894119899(119894)

119895119899(119894)

119905

+119899(119894)

119895119899(119894)

119896120575119894119895+ 120575119894119905119899(119894)

119895119899(119894)

119896)

minus

119872(119894)

3

3

120573119877120575119894119895120575119896119897]

(55)

44 The Constitutive Equation of the Multicrack Rock MassTo sum up combining the initial damage and damage

10 Mathematical Problems in Engineering

evolution tensor we can get the flexibility tensor of thefractured rock mass under seepage pressureas follows

119862119894119895119896119897

= 1198620

119894119895119896119897+ 119862119889

119894119895119896119897+ 119862119896119911

119894119895119896119897 (56)

where1198620119894119895119896119897

is the elastic flexibility tensor of the complete rock119862119889

119894119895119896119897is the initial equivalent damage flexibility tensor of the

fractured rock mass under seepage pressure and 119862119896119911119894119895119896119897

is theadditional damage flexibility tensor with crack propagationof the fractured rock mass under seepage pressure

According to generalized Hookersquos law [33]

120576119894119895= 119862119894119895119896119897120590119896119897

120590119894119895= 119864119894119895119896119897120576119896119897= (119862119894119895119896119897)

minus1

120576119896119897

(57)

and this paper sets up the constitutive relation of the multi-crack rock mass under seepage pressure

5 Conclusions

After discussing the damage and fracture evolution mecha-nism of the multicrack rock mass under compression-shearstress and seepage pressure the following main conclusionsare drawn

(1) Through proposing the fractured damage model ofthe multicrack rock mass under the combined actionof compression-shear stress and seepage pressurethis paper discusses the law of crack initiation andthe evolution law of stress intensity factor at the tipof a wing crack The interaction of multiple cracksmakes the stress intensity factor at the crack tiplarger than that of a single wing crack Regarding theseepage pressure the existence of that reinforces thewing crackrsquos propagation Moreover the high seepagepressure converts wing crack growth from stable formto unstable form

(2) The wing crack initiation from microcracks to thecompletely damage of the rock mass is a processof evolution and accumulation of the rock massdamage as well as a process of the fracture of theconnection between cracks Based on the criterionand mechanism for crack initiation and propagationthis paper puts forward the mechanical model fordifferent fracture transfixion failuremodes of the cragbridge under the combined action of seepage pressureand compression-shear stress

(3) According to the strain energy equivalent principlethis paper applies Bettirsquos reciprocal work theorem ofthe fractured rock mass to study the flexibility tensorof the rock massrsquos initial damage and its damage evo-lution and deduces the damage constitutive equationsfor the elastic-plastic fracture and damage evolutionThe theory provides a reliable theoretical principlefor quantitative research of the fractured rock massfailure under seepage pressure

Acknowledgments

This paper gets its funding from Projects (51174228 5127424941002090) supported by the National Natural ScienceFoundation of China Project supported by the HunanProvincial Innovation Foundation for Postgraduate Project(20120162120014) supported by the Specialized ResearchFund for the Doctoral Program of Higher Education ofChina Project (12KF01) supported by the Open Projects ofState Key Laboratory of Coal Resources and Safe MiningCUMTThe authors wish to acknowledge these supports

References

[1] V Kuksenko N Tomilin and A Chmel ldquoThe rock fractureexperiment with a drive control a spatial aspectrdquo Tectono-physics vol 431 no 1ndash4 pp 123ndash129 2007

[2] S Marfia and E Sacco ldquoA fracture evolution procedure forcohesive materialsrdquo International Journal of Fracture vol 110no 3 pp 241ndash261 2001

[3] L N Lens E Bittencourt and V M R drsquoAvila ldquoConstitutivemodels for cohesive zones in mixed-mode fracture of plainconcreterdquo Engineering Fracture Mechanics vol 76 no 14 pp2281ndash2297 2009

[4] M K Rahman Y A Suarez Z Chen and S S RahmanldquoUnsuccessful hydraulic fracturing cases in Australia investi-gation into causes of failures and their remediesrdquo Journal ofPetroleum Science and Engineering vol 57 no 1-2 pp 70ndash812007

[5] R P Orense S Shimoma K Maeda and I Towhata ldquoInstru-mented model slope failure due to water seepagerdquo Journal ofNatural Disaster Science vol 26 pp 15ndash26 2004

[6] M M Hossain and M K Rahman ldquoNumerical simulationof complex fracture growth during tight reservoir stimulationby hydraulic fracturingrdquo Journal of Petroleum Science andEngineering vol 60 no 2 pp 86ndash104 2008

[7] D Hoxha M Lespinasse J Sausse and F Homand ldquoAmicrostructural study of natural and experimentally inducedcracks in a granodioriterdquo Tectonophysics vol 395 no 1-2 pp99ndash112 2005

[8] M Sagong and A Bobet ldquoCoalescence of multiple flaws ina rock-model material in uniaxial compressionrdquo InternationalJournal of Rock Mechanics and Mining Sciences vol 39 no 2pp 229ndash241 2002

[9] R H C Wong K T Chau C A Tang and P Lin ldquoAnalysisof crack coalescence in rock-like materials containing threeflawsmdashpart I experimental approachrdquo International Journal ofRockMechanics andMining Sciences vol 38 no 7 pp 909ndash9242001

[10] S YWang SW SloanD C Sheng andCA Tang ldquoNumericalanalysis of the failure process around a circular opening in rockrdquoComputers and Geotechnics vol 39 pp 8ndash16 2012

[11] O Mutlu and A Bobet ldquoSlip initiation on frictional fracturesrdquoEngineering FractureMechanics vol 72 no 5 pp 729ndash747 2005

[12] Y Nara and K Kaneko ldquoStudy of subcritical crack growth inandesite using the Double Torsion testrdquo International Journal ofRock Mechanics and Mining Sciences vol 42 no 4 pp 521ndash5302005

[13] M Haggerty Q Lin and J F Labuz ldquoObserving deformationand fracture of rock with speckle patternsrdquo Rock Mechanics andRock Engineering vol 43 pp 417ndash426 2010

Mathematical Problems in Engineering 11

[14] C Dascalu B Francois and O Keita ldquoA two-scale model forsubcritical damage propagationrdquo International Journal of Solidsand Structures vol 47 no 3-4 pp 493ndash502 2010

[15] SMisra ldquoDeformation localization at the tips of shear fracturesan analytical approachrdquo Tectonophysics vol 503 no 1-2 pp182ndash187 2011

[16] M F Ashby and S D Hallam ldquoThe failure of brittle solidscontaining small cracks under compressive stress statesrdquo ActaMetallurgica vol 34 no 3 pp 497ndash510 1986

[17] C Fairhurst and N G W Cood ldquoThe phenomenon of rocksplitting parallel to the direction of maximum compression inthe neighbourhood of a surfacerdquo in Proceedings of the Congressof the International Society of RockMechanics L Muller Ed pp687ndash692 Balkema Rotterdam The Netherlands 1966

[18] H P Yuan P Cao and W Z Xu ldquoMechanism study onsubcritical crack growth of flabby and intricate ore rockrdquoTransactions of Nonferrous Metals Society of China vol 16 no3 pp 723ndash727 2006

[19] E Sahouryeh A V Dyskin and L N Germanovich ldquoCrackgrowth under biaxial compressionrdquo Engineering FractureMechanics vol 69 no 18 pp 2187ndash2198 2002

[20] S-H Zheng Research on Coupling Theory Between Seepageand Damage of Fractured Rock Mass and Its Application toEngineering Wuhan Institute of Rock and Soil MechanicsChinese Academy of Scinces Wuhan China 2000

[21] T Liu P Cao and H Lin ldquoAnalytical solutions for the seepageinduced fracture failure propagation in rock massrdquo Journal ofDisaster Advances vol 3 supplement 1 pp 307ndash314 2013

[22] L N Y Wong and H H Einstein ldquoUsing high speed videoimaging in the study of cracking processes in rockrdquoGeotechnicalTesting Journal vol 32 no 2 pp 164ndash180 2009

[23] W Zhu and Q Zhang ldquoBrittle elastic fracture damage constitu-tive model of jointed rockmass and its application to engineer-ingrdquoChinese Journal of RockMechanics and Engineering vol 18no 3 pp 245ndash249 1999

[24] K Y Lam and S P Phua ldquoMultiple crack interaction and itseffect on stress intensity factorrdquoEngineering FractureMechanicsvol 40 no 3 pp 585ndash592 1991

[25] M Kachanov ldquoOn the problems of crack interactions and crackcoalescencerdquo International Journal of Fracture vol 120 no 3pp 537ndash543 2003

[26] H Qing and W Yang ldquoCharacterization of strongly interactedmultiple cracks in an infinite platerdquo Theoretical and AppliedFracture Mechanics vol 46 no 3 pp 209ndash216 2006

[27] A P S Selvadurai and Q Yu ldquoMechanics of a discontinuity in ageomaterialrdquo Computers and Geotechnics vol 32 no 2 pp 92ndash106 2005

[28] X P Yuan H Y Liu and Z Q Wang ldquoAn interacting crack-mechanics basedmodel for elastoplastic damagemodel of rock-likematerials under compressionrdquo International Journal of RockMechanics and Mining Sciences vol 58 pp 92ndash102 2013

[29] B Franois and C Dascalu ldquoA two-scale time-dependentdamage model based on non-planar growth of micro-cracksrdquoJournal of the Mechanics and Physics of Solids vol 58 no 11 pp1928ndash1946 2010

[30] V Palchik andYHHatzor ldquoCrack damage stress as a compositefunction of porosity and elasticmatrix stiffness in dolomites andlimestonesrdquo Engineering Geology vol 63 no 3-4 pp 233ndash2452002

[31] G Alfano S Marfia and E Sacco ldquoA cohesive damage-friction interface model accounting for water pressure on crack

propagationrdquo Computer Methods in Applied Mechanics andEngineering vol 196 no 1ndash3 pp 192ndash209 2006

[32] V Monchiet C Gruescu O Cazacu and D Kondo ldquoAmicromechanical approach of crack-induced damage inorthotropic media application to a brittle matrix compositerdquoEngineering Fracture Mechanics vol 83 pp 40ndash53 2012

[33] A Golshani Y Okui M Oda and T Takemura ldquoA microme-chanical model for brittle failure of rock and its relation tocrack growth observed in triaxial compression tests of graniterdquoMechanics of Materials vol 38 no 4 pp 287ndash303 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

42TheEquivalent Damage Flexibility Tensor Based onCrackrsquosElastic Strain Energy The strain energy of single crack isassumed as [29]

119880(119894)

119889= 2int

119886

0

119866119889Γ = 2

1 minus ]20

1198640

int

119886

0

(1198702

119868+ 1198702

119868119868) 119889119886 (35)

where Γ is the line length of cracksIntegrating over the line length the stress intensity factors

of type 119868 or type 119868119868 for the crack under seepage pressure are

119870(119894)

119868= 120590(119894)

119891

radic120587119886(119894) 119870

(119894)

119868119868= 120591(119894)

119902radic120587119886(119894) (36)

where 120590(119894)119891

is the normal stress of the crack and 120591(119894)

119902is the

tangential shear stress Combining the former equation thenwe get

119880(119894)

119889=

1 minus ]20

1198640

120587119886(119894)2

[120590(119894)2

119891+ 120591(119894)2

119902] (37)

By assuming there are 119899 groups of cracks in the fracturedrock mass the strain energy of the fractured rock mass willbe

119880119889=

119899

sum

119894=1

119880(119894)

119889=

1 minus ]20

1198640

120587

119899

sum

119894=1

119886(119894)2120588(119894)[120590(119894)2

119891+ 120591(119894)2

119902] (38)

Set the equation 119867 = ((1 minus ]20)1198640)120587 119860(119894) = 119886

(119894)2120588(119894) in

which 120588(119894) is the density of cracks then

119880119889= 119867

119899

sum

119894=1

119860(119894)[120590(119894)2

119891+ 120591(119894)2

119902] (39)

Rewrite (39) into the tensor expression

119880119889=

1

2

120590119894119895119862119889

119894119895119896119897120590119896119897= 119867

119899

sum

119894=1

119860(119894)[120590(119894)

119891120590(119894)

119891+ 120591(119894)

119902120591(119894)

119902] (40)

According to the hydrostatic pressure principle andCauchy criterion the remote field stress is denoted as 120590

119894119895 and

the stress matrix on the surface can be defined as [31]

[

[

119879V119909

119879V119910

]

]

= [

120590119909119909minus 119901 120590

119909119910

120590119910119909

120590119909119909minus 119901

] [

]119909

]119910

] (41)

Defining 119899119894119899119894= 1 authors rewrite the equation into the

tensor expression

119879(119894)

119894= (120590(119894)

119894119895minus 120575119894119895119901) 119899(119894)

119895= 120590(119894)

119894119895119899(119894)

119895minus 119901119899(119894)

119894 (42)

Considering the compression transferring coefficient 119862119899

the equivalent normal stress on the surface of the crack canbe expressed as

120590(119894)

119891= 119879(119894)

119894119899(119894)

119895= (1 minus 119862

(119894)

119899) 120590(119894)

119894119895119899(119894)

119895119899(119894)

119894minus 120573(119894)119901

(120590(119894)

119891)119894= (1 minus 119862

119899) 120590(119894)

119895119896119899(119894)

119895119899(119894)

119896119899(119894)

119894minus 120573(119894)119901119899(119894)

119894

(43)

Considering the shear transferring coefficient 119862119904again

the shear stress on the crack surface can be expressed as

(120591(119894)

119891)119894= 119879(119894)

119894minus (120590(119894)

119891)119894= (1 minus 119862

(119894)

119899) 120590(119894)

119895119896119899(119894)

119895(120575119896119897minus 119899(119894)

119896119899(119894)

119894)

(44)

Then we get

(120590(119894)

119891)119894(120590(119894)

119891)119894= ((1 minus 119862

(119894)

119899) 120590(119894)

119895119896119899(119894)

119895119899(119894)

119896119899(119894)

119894minus 120573(119894)119901119899(119894)

119894)

times ((1 minus 119862(119894)

119899) 120590(119894)

119904119905119899(119894)

119904119899(119894)

119905119899(119894)

119894minus 120573(119894)119901119899(119894)

119894)

(45)

Considering the symmetry of 120590119894119895 the following equations

can be deduced

(120590(119894)

119891)119894(120590(119894)

119891)119894= (1 minus 119862

(119894)

119899)

2

120590(119894)

119895119896119899(119894)

119895119899(119894)

119896119899(119894)

119904119899(119894)

119905120590(119894)

119904119905

minus (1 minus 119862(119894)

119899) 120590(119894)

119895119896119899(119894)

119895119899(119894)

119896119901

minus (1 minus 119862(119894)

119899) 120590(119894)

119904119905119899(119894)

119904119899(119894)

119905119901 + 120573(119894)2

1199012

(120591(119894)

119891)119894(120591(119894)

119891)119894= (1 minus 119862

(119894)

119904)

2

120590(119894)

119895119896119899(119894)

119895(120575119896119897minus 119899(119894)

119896119899(119894)

119894) 120590(119894)

119904119905119899(119894)

119904

times (120575119905119894minus 119899(119894)

119894119899(119894)

119894)

= (1 minus 119862(119894)

119904)

2

120590(119894)

119895119896

times (120575119896119897119899(119894)

119895119899(119894)

119904minus 119899(119894)

119895119899(119894)

119896119899(119894)

119904119899(119894)

119905) 120590(119894)

119904119905

(46)

Supposing the proportion coefficient 119877 = 119901120590 120590 is theaverage stress 120590 = (12)120590

119894119894 120590119894119894is the first invariant stress

Then the seepage pressure can be transformed to

119901 = 119901

120590119904119904

120590119904119904

= 120590119904119905120575119904119905

119901

2120590

=

1

2

120590119904119905120575119904119905119877

1199012= 1199012 120590119896119896

120590119896119896

120590119904119904

120590119904119904

= 120590119896119895120575119896119895120590119904119905120575119905119904

119901

2120590

119901

2120590

=

1

4

1205901198961198951205751198961198951205901199041199051205751199051199041198772

(47)

Then we get

(120590(119894)

119891)119894(120590(119894)

119891)119894

= (1 minus 119862(119894)

119899)

2

120590(119894)

119895119896119899(119894)

119895119899(119894)

119896119899(119894)

119904119899(119894)

119905120590(119894)

119904119905

minus

1

2

(1 minus 119862(119894)

119899) 119877120590(119894)

119895119896

times (119899(119894)

119895119899(119894)

119896120590(119894)

119904119905+ 120575119895119896119899(119894)

119904119899(119894)

119905)

+

1

4

(120573(119894)119877)

2

120590(119894)

119895119896120590(119894)

119904119905120590119905119904

8 Mathematical Problems in Engineering

Tension crack

1205901

1205901

12059031205903

ℎ

119863

2119886

120595

2119886

Figure 5 Failure characteristic of crag bridge shearing

119880119889=

1

2

120590119895119896119862119889

119895119896119904119905120590119904119894

= 119867

119899

sum

119894=1

119860(119894)120590(119894)

119895119896[(1 minus 119862

(119894)

119899)

2

119899(119894)

119895119899(119894)

119896119899(119894)

119904119899(119894)

119905

minus

1

2

(1 minus 119862(119894)

119899) 120573(119894)119877120590(119894)

119895119896

times (119899(119894)

119895119899(119894)

119896120590(119894)

119904119905+ 120575119895119896119899(119894)

119904119899(119894)

119905)

+

1

4

(119877120573(119894))

2

120590(119894)

119895119896120590(119894)

119904119905120590119905119904+ (1 minus 119862

(119894)

119904)

2

times (120575119896119905119899(119894)

119895119899(119894)

119904minus 119899(119894)

119895119899(119894)

119896119899(119894)

119904119899(119894)

119905) ] 120590(119894)

119904119905

(48)

Finally it is clear to conclude the additional flexibilitytensor of fractured rock mass under seepage pressure asfollows

119862119889

119894119895119896119897= (1 minus 119862

(119894)

119899)

2

119899(119894)

119895119899(119894)

119896119899(119894)

119904119899(119894)

119905

minus

1

2

(1 minus 119862(119894)

119899) 120573(119894)119877120590(119894)

119895119896(119899(119894)

119895119899(119894)

119896120590(119894)

119904119905+ 120575119895119896119899(119894)

119904119899(119894)

119905)

+

1

4

(119877120573(119894))

2

120590(119894)

119895119896120590(119894)

119904119905120590119905119904

+ (1 minus 119862(119894)

119904)

2

(120575119896119905119899(119894)

119895119899(119894)

119904minus 119899(119894)

119895119899(119894)

119896119899(119894)

119904119899(119894)

119905)

(49)

43The Fracture Damage Evolution Equation The additionalelastic strain energy density 119906119894

119896119911

produced by the wing crackis [32]

119906(119894)

119896119911=

4

1198640

120588(119894)

V 119886(119894)2(

5

2

120590(119894)

eff119871(119894)+

2

radic3

120591(119894)

eff)2

(50)

where 119871(119894) is the equivalent length for the crack

Assuming there are 119899 groups of cracks in the fracturedrock mass the additional elastic strain energy density of thefractured rock 119880

119896119911is

119880119896119911=

119899

sum

119894=1

119906(119894)

119896119911=

1

2

120590119894119895119862119896119911

119894119895119896119897120590119896119897 (51)

Crack extension reduces the stiffness of rock masses andincreases its flexibility Through the derivative of 119880

119896119911with

respect to the stress tensor we can get the additional flexibilitytensor of fractured rock masses in the damage evolutionprocess

120597119880119896119911

120597120590119894119895

= 119862119896119911

119894119895119896119897120590119896119897 (52)

At the same time by calculating the partial derivativeof 119871(119894) 120590(119894)eff and 120591

(119894)

eff with respect to 120590119894119895 respectively and

considering the symmetry of 119862119896119911119894119895119896119897

we can get the followingresults

120597119880119896119911

120597120590119894119895

=

1

1198640

119899

sum

119894=1

119886(119894)2120588(119894)

V [119872

(119894)

1119899(119894)

119895119899(119894)

119896119899(119894)

119896119899(119894)

119905minus119872(119894)

2

times (119899(119894)

119895119899(119894)

119896120575119896119895+ 120575119896119894119899(119894)

119895119899(119894)

119905

+119899(119894)

119895119899(119894)

119896120575119894119895+ 120575119894119905119899(119894)

119895119899(119894)

119896)

minus

119872(119894)

3

3

120573119877120575119894119895120575119896119897]120590119896119897

(53)

Mathematical Problems in Engineering 9

Shear crack

Tension crack

1199031

1199033

120595

119862

119864

119865

119861

119863119897

120591CD

120590ne120591ne

120579

119903 2

119901

119901

1205901

1205903

119897

119886119860

2119886

120590CD119899

Figure 6 Failure characteristic of crag bridge shearing

where

119872(119894)

1= (20120590

(119894)

eff119871(119894)+

16

radic3

120591(119894)

eff)

times [

1

120590(119894)

119899

(

5

2

119871(119894)+

2

radic3

119891)

minus

1

120591(119894)

119899

(

5

2

cot 120579(119894)119871(119894) + 2

radic3

119891) +

5

2

120590(119894)

eff119873(119894)

1]

119872(119894)

2= (5120590

(119894)

eff119871(119894)+

4

radic3

120591(119894)

eff)

times [

1

120591(119894)

119899

(

5

2

cot 120579(119894)119871(119894) + 2

radic3

119891) +

5

2

120590(119894)

eff119873(119894)

2]

119872(119894)

3= (20120590

(119894)

eff119871(119894)+

16

radic3

120591(119894)

eff)

times [

2

radic3

119891 +

5

2

(120590(119894)

eff119873(119894)

3+ 119871(119894))]

119873(119894)

1=radic119871(119894)[minus4119886(119894) cos 120579(119894)120590(119894)2eff (119891120591

(119894)

119899minus 120590(119894)

119899)

minus (119870(119894)2

119868119862minus 4119886(119894) cos 120579(119894)120590(119894)eff

minus 2radic120587119886(119894)119860(119894)119870(119894)

119868119862120590(119894)

eff )

times (120591(119894)

119899minus 120590(119894)

119899cot120579(119894))]

times (2120587119886(119894)119860(119894)120590(119894)

eff120590(119894)

119899120591(119894)

119899)

minus1

119873(119894)

2=radic119871(119894)[minus4119886(119894) cos 120579(119894)120590(119894)2eff

minus (119870(119894)

119868119862

2

minus 4119886(119894) cos 120579(119894)120590(119894)eff120591

(119894)

eff

minus2radic120587119886(119894)119860(119894)119870(119894)

119868119862120590(119894)

eff) cot 120579(119894)]

times (2120587119886(119894)119860(119894)120590(119894)

eff120590(119894)

119899120591(119894)

119899)

minus1

119873(119894)

3

=[

[

minus(minus2(

119870(119894)

119868119862

radic120587119886(119894)

)

2

120590(119894)3

eff minus2

120587

cos 120579(119894)(119891

120590(119894)

eff

minus

120591(119894)

eff

120590(119894)2

eff

))]

]

times

radic119871(119894)

119860(119894)

minus

2119870(119894)

119868119862radic119871(119894)

120590(119894)2

effradic2120587119886

(119894)

119860(119894)= radic

119870(119894)

119868119862

2120590(119894)2

effradic2120587119886

(119894)

minus

2

120587

cos 120579(119894)120591(119894)

eff

120590(119894)2

eff

(54)

Then we can get the flexibility tensor 119862119896119911119894119895119896119897

due to thedamage evolution of fractured rock masses as follows

119862119896119911

119894119895119896119897=

1

1198640

119899

sum

119894=1

119886(119894)2120588(119894)

V [119872(119894)

1119899(119894)

119895119899(119894)

119896119899(119894)

119896119899(119894)

119905minus119872(119894)

2

times (119899(119894)

119895119899(119894)

119896120575119896119895+ 120575119896119894119899(119894)

119895119899(119894)

119905

+119899(119894)

119895119899(119894)

119896120575119894119895+ 120575119894119905119899(119894)

119895119899(119894)

119896)

minus

119872(119894)

3

3

120573119877120575119894119895120575119896119897]

(55)

44 The Constitutive Equation of the Multicrack Rock MassTo sum up combining the initial damage and damage

10 Mathematical Problems in Engineering

evolution tensor we can get the flexibility tensor of thefractured rock mass under seepage pressureas follows

119862119894119895119896119897

= 1198620

119894119895119896119897+ 119862119889

119894119895119896119897+ 119862119896119911

119894119895119896119897 (56)

where1198620119894119895119896119897

is the elastic flexibility tensor of the complete rock119862119889

119894119895119896119897is the initial equivalent damage flexibility tensor of the

fractured rock mass under seepage pressure and 119862119896119911119894119895119896119897

is theadditional damage flexibility tensor with crack propagationof the fractured rock mass under seepage pressure

According to generalized Hookersquos law [33]

120576119894119895= 119862119894119895119896119897120590119896119897

120590119894119895= 119864119894119895119896119897120576119896119897= (119862119894119895119896119897)

minus1

120576119896119897

(57)

and this paper sets up the constitutive relation of the multi-crack rock mass under seepage pressure

5 Conclusions

After discussing the damage and fracture evolution mecha-nism of the multicrack rock mass under compression-shearstress and seepage pressure the following main conclusionsare drawn

(1) Through proposing the fractured damage model ofthe multicrack rock mass under the combined actionof compression-shear stress and seepage pressurethis paper discusses the law of crack initiation andthe evolution law of stress intensity factor at the tipof a wing crack The interaction of multiple cracksmakes the stress intensity factor at the crack tiplarger than that of a single wing crack Regarding theseepage pressure the existence of that reinforces thewing crackrsquos propagation Moreover the high seepagepressure converts wing crack growth from stable formto unstable form

(2) The wing crack initiation from microcracks to thecompletely damage of the rock mass is a processof evolution and accumulation of the rock massdamage as well as a process of the fracture of theconnection between cracks Based on the criterionand mechanism for crack initiation and propagationthis paper puts forward the mechanical model fordifferent fracture transfixion failuremodes of the cragbridge under the combined action of seepage pressureand compression-shear stress

(3) According to the strain energy equivalent principlethis paper applies Bettirsquos reciprocal work theorem ofthe fractured rock mass to study the flexibility tensorof the rock massrsquos initial damage and its damage evo-lution and deduces the damage constitutive equationsfor the elastic-plastic fracture and damage evolutionThe theory provides a reliable theoretical principlefor quantitative research of the fractured rock massfailure under seepage pressure

Acknowledgments

This paper gets its funding from Projects (51174228 5127424941002090) supported by the National Natural ScienceFoundation of China Project supported by the HunanProvincial Innovation Foundation for Postgraduate Project(20120162120014) supported by the Specialized ResearchFund for the Doctoral Program of Higher Education ofChina Project (12KF01) supported by the Open Projects ofState Key Laboratory of Coal Resources and Safe MiningCUMTThe authors wish to acknowledge these supports

References

[1] V Kuksenko N Tomilin and A Chmel ldquoThe rock fractureexperiment with a drive control a spatial aspectrdquo Tectono-physics vol 431 no 1ndash4 pp 123ndash129 2007

[2] S Marfia and E Sacco ldquoA fracture evolution procedure forcohesive materialsrdquo International Journal of Fracture vol 110no 3 pp 241ndash261 2001

[3] L N Lens E Bittencourt and V M R drsquoAvila ldquoConstitutivemodels for cohesive zones in mixed-mode fracture of plainconcreterdquo Engineering Fracture Mechanics vol 76 no 14 pp2281ndash2297 2009

[4] M K Rahman Y A Suarez Z Chen and S S RahmanldquoUnsuccessful hydraulic fracturing cases in Australia investi-gation into causes of failures and their remediesrdquo Journal ofPetroleum Science and Engineering vol 57 no 1-2 pp 70ndash812007

[5] R P Orense S Shimoma K Maeda and I Towhata ldquoInstru-mented model slope failure due to water seepagerdquo Journal ofNatural Disaster Science vol 26 pp 15ndash26 2004

[6] M M Hossain and M K Rahman ldquoNumerical simulationof complex fracture growth during tight reservoir stimulationby hydraulic fracturingrdquo Journal of Petroleum Science andEngineering vol 60 no 2 pp 86ndash104 2008

[7] D Hoxha M Lespinasse J Sausse and F Homand ldquoAmicrostructural study of natural and experimentally inducedcracks in a granodioriterdquo Tectonophysics vol 395 no 1-2 pp99ndash112 2005

[8] M Sagong and A Bobet ldquoCoalescence of multiple flaws ina rock-model material in uniaxial compressionrdquo InternationalJournal of Rock Mechanics and Mining Sciences vol 39 no 2pp 229ndash241 2002

[9] R H C Wong K T Chau C A Tang and P Lin ldquoAnalysisof crack coalescence in rock-like materials containing threeflawsmdashpart I experimental approachrdquo International Journal ofRockMechanics andMining Sciences vol 38 no 7 pp 909ndash9242001

[10] S YWang SW SloanD C Sheng andCA Tang ldquoNumericalanalysis of the failure process around a circular opening in rockrdquoComputers and Geotechnics vol 39 pp 8ndash16 2012

[11] O Mutlu and A Bobet ldquoSlip initiation on frictional fracturesrdquoEngineering FractureMechanics vol 72 no 5 pp 729ndash747 2005

[12] Y Nara and K Kaneko ldquoStudy of subcritical crack growth inandesite using the Double Torsion testrdquo International Journal ofRock Mechanics and Mining Sciences vol 42 no 4 pp 521ndash5302005

[13] M Haggerty Q Lin and J F Labuz ldquoObserving deformationand fracture of rock with speckle patternsrdquo Rock Mechanics andRock Engineering vol 43 pp 417ndash426 2010

Mathematical Problems in Engineering 11

[14] C Dascalu B Francois and O Keita ldquoA two-scale model forsubcritical damage propagationrdquo International Journal of Solidsand Structures vol 47 no 3-4 pp 493ndash502 2010

[15] SMisra ldquoDeformation localization at the tips of shear fracturesan analytical approachrdquo Tectonophysics vol 503 no 1-2 pp182ndash187 2011

[16] M F Ashby and S D Hallam ldquoThe failure of brittle solidscontaining small cracks under compressive stress statesrdquo ActaMetallurgica vol 34 no 3 pp 497ndash510 1986

[17] C Fairhurst and N G W Cood ldquoThe phenomenon of rocksplitting parallel to the direction of maximum compression inthe neighbourhood of a surfacerdquo in Proceedings of the Congressof the International Society of RockMechanics L Muller Ed pp687ndash692 Balkema Rotterdam The Netherlands 1966

[18] H P Yuan P Cao and W Z Xu ldquoMechanism study onsubcritical crack growth of flabby and intricate ore rockrdquoTransactions of Nonferrous Metals Society of China vol 16 no3 pp 723ndash727 2006

[19] E Sahouryeh A V Dyskin and L N Germanovich ldquoCrackgrowth under biaxial compressionrdquo Engineering FractureMechanics vol 69 no 18 pp 2187ndash2198 2002

[20] S-H Zheng Research on Coupling Theory Between Seepageand Damage of Fractured Rock Mass and Its Application toEngineering Wuhan Institute of Rock and Soil MechanicsChinese Academy of Scinces Wuhan China 2000

[21] T Liu P Cao and H Lin ldquoAnalytical solutions for the seepageinduced fracture failure propagation in rock massrdquo Journal ofDisaster Advances vol 3 supplement 1 pp 307ndash314 2013

[22] L N Y Wong and H H Einstein ldquoUsing high speed videoimaging in the study of cracking processes in rockrdquoGeotechnicalTesting Journal vol 32 no 2 pp 164ndash180 2009

[23] W Zhu and Q Zhang ldquoBrittle elastic fracture damage constitu-tive model of jointed rockmass and its application to engineer-ingrdquoChinese Journal of RockMechanics and Engineering vol 18no 3 pp 245ndash249 1999

[24] K Y Lam and S P Phua ldquoMultiple crack interaction and itseffect on stress intensity factorrdquoEngineering FractureMechanicsvol 40 no 3 pp 585ndash592 1991

[25] M Kachanov ldquoOn the problems of crack interactions and crackcoalescencerdquo International Journal of Fracture vol 120 no 3pp 537ndash543 2003

[26] H Qing and W Yang ldquoCharacterization of strongly interactedmultiple cracks in an infinite platerdquo Theoretical and AppliedFracture Mechanics vol 46 no 3 pp 209ndash216 2006

[27] A P S Selvadurai and Q Yu ldquoMechanics of a discontinuity in ageomaterialrdquo Computers and Geotechnics vol 32 no 2 pp 92ndash106 2005

[28] X P Yuan H Y Liu and Z Q Wang ldquoAn interacting crack-mechanics basedmodel for elastoplastic damagemodel of rock-likematerials under compressionrdquo International Journal of RockMechanics and Mining Sciences vol 58 pp 92ndash102 2013

[29] B Franois and C Dascalu ldquoA two-scale time-dependentdamage model based on non-planar growth of micro-cracksrdquoJournal of the Mechanics and Physics of Solids vol 58 no 11 pp1928ndash1946 2010

[30] V Palchik andYHHatzor ldquoCrack damage stress as a compositefunction of porosity and elasticmatrix stiffness in dolomites andlimestonesrdquo Engineering Geology vol 63 no 3-4 pp 233ndash2452002

[31] G Alfano S Marfia and E Sacco ldquoA cohesive damage-friction interface model accounting for water pressure on crack

propagationrdquo Computer Methods in Applied Mechanics andEngineering vol 196 no 1ndash3 pp 192ndash209 2006

[32] V Monchiet C Gruescu O Cazacu and D Kondo ldquoAmicromechanical approach of crack-induced damage inorthotropic media application to a brittle matrix compositerdquoEngineering Fracture Mechanics vol 83 pp 40ndash53 2012

[33] A Golshani Y Okui M Oda and T Takemura ldquoA microme-chanical model for brittle failure of rock and its relation tocrack growth observed in triaxial compression tests of graniterdquoMechanics of Materials vol 38 no 4 pp 287ndash303 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

Tension crack

1205901

1205901

12059031205903

ℎ

119863

2119886

120595

2119886

Figure 5 Failure characteristic of crag bridge shearing

119880119889=

1

2

120590119895119896119862119889

119895119896119904119905120590119904119894

= 119867

119899

sum

119894=1

119860(119894)120590(119894)

119895119896[(1 minus 119862

(119894)

119899)

2

119899(119894)

119895119899(119894)

119896119899(119894)

119904119899(119894)

119905

minus

1

2

(1 minus 119862(119894)

119899) 120573(119894)119877120590(119894)

119895119896

times (119899(119894)

119895119899(119894)

119896120590(119894)

119904119905+ 120575119895119896119899(119894)

119904119899(119894)

119905)

+

1

4

(119877120573(119894))

2

120590(119894)

119895119896120590(119894)

119904119905120590119905119904+ (1 minus 119862

(119894)

119904)

2

times (120575119896119905119899(119894)

119895119899(119894)

119904minus 119899(119894)

119895119899(119894)

119896119899(119894)

119904119899(119894)

119905) ] 120590(119894)

119904119905

(48)

Finally it is clear to conclude the additional flexibilitytensor of fractured rock mass under seepage pressure asfollows

119862119889

119894119895119896119897= (1 minus 119862

(119894)

119899)

2

119899(119894)

119895119899(119894)

119896119899(119894)

119904119899(119894)

119905

minus

1

2

(1 minus 119862(119894)

119899) 120573(119894)119877120590(119894)

119895119896(119899(119894)

119895119899(119894)

119896120590(119894)

119904119905+ 120575119895119896119899(119894)

119904119899(119894)

119905)

+

1

4

(119877120573(119894))

2

120590(119894)

119895119896120590(119894)

119904119905120590119905119904

+ (1 minus 119862(119894)

119904)

2

(120575119896119905119899(119894)

119895119899(119894)

119904minus 119899(119894)

119895119899(119894)

119896119899(119894)

119904119899(119894)

119905)

(49)

43The Fracture Damage Evolution Equation The additionalelastic strain energy density 119906119894

119896119911

produced by the wing crackis [32]

119906(119894)

119896119911=

4

1198640

120588(119894)

V 119886(119894)2(

5

2

120590(119894)

eff119871(119894)+

2

radic3

120591(119894)

eff)2

(50)

where 119871(119894) is the equivalent length for the crack

Assuming there are 119899 groups of cracks in the fracturedrock mass the additional elastic strain energy density of thefractured rock 119880

119896119911is

119880119896119911=

119899

sum

119894=1

119906(119894)

119896119911=

1

2

120590119894119895119862119896119911

119894119895119896119897120590119896119897 (51)

Crack extension reduces the stiffness of rock masses andincreases its flexibility Through the derivative of 119880

119896119911with

respect to the stress tensor we can get the additional flexibilitytensor of fractured rock masses in the damage evolutionprocess

120597119880119896119911

120597120590119894119895

= 119862119896119911

119894119895119896119897120590119896119897 (52)

At the same time by calculating the partial derivativeof 119871(119894) 120590(119894)eff and 120591

(119894)

eff with respect to 120590119894119895 respectively and

considering the symmetry of 119862119896119911119894119895119896119897

we can get the followingresults

120597119880119896119911

120597120590119894119895

=

1

1198640

119899

sum

119894=1

119886(119894)2120588(119894)

V [119872

(119894)

1119899(119894)

119895119899(119894)

119896119899(119894)

119896119899(119894)

119905minus119872(119894)

2

times (119899(119894)

119895119899(119894)

119896120575119896119895+ 120575119896119894119899(119894)

119895119899(119894)

119905

+119899(119894)

119895119899(119894)

119896120575119894119895+ 120575119894119905119899(119894)

119895119899(119894)

119896)

minus

119872(119894)

3

3

120573119877120575119894119895120575119896119897]120590119896119897

(53)

Mathematical Problems in Engineering 9

Shear crack

Tension crack

1199031

1199033

120595

119862

119864

119865

119861

119863119897

120591CD

120590ne120591ne

120579

119903 2

119901

119901

1205901

1205903

119897

119886119860

2119886

120590CD119899

Figure 6 Failure characteristic of crag bridge shearing

where

119872(119894)

1= (20120590

(119894)

eff119871(119894)+

16

radic3

120591(119894)

eff)

times [

1

120590(119894)

119899

(

5

2

119871(119894)+

2

radic3

119891)

minus

1

120591(119894)

119899

(

5

2

cot 120579(119894)119871(119894) + 2

radic3

119891) +

5

2

120590(119894)

eff119873(119894)

1]

119872(119894)

2= (5120590

(119894)

eff119871(119894)+

4

radic3

120591(119894)

eff)

times [

1

120591(119894)

119899

(

5

2

cot 120579(119894)119871(119894) + 2

radic3

119891) +

5

2

120590(119894)

eff119873(119894)

2]

119872(119894)

3= (20120590

(119894)

eff119871(119894)+

16

radic3

120591(119894)

eff)

times [

2

radic3

119891 +

5

2

(120590(119894)

eff119873(119894)

3+ 119871(119894))]

119873(119894)

1=radic119871(119894)[minus4119886(119894) cos 120579(119894)120590(119894)2eff (119891120591

(119894)

119899minus 120590(119894)

119899)

minus (119870(119894)2

119868119862minus 4119886(119894) cos 120579(119894)120590(119894)eff

minus 2radic120587119886(119894)119860(119894)119870(119894)

119868119862120590(119894)

eff )

times (120591(119894)

119899minus 120590(119894)

119899cot120579(119894))]

times (2120587119886(119894)119860(119894)120590(119894)

eff120590(119894)

119899120591(119894)

119899)

minus1

119873(119894)

2=radic119871(119894)[minus4119886(119894) cos 120579(119894)120590(119894)2eff

minus (119870(119894)

119868119862

2

minus 4119886(119894) cos 120579(119894)120590(119894)eff120591

(119894)

eff

minus2radic120587119886(119894)119860(119894)119870(119894)

119868119862120590(119894)

eff) cot 120579(119894)]

times (2120587119886(119894)119860(119894)120590(119894)

eff120590(119894)

119899120591(119894)

119899)

minus1

119873(119894)

3

=[

[

minus(minus2(

119870(119894)

119868119862

radic120587119886(119894)

)

2

120590(119894)3

eff minus2

120587

cos 120579(119894)(119891

120590(119894)

eff

minus

120591(119894)

eff

120590(119894)2

eff

))]

]

times

radic119871(119894)

119860(119894)

minus

2119870(119894)

119868119862radic119871(119894)

120590(119894)2

effradic2120587119886

(119894)

119860(119894)= radic

119870(119894)

119868119862

2120590(119894)2

effradic2120587119886

(119894)

minus

2

120587

cos 120579(119894)120591(119894)

eff

120590(119894)2

eff

(54)

Then we can get the flexibility tensor 119862119896119911119894119895119896119897

due to thedamage evolution of fractured rock masses as follows

119862119896119911

119894119895119896119897=

1

1198640

119899

sum

119894=1

119886(119894)2120588(119894)

V [119872(119894)

1119899(119894)

119895119899(119894)

119896119899(119894)

119896119899(119894)

119905minus119872(119894)

2

times (119899(119894)

119895119899(119894)

119896120575119896119895+ 120575119896119894119899(119894)

119895119899(119894)

119905

+119899(119894)

119895119899(119894)

119896120575119894119895+ 120575119894119905119899(119894)

119895119899(119894)

119896)

minus

119872(119894)

3

3

120573119877120575119894119895120575119896119897]

(55)

44 The Constitutive Equation of the Multicrack Rock MassTo sum up combining the initial damage and damage

10 Mathematical Problems in Engineering

evolution tensor we can get the flexibility tensor of thefractured rock mass under seepage pressureas follows

119862119894119895119896119897

= 1198620

119894119895119896119897+ 119862119889

119894119895119896119897+ 119862119896119911

119894119895119896119897 (56)

where1198620119894119895119896119897

is the elastic flexibility tensor of the complete rock119862119889

119894119895119896119897is the initial equivalent damage flexibility tensor of the

fractured rock mass under seepage pressure and 119862119896119911119894119895119896119897

is theadditional damage flexibility tensor with crack propagationof the fractured rock mass under seepage pressure

According to generalized Hookersquos law [33]

120576119894119895= 119862119894119895119896119897120590119896119897

120590119894119895= 119864119894119895119896119897120576119896119897= (119862119894119895119896119897)

minus1

120576119896119897

(57)

and this paper sets up the constitutive relation of the multi-crack rock mass under seepage pressure

5 Conclusions

After discussing the damage and fracture evolution mecha-nism of the multicrack rock mass under compression-shearstress and seepage pressure the following main conclusionsare drawn

(1) Through proposing the fractured damage model ofthe multicrack rock mass under the combined actionof compression-shear stress and seepage pressurethis paper discusses the law of crack initiation andthe evolution law of stress intensity factor at the tipof a wing crack The interaction of multiple cracksmakes the stress intensity factor at the crack tiplarger than that of a single wing crack Regarding theseepage pressure the existence of that reinforces thewing crackrsquos propagation Moreover the high seepagepressure converts wing crack growth from stable formto unstable form

(2) The wing crack initiation from microcracks to thecompletely damage of the rock mass is a processof evolution and accumulation of the rock massdamage as well as a process of the fracture of theconnection between cracks Based on the criterionand mechanism for crack initiation and propagationthis paper puts forward the mechanical model fordifferent fracture transfixion failuremodes of the cragbridge under the combined action of seepage pressureand compression-shear stress

(3) According to the strain energy equivalent principlethis paper applies Bettirsquos reciprocal work theorem ofthe fractured rock mass to study the flexibility tensorof the rock massrsquos initial damage and its damage evo-lution and deduces the damage constitutive equationsfor the elastic-plastic fracture and damage evolutionThe theory provides a reliable theoretical principlefor quantitative research of the fractured rock massfailure under seepage pressure

Acknowledgments

This paper gets its funding from Projects (51174228 5127424941002090) supported by the National Natural ScienceFoundation of China Project supported by the HunanProvincial Innovation Foundation for Postgraduate Project(20120162120014) supported by the Specialized ResearchFund for the Doctoral Program of Higher Education ofChina Project (12KF01) supported by the Open Projects ofState Key Laboratory of Coal Resources and Safe MiningCUMTThe authors wish to acknowledge these supports

References

[1] V Kuksenko N Tomilin and A Chmel ldquoThe rock fractureexperiment with a drive control a spatial aspectrdquo Tectono-physics vol 431 no 1ndash4 pp 123ndash129 2007

[2] S Marfia and E Sacco ldquoA fracture evolution procedure forcohesive materialsrdquo International Journal of Fracture vol 110no 3 pp 241ndash261 2001

[3] L N Lens E Bittencourt and V M R drsquoAvila ldquoConstitutivemodels for cohesive zones in mixed-mode fracture of plainconcreterdquo Engineering Fracture Mechanics vol 76 no 14 pp2281ndash2297 2009

[4] M K Rahman Y A Suarez Z Chen and S S RahmanldquoUnsuccessful hydraulic fracturing cases in Australia investi-gation into causes of failures and their remediesrdquo Journal ofPetroleum Science and Engineering vol 57 no 1-2 pp 70ndash812007

[5] R P Orense S Shimoma K Maeda and I Towhata ldquoInstru-mented model slope failure due to water seepagerdquo Journal ofNatural Disaster Science vol 26 pp 15ndash26 2004

[6] M M Hossain and M K Rahman ldquoNumerical simulationof complex fracture growth during tight reservoir stimulationby hydraulic fracturingrdquo Journal of Petroleum Science andEngineering vol 60 no 2 pp 86ndash104 2008

[7] D Hoxha M Lespinasse J Sausse and F Homand ldquoAmicrostructural study of natural and experimentally inducedcracks in a granodioriterdquo Tectonophysics vol 395 no 1-2 pp99ndash112 2005

[8] M Sagong and A Bobet ldquoCoalescence of multiple flaws ina rock-model material in uniaxial compressionrdquo InternationalJournal of Rock Mechanics and Mining Sciences vol 39 no 2pp 229ndash241 2002

[9] R H C Wong K T Chau C A Tang and P Lin ldquoAnalysisof crack coalescence in rock-like materials containing threeflawsmdashpart I experimental approachrdquo International Journal ofRockMechanics andMining Sciences vol 38 no 7 pp 909ndash9242001

[10] S YWang SW SloanD C Sheng andCA Tang ldquoNumericalanalysis of the failure process around a circular opening in rockrdquoComputers and Geotechnics vol 39 pp 8ndash16 2012

[11] O Mutlu and A Bobet ldquoSlip initiation on frictional fracturesrdquoEngineering FractureMechanics vol 72 no 5 pp 729ndash747 2005

[12] Y Nara and K Kaneko ldquoStudy of subcritical crack growth inandesite using the Double Torsion testrdquo International Journal ofRock Mechanics and Mining Sciences vol 42 no 4 pp 521ndash5302005

[13] M Haggerty Q Lin and J F Labuz ldquoObserving deformationand fracture of rock with speckle patternsrdquo Rock Mechanics andRock Engineering vol 43 pp 417ndash426 2010

Mathematical Problems in Engineering 11

[14] C Dascalu B Francois and O Keita ldquoA two-scale model forsubcritical damage propagationrdquo International Journal of Solidsand Structures vol 47 no 3-4 pp 493ndash502 2010

[15] SMisra ldquoDeformation localization at the tips of shear fracturesan analytical approachrdquo Tectonophysics vol 503 no 1-2 pp182ndash187 2011

[16] M F Ashby and S D Hallam ldquoThe failure of brittle solidscontaining small cracks under compressive stress statesrdquo ActaMetallurgica vol 34 no 3 pp 497ndash510 1986

[17] C Fairhurst and N G W Cood ldquoThe phenomenon of rocksplitting parallel to the direction of maximum compression inthe neighbourhood of a surfacerdquo in Proceedings of the Congressof the International Society of RockMechanics L Muller Ed pp687ndash692 Balkema Rotterdam The Netherlands 1966

[18] H P Yuan P Cao and W Z Xu ldquoMechanism study onsubcritical crack growth of flabby and intricate ore rockrdquoTransactions of Nonferrous Metals Society of China vol 16 no3 pp 723ndash727 2006

[19] E Sahouryeh A V Dyskin and L N Germanovich ldquoCrackgrowth under biaxial compressionrdquo Engineering FractureMechanics vol 69 no 18 pp 2187ndash2198 2002

[20] S-H Zheng Research on Coupling Theory Between Seepageand Damage of Fractured Rock Mass and Its Application toEngineering Wuhan Institute of Rock and Soil MechanicsChinese Academy of Scinces Wuhan China 2000

[21] T Liu P Cao and H Lin ldquoAnalytical solutions for the seepageinduced fracture failure propagation in rock massrdquo Journal ofDisaster Advances vol 3 supplement 1 pp 307ndash314 2013

[22] L N Y Wong and H H Einstein ldquoUsing high speed videoimaging in the study of cracking processes in rockrdquoGeotechnicalTesting Journal vol 32 no 2 pp 164ndash180 2009

[23] W Zhu and Q Zhang ldquoBrittle elastic fracture damage constitu-tive model of jointed rockmass and its application to engineer-ingrdquoChinese Journal of RockMechanics and Engineering vol 18no 3 pp 245ndash249 1999

[24] K Y Lam and S P Phua ldquoMultiple crack interaction and itseffect on stress intensity factorrdquoEngineering FractureMechanicsvol 40 no 3 pp 585ndash592 1991

[25] M Kachanov ldquoOn the problems of crack interactions and crackcoalescencerdquo International Journal of Fracture vol 120 no 3pp 537ndash543 2003

[26] H Qing and W Yang ldquoCharacterization of strongly interactedmultiple cracks in an infinite platerdquo Theoretical and AppliedFracture Mechanics vol 46 no 3 pp 209ndash216 2006

[27] A P S Selvadurai and Q Yu ldquoMechanics of a discontinuity in ageomaterialrdquo Computers and Geotechnics vol 32 no 2 pp 92ndash106 2005

[28] X P Yuan H Y Liu and Z Q Wang ldquoAn interacting crack-mechanics basedmodel for elastoplastic damagemodel of rock-likematerials under compressionrdquo International Journal of RockMechanics and Mining Sciences vol 58 pp 92ndash102 2013

[29] B Franois and C Dascalu ldquoA two-scale time-dependentdamage model based on non-planar growth of micro-cracksrdquoJournal of the Mechanics and Physics of Solids vol 58 no 11 pp1928ndash1946 2010

[30] V Palchik andYHHatzor ldquoCrack damage stress as a compositefunction of porosity and elasticmatrix stiffness in dolomites andlimestonesrdquo Engineering Geology vol 63 no 3-4 pp 233ndash2452002

[31] G Alfano S Marfia and E Sacco ldquoA cohesive damage-friction interface model accounting for water pressure on crack

propagationrdquo Computer Methods in Applied Mechanics andEngineering vol 196 no 1ndash3 pp 192ndash209 2006

[32] V Monchiet C Gruescu O Cazacu and D Kondo ldquoAmicromechanical approach of crack-induced damage inorthotropic media application to a brittle matrix compositerdquoEngineering Fracture Mechanics vol 83 pp 40ndash53 2012

[33] A Golshani Y Okui M Oda and T Takemura ldquoA microme-chanical model for brittle failure of rock and its relation tocrack growth observed in triaxial compression tests of graniterdquoMechanics of Materials vol 38 no 4 pp 287ndash303 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

Shear crack

Tension crack

1199031

1199033

120595

119862

119864

119865

119861

119863119897

120591CD

120590ne120591ne

120579

119903 2

119901

119901

1205901

1205903

119897

119886119860

2119886

120590CD119899

Figure 6 Failure characteristic of crag bridge shearing

where

119872(119894)

1= (20120590

(119894)

eff119871(119894)+

16

radic3

120591(119894)

eff)

times [

1

120590(119894)

119899

(

5

2

119871(119894)+

2

radic3

119891)

minus

1

120591(119894)

119899

(

5

2

cot 120579(119894)119871(119894) + 2

radic3

119891) +

5

2

120590(119894)

eff119873(119894)

1]

119872(119894)

2= (5120590

(119894)

eff119871(119894)+

4

radic3

120591(119894)

eff)

times [

1

120591(119894)

119899

(

5

2

cot 120579(119894)119871(119894) + 2

radic3

119891) +

5

2

120590(119894)

eff119873(119894)

2]

119872(119894)

3= (20120590

(119894)

eff119871(119894)+

16

radic3

120591(119894)

eff)

times [

2

radic3

119891 +

5

2

(120590(119894)

eff119873(119894)

3+ 119871(119894))]

119873(119894)

1=radic119871(119894)[minus4119886(119894) cos 120579(119894)120590(119894)2eff (119891120591

(119894)

119899minus 120590(119894)

119899)

minus (119870(119894)2

119868119862minus 4119886(119894) cos 120579(119894)120590(119894)eff

minus 2radic120587119886(119894)119860(119894)119870(119894)

119868119862120590(119894)

eff )

times (120591(119894)

119899minus 120590(119894)

119899cot120579(119894))]

times (2120587119886(119894)119860(119894)120590(119894)

eff120590(119894)

119899120591(119894)

119899)

minus1

119873(119894)

2=radic119871(119894)[minus4119886(119894) cos 120579(119894)120590(119894)2eff

minus (119870(119894)

119868119862

2

minus 4119886(119894) cos 120579(119894)120590(119894)eff120591

(119894)

eff

minus2radic120587119886(119894)119860(119894)119870(119894)

119868119862120590(119894)

eff) cot 120579(119894)]

times (2120587119886(119894)119860(119894)120590(119894)

eff120590(119894)

119899120591(119894)

119899)

minus1

119873(119894)

3

=[

[

minus(minus2(

119870(119894)

119868119862

radic120587119886(119894)

)

2

120590(119894)3

eff minus2

120587

cos 120579(119894)(119891

120590(119894)

eff

minus

120591(119894)

eff

120590(119894)2

eff

))]

]

times

radic119871(119894)

119860(119894)

minus

2119870(119894)

119868119862radic119871(119894)

120590(119894)2

effradic2120587119886

(119894)

119860(119894)= radic

119870(119894)

119868119862

2120590(119894)2

effradic2120587119886

(119894)

minus

2

120587

cos 120579(119894)120591(119894)

eff

120590(119894)2

eff

(54)

Then we can get the flexibility tensor 119862119896119911119894119895119896119897

due to thedamage evolution of fractured rock masses as follows

119862119896119911

119894119895119896119897=

1

1198640

119899

sum

119894=1

119886(119894)2120588(119894)

V [119872(119894)

1119899(119894)

119895119899(119894)

119896119899(119894)

119896119899(119894)

119905minus119872(119894)

2

times (119899(119894)

119895119899(119894)

119896120575119896119895+ 120575119896119894119899(119894)

119895119899(119894)

119905

+119899(119894)

119895119899(119894)

119896120575119894119895+ 120575119894119905119899(119894)

119895119899(119894)

119896)

minus

119872(119894)

3

3

120573119877120575119894119895120575119896119897]

(55)

44 The Constitutive Equation of the Multicrack Rock MassTo sum up combining the initial damage and damage

10 Mathematical Problems in Engineering

evolution tensor we can get the flexibility tensor of thefractured rock mass under seepage pressureas follows

119862119894119895119896119897

= 1198620

119894119895119896119897+ 119862119889

119894119895119896119897+ 119862119896119911

119894119895119896119897 (56)

where1198620119894119895119896119897

is the elastic flexibility tensor of the complete rock119862119889

119894119895119896119897is the initial equivalent damage flexibility tensor of the

fractured rock mass under seepage pressure and 119862119896119911119894119895119896119897

is theadditional damage flexibility tensor with crack propagationof the fractured rock mass under seepage pressure

According to generalized Hookersquos law [33]

120576119894119895= 119862119894119895119896119897120590119896119897

120590119894119895= 119864119894119895119896119897120576119896119897= (119862119894119895119896119897)

minus1

120576119896119897

(57)

and this paper sets up the constitutive relation of the multi-crack rock mass under seepage pressure

5 Conclusions

After discussing the damage and fracture evolution mecha-nism of the multicrack rock mass under compression-shearstress and seepage pressure the following main conclusionsare drawn

(1) Through proposing the fractured damage model ofthe multicrack rock mass under the combined actionof compression-shear stress and seepage pressurethis paper discusses the law of crack initiation andthe evolution law of stress intensity factor at the tipof a wing crack The interaction of multiple cracksmakes the stress intensity factor at the crack tiplarger than that of a single wing crack Regarding theseepage pressure the existence of that reinforces thewing crackrsquos propagation Moreover the high seepagepressure converts wing crack growth from stable formto unstable form

(2) The wing crack initiation from microcracks to thecompletely damage of the rock mass is a processof evolution and accumulation of the rock massdamage as well as a process of the fracture of theconnection between cracks Based on the criterionand mechanism for crack initiation and propagationthis paper puts forward the mechanical model fordifferent fracture transfixion failuremodes of the cragbridge under the combined action of seepage pressureand compression-shear stress

(3) According to the strain energy equivalent principlethis paper applies Bettirsquos reciprocal work theorem ofthe fractured rock mass to study the flexibility tensorof the rock massrsquos initial damage and its damage evo-lution and deduces the damage constitutive equationsfor the elastic-plastic fracture and damage evolutionThe theory provides a reliable theoretical principlefor quantitative research of the fractured rock massfailure under seepage pressure

Acknowledgments

This paper gets its funding from Projects (51174228 5127424941002090) supported by the National Natural ScienceFoundation of China Project supported by the HunanProvincial Innovation Foundation for Postgraduate Project(20120162120014) supported by the Specialized ResearchFund for the Doctoral Program of Higher Education ofChina Project (12KF01) supported by the Open Projects ofState Key Laboratory of Coal Resources and Safe MiningCUMTThe authors wish to acknowledge these supports

References

[1] V Kuksenko N Tomilin and A Chmel ldquoThe rock fractureexperiment with a drive control a spatial aspectrdquo Tectono-physics vol 431 no 1ndash4 pp 123ndash129 2007

[2] S Marfia and E Sacco ldquoA fracture evolution procedure forcohesive materialsrdquo International Journal of Fracture vol 110no 3 pp 241ndash261 2001

[3] L N Lens E Bittencourt and V M R drsquoAvila ldquoConstitutivemodels for cohesive zones in mixed-mode fracture of plainconcreterdquo Engineering Fracture Mechanics vol 76 no 14 pp2281ndash2297 2009

[4] M K Rahman Y A Suarez Z Chen and S S RahmanldquoUnsuccessful hydraulic fracturing cases in Australia investi-gation into causes of failures and their remediesrdquo Journal ofPetroleum Science and Engineering vol 57 no 1-2 pp 70ndash812007

[5] R P Orense S Shimoma K Maeda and I Towhata ldquoInstru-mented model slope failure due to water seepagerdquo Journal ofNatural Disaster Science vol 26 pp 15ndash26 2004

[6] M M Hossain and M K Rahman ldquoNumerical simulationof complex fracture growth during tight reservoir stimulationby hydraulic fracturingrdquo Journal of Petroleum Science andEngineering vol 60 no 2 pp 86ndash104 2008

[7] D Hoxha M Lespinasse J Sausse and F Homand ldquoAmicrostructural study of natural and experimentally inducedcracks in a granodioriterdquo Tectonophysics vol 395 no 1-2 pp99ndash112 2005

[8] M Sagong and A Bobet ldquoCoalescence of multiple flaws ina rock-model material in uniaxial compressionrdquo InternationalJournal of Rock Mechanics and Mining Sciences vol 39 no 2pp 229ndash241 2002

[9] R H C Wong K T Chau C A Tang and P Lin ldquoAnalysisof crack coalescence in rock-like materials containing threeflawsmdashpart I experimental approachrdquo International Journal ofRockMechanics andMining Sciences vol 38 no 7 pp 909ndash9242001

[10] S YWang SW SloanD C Sheng andCA Tang ldquoNumericalanalysis of the failure process around a circular opening in rockrdquoComputers and Geotechnics vol 39 pp 8ndash16 2012

[11] O Mutlu and A Bobet ldquoSlip initiation on frictional fracturesrdquoEngineering FractureMechanics vol 72 no 5 pp 729ndash747 2005

[12] Y Nara and K Kaneko ldquoStudy of subcritical crack growth inandesite using the Double Torsion testrdquo International Journal ofRock Mechanics and Mining Sciences vol 42 no 4 pp 521ndash5302005

[13] M Haggerty Q Lin and J F Labuz ldquoObserving deformationand fracture of rock with speckle patternsrdquo Rock Mechanics andRock Engineering vol 43 pp 417ndash426 2010

Mathematical Problems in Engineering 11

[14] C Dascalu B Francois and O Keita ldquoA two-scale model forsubcritical damage propagationrdquo International Journal of Solidsand Structures vol 47 no 3-4 pp 493ndash502 2010

[15] SMisra ldquoDeformation localization at the tips of shear fracturesan analytical approachrdquo Tectonophysics vol 503 no 1-2 pp182ndash187 2011

[16] M F Ashby and S D Hallam ldquoThe failure of brittle solidscontaining small cracks under compressive stress statesrdquo ActaMetallurgica vol 34 no 3 pp 497ndash510 1986

[17] C Fairhurst and N G W Cood ldquoThe phenomenon of rocksplitting parallel to the direction of maximum compression inthe neighbourhood of a surfacerdquo in Proceedings of the Congressof the International Society of RockMechanics L Muller Ed pp687ndash692 Balkema Rotterdam The Netherlands 1966

[18] H P Yuan P Cao and W Z Xu ldquoMechanism study onsubcritical crack growth of flabby and intricate ore rockrdquoTransactions of Nonferrous Metals Society of China vol 16 no3 pp 723ndash727 2006

[19] E Sahouryeh A V Dyskin and L N Germanovich ldquoCrackgrowth under biaxial compressionrdquo Engineering FractureMechanics vol 69 no 18 pp 2187ndash2198 2002

[20] S-H Zheng Research on Coupling Theory Between Seepageand Damage of Fractured Rock Mass and Its Application toEngineering Wuhan Institute of Rock and Soil MechanicsChinese Academy of Scinces Wuhan China 2000

[21] T Liu P Cao and H Lin ldquoAnalytical solutions for the seepageinduced fracture failure propagation in rock massrdquo Journal ofDisaster Advances vol 3 supplement 1 pp 307ndash314 2013

[22] L N Y Wong and H H Einstein ldquoUsing high speed videoimaging in the study of cracking processes in rockrdquoGeotechnicalTesting Journal vol 32 no 2 pp 164ndash180 2009

[23] W Zhu and Q Zhang ldquoBrittle elastic fracture damage constitu-tive model of jointed rockmass and its application to engineer-ingrdquoChinese Journal of RockMechanics and Engineering vol 18no 3 pp 245ndash249 1999

[24] K Y Lam and S P Phua ldquoMultiple crack interaction and itseffect on stress intensity factorrdquoEngineering FractureMechanicsvol 40 no 3 pp 585ndash592 1991

[25] M Kachanov ldquoOn the problems of crack interactions and crackcoalescencerdquo International Journal of Fracture vol 120 no 3pp 537ndash543 2003

[26] H Qing and W Yang ldquoCharacterization of strongly interactedmultiple cracks in an infinite platerdquo Theoretical and AppliedFracture Mechanics vol 46 no 3 pp 209ndash216 2006

[27] A P S Selvadurai and Q Yu ldquoMechanics of a discontinuity in ageomaterialrdquo Computers and Geotechnics vol 32 no 2 pp 92ndash106 2005

[28] X P Yuan H Y Liu and Z Q Wang ldquoAn interacting crack-mechanics basedmodel for elastoplastic damagemodel of rock-likematerials under compressionrdquo International Journal of RockMechanics and Mining Sciences vol 58 pp 92ndash102 2013

[29] B Franois and C Dascalu ldquoA two-scale time-dependentdamage model based on non-planar growth of micro-cracksrdquoJournal of the Mechanics and Physics of Solids vol 58 no 11 pp1928ndash1946 2010

[30] V Palchik andYHHatzor ldquoCrack damage stress as a compositefunction of porosity and elasticmatrix stiffness in dolomites andlimestonesrdquo Engineering Geology vol 63 no 3-4 pp 233ndash2452002

[31] G Alfano S Marfia and E Sacco ldquoA cohesive damage-friction interface model accounting for water pressure on crack

propagationrdquo Computer Methods in Applied Mechanics andEngineering vol 196 no 1ndash3 pp 192ndash209 2006

[32] V Monchiet C Gruescu O Cazacu and D Kondo ldquoAmicromechanical approach of crack-induced damage inorthotropic media application to a brittle matrix compositerdquoEngineering Fracture Mechanics vol 83 pp 40ndash53 2012

[33] A Golshani Y Okui M Oda and T Takemura ldquoA microme-chanical model for brittle failure of rock and its relation tocrack growth observed in triaxial compression tests of graniterdquoMechanics of Materials vol 38 no 4 pp 287ndash303 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Mathematical Problems in Engineering

evolution tensor we can get the flexibility tensor of thefractured rock mass under seepage pressureas follows

119862119894119895119896119897

= 1198620

119894119895119896119897+ 119862119889

119894119895119896119897+ 119862119896119911

119894119895119896119897 (56)

where1198620119894119895119896119897

is the elastic flexibility tensor of the complete rock119862119889

119894119895119896119897is the initial equivalent damage flexibility tensor of the

fractured rock mass under seepage pressure and 119862119896119911119894119895119896119897

is theadditional damage flexibility tensor with crack propagationof the fractured rock mass under seepage pressure

According to generalized Hookersquos law [33]

120576119894119895= 119862119894119895119896119897120590119896119897

120590119894119895= 119864119894119895119896119897120576119896119897= (119862119894119895119896119897)

minus1

120576119896119897

(57)

and this paper sets up the constitutive relation of the multi-crack rock mass under seepage pressure

5 Conclusions

After discussing the damage and fracture evolution mecha-nism of the multicrack rock mass under compression-shearstress and seepage pressure the following main conclusionsare drawn

(1) Through proposing the fractured damage model ofthe multicrack rock mass under the combined actionof compression-shear stress and seepage pressurethis paper discusses the law of crack initiation andthe evolution law of stress intensity factor at the tipof a wing crack The interaction of multiple cracksmakes the stress intensity factor at the crack tiplarger than that of a single wing crack Regarding theseepage pressure the existence of that reinforces thewing crackrsquos propagation Moreover the high seepagepressure converts wing crack growth from stable formto unstable form

(2) The wing crack initiation from microcracks to thecompletely damage of the rock mass is a processof evolution and accumulation of the rock massdamage as well as a process of the fracture of theconnection between cracks Based on the criterionand mechanism for crack initiation and propagationthis paper puts forward the mechanical model fordifferent fracture transfixion failuremodes of the cragbridge under the combined action of seepage pressureand compression-shear stress

(3) According to the strain energy equivalent principlethis paper applies Bettirsquos reciprocal work theorem ofthe fractured rock mass to study the flexibility tensorof the rock massrsquos initial damage and its damage evo-lution and deduces the damage constitutive equationsfor the elastic-plastic fracture and damage evolutionThe theory provides a reliable theoretical principlefor quantitative research of the fractured rock massfailure under seepage pressure

Acknowledgments

This paper gets its funding from Projects (51174228 5127424941002090) supported by the National Natural ScienceFoundation of China Project supported by the HunanProvincial Innovation Foundation for Postgraduate Project(20120162120014) supported by the Specialized ResearchFund for the Doctoral Program of Higher Education ofChina Project (12KF01) supported by the Open Projects ofState Key Laboratory of Coal Resources and Safe MiningCUMTThe authors wish to acknowledge these supports

References

[1] V Kuksenko N Tomilin and A Chmel ldquoThe rock fractureexperiment with a drive control a spatial aspectrdquo Tectono-physics vol 431 no 1ndash4 pp 123ndash129 2007

[2] S Marfia and E Sacco ldquoA fracture evolution procedure forcohesive materialsrdquo International Journal of Fracture vol 110no 3 pp 241ndash261 2001

[3] L N Lens E Bittencourt and V M R drsquoAvila ldquoConstitutivemodels for cohesive zones in mixed-mode fracture of plainconcreterdquo Engineering Fracture Mechanics vol 76 no 14 pp2281ndash2297 2009

[4] M K Rahman Y A Suarez Z Chen and S S RahmanldquoUnsuccessful hydraulic fracturing cases in Australia investi-gation into causes of failures and their remediesrdquo Journal ofPetroleum Science and Engineering vol 57 no 1-2 pp 70ndash812007

[5] R P Orense S Shimoma K Maeda and I Towhata ldquoInstru-mented model slope failure due to water seepagerdquo Journal ofNatural Disaster Science vol 26 pp 15ndash26 2004

[6] M M Hossain and M K Rahman ldquoNumerical simulationof complex fracture growth during tight reservoir stimulationby hydraulic fracturingrdquo Journal of Petroleum Science andEngineering vol 60 no 2 pp 86ndash104 2008

[7] D Hoxha M Lespinasse J Sausse and F Homand ldquoAmicrostructural study of natural and experimentally inducedcracks in a granodioriterdquo Tectonophysics vol 395 no 1-2 pp99ndash112 2005

[8] M Sagong and A Bobet ldquoCoalescence of multiple flaws ina rock-model material in uniaxial compressionrdquo InternationalJournal of Rock Mechanics and Mining Sciences vol 39 no 2pp 229ndash241 2002

[9] R H C Wong K T Chau C A Tang and P Lin ldquoAnalysisof crack coalescence in rock-like materials containing threeflawsmdashpart I experimental approachrdquo International Journal ofRockMechanics andMining Sciences vol 38 no 7 pp 909ndash9242001

[10] S YWang SW SloanD C Sheng andCA Tang ldquoNumericalanalysis of the failure process around a circular opening in rockrdquoComputers and Geotechnics vol 39 pp 8ndash16 2012

[11] O Mutlu and A Bobet ldquoSlip initiation on frictional fracturesrdquoEngineering FractureMechanics vol 72 no 5 pp 729ndash747 2005

[12] Y Nara and K Kaneko ldquoStudy of subcritical crack growth inandesite using the Double Torsion testrdquo International Journal ofRock Mechanics and Mining Sciences vol 42 no 4 pp 521ndash5302005

[13] M Haggerty Q Lin and J F Labuz ldquoObserving deformationand fracture of rock with speckle patternsrdquo Rock Mechanics andRock Engineering vol 43 pp 417ndash426 2010

Mathematical Problems in Engineering 11

[14] C Dascalu B Francois and O Keita ldquoA two-scale model forsubcritical damage propagationrdquo International Journal of Solidsand Structures vol 47 no 3-4 pp 493ndash502 2010

[15] SMisra ldquoDeformation localization at the tips of shear fracturesan analytical approachrdquo Tectonophysics vol 503 no 1-2 pp182ndash187 2011

[16] M F Ashby and S D Hallam ldquoThe failure of brittle solidscontaining small cracks under compressive stress statesrdquo ActaMetallurgica vol 34 no 3 pp 497ndash510 1986

[17] C Fairhurst and N G W Cood ldquoThe phenomenon of rocksplitting parallel to the direction of maximum compression inthe neighbourhood of a surfacerdquo in Proceedings of the Congressof the International Society of RockMechanics L Muller Ed pp687ndash692 Balkema Rotterdam The Netherlands 1966

[18] H P Yuan P Cao and W Z Xu ldquoMechanism study onsubcritical crack growth of flabby and intricate ore rockrdquoTransactions of Nonferrous Metals Society of China vol 16 no3 pp 723ndash727 2006

[19] E Sahouryeh A V Dyskin and L N Germanovich ldquoCrackgrowth under biaxial compressionrdquo Engineering FractureMechanics vol 69 no 18 pp 2187ndash2198 2002

[20] S-H Zheng Research on Coupling Theory Between Seepageand Damage of Fractured Rock Mass and Its Application toEngineering Wuhan Institute of Rock and Soil MechanicsChinese Academy of Scinces Wuhan China 2000

[21] T Liu P Cao and H Lin ldquoAnalytical solutions for the seepageinduced fracture failure propagation in rock massrdquo Journal ofDisaster Advances vol 3 supplement 1 pp 307ndash314 2013

[22] L N Y Wong and H H Einstein ldquoUsing high speed videoimaging in the study of cracking processes in rockrdquoGeotechnicalTesting Journal vol 32 no 2 pp 164ndash180 2009

[23] W Zhu and Q Zhang ldquoBrittle elastic fracture damage constitu-tive model of jointed rockmass and its application to engineer-ingrdquoChinese Journal of RockMechanics and Engineering vol 18no 3 pp 245ndash249 1999

[24] K Y Lam and S P Phua ldquoMultiple crack interaction and itseffect on stress intensity factorrdquoEngineering FractureMechanicsvol 40 no 3 pp 585ndash592 1991

[25] M Kachanov ldquoOn the problems of crack interactions and crackcoalescencerdquo International Journal of Fracture vol 120 no 3pp 537ndash543 2003

[26] H Qing and W Yang ldquoCharacterization of strongly interactedmultiple cracks in an infinite platerdquo Theoretical and AppliedFracture Mechanics vol 46 no 3 pp 209ndash216 2006

[27] A P S Selvadurai and Q Yu ldquoMechanics of a discontinuity in ageomaterialrdquo Computers and Geotechnics vol 32 no 2 pp 92ndash106 2005

[28] X P Yuan H Y Liu and Z Q Wang ldquoAn interacting crack-mechanics basedmodel for elastoplastic damagemodel of rock-likematerials under compressionrdquo International Journal of RockMechanics and Mining Sciences vol 58 pp 92ndash102 2013

[29] B Franois and C Dascalu ldquoA two-scale time-dependentdamage model based on non-planar growth of micro-cracksrdquoJournal of the Mechanics and Physics of Solids vol 58 no 11 pp1928ndash1946 2010

[30] V Palchik andYHHatzor ldquoCrack damage stress as a compositefunction of porosity and elasticmatrix stiffness in dolomites andlimestonesrdquo Engineering Geology vol 63 no 3-4 pp 233ndash2452002

[31] G Alfano S Marfia and E Sacco ldquoA cohesive damage-friction interface model accounting for water pressure on crack

propagationrdquo Computer Methods in Applied Mechanics andEngineering vol 196 no 1ndash3 pp 192ndash209 2006

[32] V Monchiet C Gruescu O Cazacu and D Kondo ldquoAmicromechanical approach of crack-induced damage inorthotropic media application to a brittle matrix compositerdquoEngineering Fracture Mechanics vol 83 pp 40ndash53 2012

[33] A Golshani Y Okui M Oda and T Takemura ldquoA microme-chanical model for brittle failure of rock and its relation tocrack growth observed in triaxial compression tests of graniterdquoMechanics of Materials vol 38 no 4 pp 287ndash303 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 11

[14] C Dascalu B Francois and O Keita ldquoA two-scale model forsubcritical damage propagationrdquo International Journal of Solidsand Structures vol 47 no 3-4 pp 493ndash502 2010

[15] SMisra ldquoDeformation localization at the tips of shear fracturesan analytical approachrdquo Tectonophysics vol 503 no 1-2 pp182ndash187 2011

[16] M F Ashby and S D Hallam ldquoThe failure of brittle solidscontaining small cracks under compressive stress statesrdquo ActaMetallurgica vol 34 no 3 pp 497ndash510 1986

[17] C Fairhurst and N G W Cood ldquoThe phenomenon of rocksplitting parallel to the direction of maximum compression inthe neighbourhood of a surfacerdquo in Proceedings of the Congressof the International Society of RockMechanics L Muller Ed pp687ndash692 Balkema Rotterdam The Netherlands 1966

[18] H P Yuan P Cao and W Z Xu ldquoMechanism study onsubcritical crack growth of flabby and intricate ore rockrdquoTransactions of Nonferrous Metals Society of China vol 16 no3 pp 723ndash727 2006

[19] E Sahouryeh A V Dyskin and L N Germanovich ldquoCrackgrowth under biaxial compressionrdquo Engineering FractureMechanics vol 69 no 18 pp 2187ndash2198 2002

[20] S-H Zheng Research on Coupling Theory Between Seepageand Damage of Fractured Rock Mass and Its Application toEngineering Wuhan Institute of Rock and Soil MechanicsChinese Academy of Scinces Wuhan China 2000

[21] T Liu P Cao and H Lin ldquoAnalytical solutions for the seepageinduced fracture failure propagation in rock massrdquo Journal ofDisaster Advances vol 3 supplement 1 pp 307ndash314 2013

[22] L N Y Wong and H H Einstein ldquoUsing high speed videoimaging in the study of cracking processes in rockrdquoGeotechnicalTesting Journal vol 32 no 2 pp 164ndash180 2009

[23] W Zhu and Q Zhang ldquoBrittle elastic fracture damage constitu-tive model of jointed rockmass and its application to engineer-ingrdquoChinese Journal of RockMechanics and Engineering vol 18no 3 pp 245ndash249 1999

[24] K Y Lam and S P Phua ldquoMultiple crack interaction and itseffect on stress intensity factorrdquoEngineering FractureMechanicsvol 40 no 3 pp 585ndash592 1991

[25] M Kachanov ldquoOn the problems of crack interactions and crackcoalescencerdquo International Journal of Fracture vol 120 no 3pp 537ndash543 2003

[26] H Qing and W Yang ldquoCharacterization of strongly interactedmultiple cracks in an infinite platerdquo Theoretical and AppliedFracture Mechanics vol 46 no 3 pp 209ndash216 2006

[27] A P S Selvadurai and Q Yu ldquoMechanics of a discontinuity in ageomaterialrdquo Computers and Geotechnics vol 32 no 2 pp 92ndash106 2005

[28] X P Yuan H Y Liu and Z Q Wang ldquoAn interacting crack-mechanics basedmodel for elastoplastic damagemodel of rock-likematerials under compressionrdquo International Journal of RockMechanics and Mining Sciences vol 58 pp 92ndash102 2013

[29] B Franois and C Dascalu ldquoA two-scale time-dependentdamage model based on non-planar growth of micro-cracksrdquoJournal of the Mechanics and Physics of Solids vol 58 no 11 pp1928ndash1946 2010

[30] V Palchik andYHHatzor ldquoCrack damage stress as a compositefunction of porosity and elasticmatrix stiffness in dolomites andlimestonesrdquo Engineering Geology vol 63 no 3-4 pp 233ndash2452002

[31] G Alfano S Marfia and E Sacco ldquoA cohesive damage-friction interface model accounting for water pressure on crack

propagationrdquo Computer Methods in Applied Mechanics andEngineering vol 196 no 1ndash3 pp 192ndash209 2006

[32] V Monchiet C Gruescu O Cazacu and D Kondo ldquoAmicromechanical approach of crack-induced damage inorthotropic media application to a brittle matrix compositerdquoEngineering Fracture Mechanics vol 83 pp 40ndash53 2012

[33] A Golshani Y Okui M Oda and T Takemura ldquoA microme-chanical model for brittle failure of rock and its relation tocrack growth observed in triaxial compression tests of graniterdquoMechanics of Materials vol 38 no 4 pp 287ndash303 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of