Research ArticleA Multiparameter Damage Constitutive Model forRock Based on Separation of Tension and Shear

YanHui Yuan12 and Ming Xiao12

1State Key Laboratory of Water Resources and Hydropower Engineering Science Wuhan University Wuhan 430072 China2Key Laboratory of Rock Mechanics in Hydraulic Structural Engineering Wuhan University Ministry of EducationWuhan 430072 China

Correspondence should be addressed to YanHui Yuan winfredwhueducn

Received 17 March 2015 Revised 3 August 2015 Accepted 6 August 2015

Academic Editor Marek Lefik

Copyright copy 2015 Y Yuan and M XiaoThis is an open access article distributed under the Creative CommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

By analysis of the microscopic damage mechanism of rock a multiparameter elastoplastic damage constitutive model whichconsiders damage mechanism of tension and shear is established A revised general form of elastoplastic damage model containingdamage internal variable of tensor form is derived by considering the hypothesis that damage strain is induced by the degenerationof elasticmodulusWith decomposition of plastic strain introduced the forms of tension damage variable and shear damage variableare derived based on which effects of tension and shear damage on materialrsquos stiffness and strength are considered simultaneouslyThrough the utilizing of Zienkiewicz-Pande criterion with tension limit the specific form of the multiparameter damage model isderived Numerical experiments show that the established model can simulate damage behavior of rock effectively

1 Introduction

Rock is a kind of multiphase and inhomogeneous materialwithmesoscopic discontinuities randomly distributedWhensubjected to loads discontinuities emerge and develop lead-ing to the degeneration of materialrsquos strength and stiffnessThe exact simulation of the damage behavior of rock requiresdefinition of damage variables based on statistic methodsand determination of evolution of those damage variablesaccording to specific physical background so as to establishmodels like system of microstructures [1] Owing to the com-plexity of mesoscopic structure of rock the establishmentand application of mesoscopic damage model can be difficultand time consuming Constructing a macroscopic damagemodel based on continuum hypothesis will be beneficial tothe simplification of problem and application in engineering

Many works have been done on damage property ofconcrete and geomaterials Salari et al [2] established atriaxial damage model of geomaterials accounting for tensiledamage Nguyen and Korsunsky [3] established an isotropicdamage model for concrete which addresses the relationbetween local and nonlocal parameters Li et al [4] intro-duced statistical method to describe the strain softening

behavior of rock For materials like rock the basic damagetypes in macroscopic view include tension shear and crush[5] of whom some more complicated damage forms can beviewed as superposition To describe the damage behaviorof these materials the classical damage model with onlyone parameter is not enough Damage internal variable intensor form with several independent parameters includedshould be introduced to simulate the mechanical behaviorof different damage types Many works have been donealong this approach Frantziskonis and Desai [6] establisheda model suitable for geologic materials by introducing atensor form damage variable to describe the structuralchanges in such materials Krajcinovic and Mastilovic [7]considered scalar second- fourth- and sixth-order tensorrepresentations of damage and evaluated the accuracy withwhich they approximate exact micromechanical solutionsTo simulate the damage behavior of rock it is needed torelate the damage internal variable to each damage form thusallowing each part of damage internal variable representingdifferent damage types to evolve according to specific lawsResende [8] suggested a rate-independent constitutive theoryfor concrete which consider shear damage and hydrostatic

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 821093 10 pageshttpdxdoiorg1011552015821093

2 Mathematical Problems in Engineering

tension damage this model divides stress into mean valuepart and deviatoric part and considers the damage criteriaand evolution laws respectively Jirasek [9 10] studied non-local constitutive models for damage and fracture processesof quasibrittle materials and explored a formulation withaveraging of the displacement field Lee and Fenves [11]establish a plastic damage model for concrete subjected tocyclic loading using the concepts of fracture-energy-baseddamage and stiffness degradation Li and Wu [12 13] followthe approach of stress splitting and derived a damage consti-tutive model for concrete in effective stress space based onenergy principal Ju [14] established an energy-based coupledelastoplastic damage theory by considering Helmholtz freeenergy featuring strain-split formulation which leads tomorerobust algorithm

Many works on damage model are dedicated towards thenotion of effective quantities making the establishment ofdamage state relatively indirect In this paper we follow theapproach of taking damage internal variable as a two-orderthree-dimension tensor but adopt the viewpoint raised byYing [15] that plastic deformation and damage of rock canbe treated uniformly as a macroscopic representation of theemergence development and accumulation of microscopicfaults within rock In this way a revised general form ofelastoplastic damage model for rock containing damagevariable of tensor form is derived By utilizing the assumptionof strain separation the tensor form damage variable isrelated to different damage forms in an intuitive way and theevolution laws of strength and stiffness for rock materials inthese damage states are described respectively

2 A Revised General Form ofElastoplastic Model with Damage Variablesof Tensor Form Considered

Continuum damage models are usually established by intro-ducing particular damage variables and evolution laws wherethe damage variables show to what extent damage developsNowadays what is wildly used in a large amount of literatureis the concept of effective stress in which take the scalardamage variable119863 for example119863 takes value 0 or 1 denotingno damage or full damage respectively and 1 minus 119863 can beseen as the damage coefficient a model containing damagevariable119863 can be established through the principal of equiva-lence (stress equivalence strain equivalence or energy equiv-alence) However when considered based on the intrinsicmechanism damage can together with plastic deformationbe seen as the result of emergence and development ofmicro-scopic faults that only differs from plasticity by the effectson strength and stiffness [15] In this section the results inreference [15] are generalized and derivation similar to plasticinternal variables is made about damage internal variables

As exposed in previous section rock damage has severalbasic types each affecting the strength and stiffness in dif-ferent way Similar to plastic internal variables introduce inconstitutive relation a damage internal variable Ω

119898119899(119898 119899 =

1 2 3) of tensor form which leads to

120590119894119895= 120590119894119895(120598119896119897 120585120573 Ω119898119899) (1)

where120590119894119895and 120598119896119897denote stress and strain components respec-

tively and 120585120573denotes plastic internal variable Differentiate

(1) with respect to time stress rate can be expressed as

119894119895= 119890

119894119895+ 119901

119894119895+ 119889

119894119895 (2)

where 119890119894119895= (120597120590

119894119895120597120598119896119897) 120598119896119897= 119863119894119895119896119897120598119896119897 119901119894119895= (120597120590

119894119895120597120585120573) 120585120573 and

119889

119894119895= (120597120590

119894119895120597Ω119898119899)Ω119898119899

each denotes elastic stress rate plasticstress rate and damage stress rate (Einsteinrsquos summationconvention is always assumed in this paper) here 119863

119894119895119896119897

denotes elastic stiffness For static problem there is no needto differentiate with respect to real time Similarly strain ratecan be expressed as

120598119894119895= 120598119890

119894119895+ 120598119901

119894119895+ 120598119889

119894119895 (3)

where 120598119890

119894119895= (120597120598119894119895120597120590119896119897)119896119897= 119862119894119895119896119897119896119897 120598119901119894119895= (120597120598119894119895120597120585120573) 120585120573 120598119889119894119895=

(120597120598119894119895120597Ω119898119899)Ω119898119899 and 119862

119894119895119896119897is the elastic flexibility

Assume 119891(120590119894119895 120585120573 Ω119898119899) and 119892(120598

119894119895 120585120573 Ω119898119899) are the bound

for plasticity and damage in stress and strain space In factshear damage criterion for rock material is often expressed asa form similar to plastic yield function (such as functions ofDrucker-Prager type) in application this justifies the unifiedtreatment for the judgment of plasticity and damage Formaterials satisfying Ilyushinrsquos postulate ∮120590

119894119895119889120598119894119895ge 0 similar

to the general theory for elastoplastic model a generalizedassociated flow law can be obtained by constructing a circularpath in strain space

119901

119894119895+ 119889

119894119895= minus

120597119892

120597120598119894119895

(120582 ge 0) (4)

Equation (4) is also suitable for softening material Similarly(4) can be extended to nonassociated materials Still treatdamage and plasticity in a unified way and it can be assumedthat

119901

119894119895+ 119889

119894119895= minus

120597119866

120597120598119894119895

(120582 ge 0) (5)

or

120598119901

119894119895+ 120598119889

119894119895=

120597119865

120597120598119894119895

(120582 ge 0) (6)

where 119865(120590119894119895 120585120573 Ω119898119899) and 119866(120598

119894119895 120585120573 Ω119898119899) denote plastic and

damage potential function in stress and strain space respec-tively

From the consistency condition in strain space that 119892 = 0it can be derived that 120598

119901

119894119895+ 120598119889

119894119895is proportional to 119892 when

loading namely 120598119901119894119895+ 120598119889

119894119895= 120598119894119895119892 where 120598

119894119895is the function of

strain and 119892 = (120597119892120597120598119898119899) 120598119898119899 More generally to take into

account situations when it is not loading introduceMacaulaybracket ⟨119909⟩ satisfying

⟨119909⟩ =

119909 (119909 ge 0)

0 (119909 lt 0)

(7)

it can be obtained that

120598119901

119894119895+ 120598119889

119894119895= 120598119894119895⟨119892⟩ (8)

Mathematical Problems in Engineering 3

Combine (6) and (8) it can be shown that 120598119901119894119895+ 120598119889

119894119895has direction

120597119865120597120598119894119895and is proportional to ⟨119892⟩ Thus it can be assumed

that

120598119901

119894119895+ 120598119889

119894119895= ]120597119865

120597120590119894119895

⟨119892⟩ (9)

and similarly for stress rate

119901

119894119895+ 119889

119894119895= minus]

120597119866

120597120590119896119897

⟨119892⟩ (10)

Let 119891 = (120597119891120597120590119901119902)119901119902

and 119892 = (120597119892120597120598119898119899) 120598119898119899 When loading

it can be obtained that 119892 = (120597119892120597120598119898119899)(119862119898119899119901119902

119901119902+ 120598119898119899119892) or

119891 = (1minus(120597119892120597120598119898119899)120598119898119899)119892 Suppose119891 and 119892 satisfy the relation

119891 = 120601119892 it can be derived that 120601 = 1 minus (120597119892120597120598119898119899)120598119898119899= 1 minus

(120597119892120597120598119894119895)](120597119865120597120590

119894119895) = 1minus]119867 where119867 = (120597119865120597120598

119894119895)(120597119892120597120598

119894119895) =

(120597119865120597120590119894119895)119863119894119895119896119897(120597119891120597120590

119896119897) When ] gt 0 we have 1] = 120601]+119867

Let ℎ = 120601] it can be obtained that ] = 1(119867+ℎ) and togetherwith (9) and (10) we get

119894119895= (119863

119894119895119896119897minus

1

119867 + ℎ

120597119866

120597120598119894119895

120597119892

120597120598119894119895

) 120598119896119897 (11)

120598119894119895= (119862119894119895119896119897+1

ℎ

120597119865

120597120590119894119895

120597119891

120597120590119896119897

) 119896119897

(ℎ = 0) (12)

Equations (11) and (12) have the same looking withgeneral elastoplastic relation but implicitly contain the effectof damage variable Ω

119898119899on rockrsquos strength and stiffness To

clarify it the specific form of ℎ is derived Introduce twohypothesizes raised in [15]

(1) Rockrsquos stiffness matrix degenerates as damage devel-ops Damage strain is induced by the degenera-tion of elastic stiffness matrix coefficients namely120597120598119894119895120597Ω119898119899= (120597119862

119894119895119896119897120597Ω119898119899)120590119896119897 Thus damage strain

rate can be defined as

120598119889

119894119895=

120597119862119894119895119896119897

120597Ω119898119899

120590119896119897Ω119898119899 (13)

(2) Ω119898119899

can be expressed as a homogeneous linear formof 120598119901119901119902 that is

Ω119898119899= 119872119898119899119901119902

120598119901

119901119902 (14)

where 119872119898119899119901119902

is the function of plastic strain anddamage variable

Substituting (13) and (14) into (12) leads to

120598119901

119894119895+

120597119862119894119895119896119897

120597Ω119898119899

120590119896119897119872119898119899119901119902

120598119901

119901119902=1

ℎ

120597119865

120597120590119894119895

120597119891

120597120590119896119897

119896119897 (15)

The equation above can be regarded as a system of linearequations about plastic strain rates which can be solved as

120598119901

119894119895=1

ℎ119870119894119895119896119897

120597119865

120597120590119894119895

120597119891

120597120590119898119899

119898119899 (16)

where

119870119894119895119896119897=1

2(120575119894119896120575119895119897+ 120575119894119897120575119895119896)

minus

(120597119862119894119895119898119899120597Ω119901119902) 120590119898119899119872119901119902119896119897

1 + (120597119862119904119905119898119899120597Ω119901119902) 120590119898119899119872119901119902119904119905

(17)

Substitute (16) into the consistency condition in stressspace

0 = 119891 =120597119891

120597120590119894119895

120590119894119895+120597119891

120597120585120573

120585120573+120597119891

Ω119896119897

Ω119896119897 (18)

Also consider that plastic internal variables 120585120573are usually

assumed as functions of plastic strain 120598119901119896119897 It can be derived

that

ℎ = minus119870119894119895119896119897

120597119865

120590119898119899

(120597119891

120585120573

120597120585120573

120597120598119901

119896119897

+120597119891

120597Ω119894119895

119872119894119895119896119897) (19)

Specifically suppose plastic internal variable satisfies 120585 =

radic(23) 120598119896119897120598119896119897 it can be obtained that

ℎ = minus120597119891

120597120585(2

3119861119894119895119861119894119895)

12

minus120597119891

120597Ω119894119895

119872119894119895119896119897119877119896119897 (20)

where 119861119894119895= 119870119894119895119896119897(120597119865120597120590

119896119897) In conclusion hypothesize (1)

implies the effect of damage on materialrsquos stiffness while(19) implies the effects on materialrsquos strength (plasticity anddamage bound surface)

Consider the Clausius-Duhem inequality

minus120588 ( + 120578 120579) + 120590119894119895120598119894119895minus1

120579119902119894

120597120579

120597119909119894

ge 0 (21)

where 120588 denotes density 119909 denotes spatial coordinates 120579denotes temperature 119902 denotes heat flow and120595 and 120578 denotefree energy and entropy per unit mass Similar to stress120595 canbe expressed as 120595 = 120595(120579 120598

119894119895 120585120573 Ω119898119899) and thus we have

=120597120595

120597120579

120579 +120597120595

120597120598119894119895

120598119894119895+120597120595

120597120585120573

120585120573+120597120595

120597Ω119898119899

Ω119898119899 (22)

Substituting (22) into (21) leads to

(120590119894119895minus 120588120597120595

120597120598119894119895

) 120598119894119895+120597120595

120597120585120573

120585120573+120597120595

120597Ω119898119899

Ω119898119899

minus 120588(120597120595

120597120579+ 120578) 120579 minus

1

120579119902119894

120597120579

120597119909119894

ge 0

(23)

Suppose the Coleman relation 120578 = minus120597120595120597120579 is satisfiedinequality (23) is valid if the inequalities

(120590119894119895minus 120588120597120595

120597120598119894119895

) 120598119894119895+120597120595

120597120585120573

120585120573+120597120595

120597Ω119898119899

Ω119898119899ge 0

1

120579119902119894

120597120579

120597119909119894

le 0

(24)

are satisfied which represent mechanical and thermal dissi-pation respectively

4 Mathematical Problems in Engineering

3 The Form of Damage Variables Based onSeparation of Tension and Shear

Tension and shear are the main types of damage for rockSimo and Ju [14 16] separate stress (strain) tensor into twodifferent parts and consider the damage effect respectivelyaccording to the assumption of separation of tension andshear Follow this approach and omit the effect of crushdamage variableΩ

119894119895can be reduced to a simpler form relating

to tension and shear independently Let119872119894119895119896119897

in (14) be

119872119894119895119896119897= 1205721119875119894119895119896119897+ 1205722119876119894119895119896119897 (25)

where

119875119894119895119896119897=

3

sum

120574=1

1198671(120598119901

120574) 119901120574

119894119901120574

119895119901120574

119896119901120574

119897

119876119894119895119896119897=

3

sum

120574=1

1198672(120598119901

120574) 119901120574

119894119901120574

119895119901120574

119896119901120574

119897

1198671(119909) =

1 119909 ge 0

0 119909 lt 0

1198672(119909) =

0 119909 ge 0

1 119909 lt 0

(26)

Here119901120574119894denotes cosine of the angle between the 120574th principal

direction of plastic strain and each coordinate axis and 1205721

1205722are dimensionless function about plastic strain invariables

Equation (25) can be seen as a separation of damage variableinto the tension and shear part In fact the principal form of120598119901

119894119895can be obtained when applied with 119872

119894119895119896119897which deviates

according to the sign of its principal value Here 1198671and

1198672in (26) can be seen as cut-off functions that reserve

only positive and negative part of strain respectively thusseparating damage variable into the tension and shear parteach part casts back to the component form

Substitute (25) into (14) and let 119867(119909) = 12057211198671(119909) +

12057221198672(119909) it can be obtained that

Ω119894119895=

3

sum

120574=1

119867(120598119901

120574) 119901120574

119894119901120574

119895119901120574

119896119901120574

119897120598119901

119896119897 (27)

Let

Ω119894119895=

3

sum

120574=1

120596120574119901120574

119894119901120574

119895 (28)

To simplify derivation omit derivatives of principal directionwith respect to time we have

120574= 119867(120598

119901

120574) 119901120574

119896119901120574

119897120598119901

119896119897 (29)

Damage internal variable Ω119894119895is thus reduced to three vari-

ables 120596120574(120574 = 1 2 3) each relating to the three plastic

principal strains respectively It is now possible to replace thedamage variableΩ

119894119895with its simplified form120596

120574 By using (28)

we have 120597119891120597120596120574= (120597119891120597Ω

119894119895)(120597Ω119894119895120597120596120574) = (120597119891120597Ω

119894119895)119901120574

119894119901120574

119895

The damage part in (19) can thus be expressed as

120597119891

120597Ω119894119895

119872119894119895119896119897=

3

sum

120574=1

120597119891

120597120596120574

119867(120598119901

120574) 119901120574

119896119901120574

119897 (30)

4 Multiparameter Damage Model Based onZienkiewicz-Pande Criterion

To expose the model further in this section Zienkiewicz-Pande criterion with tension limit is taken as an exampleand the specific form of the multiparameter damage model isderivedThe criteria currently used very often in research andengineering include Drucker-Prager Mohr-Coulomb Hoek-Brown and unified strength theory suggested by Yu et al[17] of which theMohr-Coulomb criterion is widely adoptedThe existence of vertex singularity in bound surface will bringsome difficulties in iteration Zienkiewicz-Pande criterion is asmooth version of Mohr-Coulomb criterion [18] which usesa quadratic curve to approximate straight line in 120587-plane soas to eliminate singular vertex and benefit numerical imple-mentation also it takes intermediate principal stress intoaccount to some extentThe Zienkiewicz-Pande criterion hasbeen used in many engineering projects for the simulationlike excavation of underground caverns and shows a goodagreement with monitoring results [19]

41 The Specific Form of Multiparameter Damage Model Indamage judgment plastic yield surface is often used as thebound for shear damage and stress or strain limit as the boundfor tensionThus in this section Zienkiewicz-Pande criterionwith tension limit is assumed as the bound for plasticity anddamage namely

119891119904= 120590119898sin120601 minus 119888 cos120601

+ radic1

21198882

1

(119903120590

119892120601(120579120590))

2

+ 1198862sin2120601(31a)

119891119905= 1205901minus 120590119905 (31b)

Here the hyperbolic type of approximation in [20] is adoptedthat is

1198882

1=

2radic3

3 minus sin120601

119892120601(120579120590) =

2119896120601

(1 + 119896120601) minus (1 minus 119896

120601) sin 3120579

120590

119896120601=3 minus sin1206013 + sin120601

(32)

where 119888 is the cohesion 120601 is the angle of internal friction120590119905is the tension limit 120590

119898is the mean stress 120590

1198961198963 120579120590is

Mathematical Problems in Engineering 5

lode angle 119903120590is lode radius namely radic2119869

2 11988821is the revising

coefficient introduced in [20] and 119886 controls to what extentapproximation can be made to the Mohr-Coulomb yieldsurface Here the damage type is determined by the firstbound the state reaches when loading Let potential functionbe

119865119904= 120590119898sin120595 minus 119888 cos120595

+ radic1

21198882

1

(119903120590

119892120595(120579120590))

2

+ 1198862sin2120595(33a)

119865119905= 119891119905 (33b)

where

119892120595(120579120590) =

2119896120595

(1 + 119896120595) minus (1 minus 119896

120595) sin 3120579

120590

119896120595=3 minus sin1205953 + sin120595

(34)

Here120595 denotes the angle of dilatancy Consider that 1199032120590= 21198692

sin3120579120590= minus(3radic32)(119869

311986932

2) 1205901= 120590119898+ (2radic33)119869

12

2sin(120579120590+

(23)120587) and 120597120590119898120597120590119894119895= (13)120575

119894119895 1205971198692120597120590119894119895= 119904119894119895 and

1205971198693120597120590119894119895= 119904119894119896119904119896119895minus (23)119869

2120575119894119895 it can be derived that

120597119891119904

120597120590119894119895

= 1198601120575119894119895+ 1198602119904119894119895+ 1198603119904119894119896119904119896119895 (35a)

120597119891119905

120597120590119894119895

= 1198604120575119894119895+ 1198605119904119894119895+ 1198606119904119894119896119904119896119895 (35b)

where

1198601=1

3sin120601 minus

radic3

1198882

11198923

120601(120579120590)11986912

211988711198872

1198602=

1

21198882

11198922

120601(120579120590)1198871minus

9radic3

41198882

11198923

120601(120579120590)

1198693

11986932

2

11988711198872

1198603=

3radic3

21198882

11198923

120601(120579120590)

1

11986912

2

11988711198872

1198604=1

3(1 + 2119887

3)

1198605=radic3

3

1

11986912

2

1198874+3

2

1198693

1198692

2

1198873

1198606= minus

1

1198692

1198873

(36a)

1198871= (

1

1198882

11198922 (120579120590)1198692+ 1198862sin2120601)

minus12

1198872=

2119896120601(1 minus 119896

120601)

((1 + 119896120601) minus (1 minus 119896

120601) sin 3120579

120590)2

1198873=cos (120579

120590+ (23) 120587)

1003816100381610038161003816cos 31205791205901003816100381610038161003816

1198874= sin(120579

120590+2

3120587)

(36b)

Consider that 120597119892120597120598119894119895= (120597119891120597120590

119896119897)(120590119896119897120598119894119895) = 119863

119894119895119896119897(120597119891120597120590

119896119897)

it is easy to get 120597119892120597120598119894119895accordingly Similarly 120597119865120597120590

119894119895and

120597119866120597120598119894119895can be obtained

Let elastic flexibility be an isotropic tensor

119862119894119895119896119897= minus120583

119864120575119894119895120575119896119897+1 + 120583

2119864(120575119894119896120575119895119897+ 120575119894119897120575119895119896) (37)

where 119864 denotes Youngrsquos modulus and 120583 denotes Poissonrsquosratio Suppose only Youngrsquos modulus is affected by damageinternal variables then similar to (30) we can get

120597119862119894119895119898119899

120597Ω119901119902

120590119898119899119872119901119902119896119897= 119870(1)

119894119895119870(2)

119896119897 (38a)

120597119862119904119905119898119899

120597Ω119901119902

120590119898119899119872119901119902119904119905= 119870(1)

119904119905119870(2)

119904119905 (38b)

where

119870(1)

119894119895=3120583

1198642120590119898120575119894119895minus2 (1 + 120583)

1198642120590119894119895

119870(2)

119896119897=

3

sum

120574=1

120597119864

120597120596120574

119867(120598119901

120574) 119901120574

119896119901120574

119897

(39)

Substitute (38a) and (38b) into (17) we can get 119870119894119895119896119897

whichtogether with (20) and (35a) and (35b) (also 120597119892120597120598

119894119895 120597119865120597120590

119894119895

and 120597119866120597120598119894119895which can be obtained in a similar approach)

leads to a constitutive relation in the form as (11) and (12)

42 The Damage Evolution of Material Strength and StiffnessMaterial strength and stiffness degenerate as damage evolvesHere those strength and stiffness parameters are treated asfunctions of the reduced form of damage variables 120596

120573and

their specific forms are examined Consider that damagevariable has a degenerating effect on cohesion and angle ofinternal friction it can be derived that

120597119891119904

120597120596120574

=120597119891119904

120597119888

120597119888

120597120596120574

+120597119891119904

120597120601

120597120601

120597120596120574

(for shearing) (40a)

Similarly consider the degenerating effect damage variablehas on tension limit we have

120597119891119905

120597120596120574

= minus120597120590119905

120597120596120574

(for tension) (40b)

6 Mathematical Problems in Engineering

Since Zienkiewicz-Pande yield surface is a smooth approx-imation to the Mohr-Coulomb yield surface and the Mohr-Coulomb criterion has the form

119888 =120590119888

2

1 minus sin120601cos120601 (41)

during uniaxial compression it can be assumed that thecohesion 119888 in Zienkiewicz-Pande criterion is proportional touniaxial compressive strength 120590

119888 Instead of plastic hardening

effect damage makes materialrsquos strength degenerate To sim-plify implementation omit the effect damage variable has onangle of internal friction and also consider the function typeadopted in [21] for the degeneration of materialrsquos parameterslet 119888 and 120590

119905be linear functions for damage variables

119888 =1 minus sin1206012 cos120601

(1205901198880minus 119864119904ℎ (120596120574)) (42a)

120590119905= 1205901199050minus 119864119905ℎ (120596120574) (42b)

where 1205901198880

and 1205901199050

denote the original compression andtension strength and 119864

119904and 119864

119905are damagemodulus for shear

and tension respectivelyAs internal microscopic cracks develop very fast after

reaching damage [22] exponential function is adopted

119864 = 1198640119890minus119877ℎ(120596120574) (43)

where1198640is the original Youngrsquosmodulus119877 ismaterialrsquos dam-

age coefficient and ℎ(120596120574) is taken as the linear combination

of damage variables 120596120574 Specifically let

ℎ (120596120574) =

max (120596120574) minusmin (120596

120574) (shear)

max (120596120574) (tension)

(44)

Given the irreversibility of damage during the damageevolution of materialrsquos parameters the minima of materialparameters along the evolution path are used

5 Numerical Example

The algorithm for solving group of nonlinear equationsincludes Newton-Raphson method modified Newton-Raphson method and quasi-Newton-Raphson method ofwhich Newton-Raphson method and modified Newton-Raphson method can effectively utilize the sparsity ofstiffness matrix Consider that the model raised in this paperhas many factors interacting with each other and thus highlynonlinearized a framework for nonlinear finite elementanalysis by using Newton-Raphson method is implementedwhich calls specific constitutive model to handle iterativesolving

Typical implementation ofmaterialmodel can be reducedto the following problems

(1) The establishment of tangent stiffness matrix whichcan be implemented using (11)

0

20

40

60

80

100

120

140

160

Stre

ss

0001 0002 0003 0004 0005 00060000Strain

Figure 1 Stress-strain curve under uniaxial compression (MPa)

(2) State update namely update of stress strain andmaterial parameters of each quadrature point whenstrain increment is known Typical implementationincludes explicit method (eg Runge-Kutta method)and implicit method (eg return map method [2324]) To simplify implementation explicit method isadopted which takes the incremental form of (11) andperform update and revision for state and materialparameters at the end of each increment step

To validate the effectiveness of the model establishedconsider the cylindrical sample in [25] Let 119864 = 3143GPa120583 = 023 119888 = 2622MPa and 120601 = 447∘ Apply uniaxialcompression to the sample model The stress-strain curve isdepicted in Figure 1 Comparison with the experiment resultof uniaxial compression in [25] shows that the establishedmodel can simulate this process well

Consider a cubic finite element model that each side haslength 1m Let 119864 = 317GPa 120583 = 026 119888 = 30MPa120601 = 464

∘ and 120590119905= 35MPa Apply cyclic compression

and tension uniaxial load The stress-strain curves are shownin Figure 2 When strain reaches 00018 and 00028 forcompression or when strain reaches 000016 and 00022 fortension unload stress to zero and then apply again Thetangent stiffness during unloading is smaller than elasticstiffness as the introduction of damage The model presentedin this paper can simulate damage behavior under cycliccompression and tension effectively

6 Engineering Application

In this section the established model is used for the excava-tion simulation of Zhenrsquoan underground power station

61 Project Overview and Numerical Modeling Zhenrsquoanpumped storage power station is located in Shanxi Provincein China with installed capacity of 1400MW (4 times 350MW)The underground powerhouse is made up of the main pow-erhouse main transform house tailrace surge chamber and

Mathematical Problems in Engineering 7

Table 1 Mechanical parameters for rock

Density (gcm3) Youngrsquos modulus (GPa) Poissonrsquos ratio Cohesion (MPa) Internal friction angle (∘) Tension limit (MPa)267 130 027 10 477 045

000

00

000

05

000

10

000

15

000

20

000

25

000

30

000

45

000

40

000

35

Strain

30

25

20

15

10

5

0

Stre

ss

(a) Compression

000

000

000

005

000

010

000

015

000

020

000

025

000

030

000

040

000

035

Strain

Stre

ss

40

35

30

25

20

15

10

05

00

(b) Tension

Figure 2 Stress-strain curve under cyclic uniaxial compression and tension (MPa)

several other tunnels connecting each cavern Mechanicalparameters for rock are shown in Table 1 Finite elementmodel shown in Figure 3 contains 480608 isoparametric 8-node elements with 109011 elements to be excavated dividedinto 10 stages 119909-axis of the model is perpendicular to lon-gitudinal axis of main powerhouse with a range of 31864m119910-axis is along the longitudinal axis with a range of 3896m119911-axis is along the vertical axis with a range of 5686m Initialstress is shown in Figure 4 It is obtained by inverse analysisbased on 4 gauging points near the carven and the resultshows that it is generated mainly by gravity The establishedmodel is used for excavation simulation of the undergroundpowerhouse

62 Result Analysis

621 Damage Zone Distribution To simulate the process ofexcavation divide the computation into 10 steps each witha layer shown in Figure 3 removed and let the iterationreach convergence from the unbalanced state Figure 5 givesthe development of damage zone where the third and sixthstage are the excavation of lateral caverns resulting insome finite elements returning from damage bound Whenexcavation is complete damage zone distribution of 2 unitsectionrsquos surrounding rocks is shown in Figure 6 Depth ofdamage zone reaches a maximum of 21m near the arch roofDepth of damage zone near side wall ranges from 184m to201m and ranges from 70m to 90m near main transformhouse Tension damage zone is mostly distributed near

the intersection of tunnels and caverns where release ofinitial stress along multiple direction makes it easy to crack

622 Stress Distribution After excavation stress distribu-tions of surrounding rock of each unit section are roughlysimilar The first and third principal stress of 2 unit sectionrsquossurrounding rocks are shown in Figure 7 the direction of thethird principal stress is almost vertical which is the same asthe direction of gravity and first principal stress is along theradial direction Near the intersection of each tunnel with themain powerhouse the third principal stress concentrates andreaches a maximum of minus20MPa First principal stress in themiddle part of 2 unit sectionrsquos side wall reaches a peak ofminus237MPa and third principal stress is around minus20MPa

623 Displacement Distribution What is shown in Figure 8is the displacement of 2 unit section Release of initialstress makes the displacement directing towards the cavernsDisplacement near the intersection of each cavern is relativelylarger than that of other areas In other areas displacementfield is distributed relatively uniformly After excavation 2unit sectionrsquos arch roof rsquos displacement resiles to 30mmMaximum displacement of main powerhousersquos upstream anddownstream side wall is 421mm and 355mm respectivelyThe values are 138mm and 196mm for main transformhouse

From the analysis above it is easy to know that thedamage zone stress and displacement distribution obtained

8 Mathematical Problems in Engineering

(a) (b)

Figure 3 Computation model (a) and excavation elements (b)S1

minus05

minus9

minus85

minus8

minus75

minus7

minus65

minus6

minus55

minus5

minus45

minus4

minus35

minus3

minus25

minus2

minus15

minus1

(a)

S3

minus1

minus2

minus3

minus4

minus5

minus6

minus7

minus8

minus9

minus10

minus11

minus12

minus13

(b)

Figure 4 Initial first (a) and third (b) principal stress (MPa)

using the model established in this paper are similar tothose using Zienkiewicz-Pande or Mohr-Coulomb criterionbecause of the form of shear damage bound adopted Theintroduction of damage mechanismmakes it more natural tofigure out the shear and tension damage area

7 Conclusion

This paper approaches based on the microscopic mechanismof damage for rock material and establishes in an intuitiveway a multiparameter elastoplastic damage model for rockthat is applicable to engineering With the assumption thatdamage comes into existence as the materialrsquos strength and

stiffness degenerate and that damage is interconnected withplastic deformation a revised general form for elastoplasticdamage model containing damage variable of tensor formis established By considering plastic strain separation theexpression of damage variable reflecting the damage mech-anism for shear and tension simultaneously is derived Byadopting Zienkiewicz-Pande criterion with tension limit asthe bound for plasticity and damage the specific form for thedamage model is derived and implemented

The model established in this paper is physically intuitiveand has relativelywell-based theoretical backgroundNumer-ical experiments and engineering application show that thismodel can reflect the damage behavior of rock effectively

Mathematical Problems in Engineering 9

1 2 3 4 5 6 7 8 9 10Stage

25

20

15

10

5

0

Volu

me o

f dam

age z

one

Shear damageTension damage

Figure 5 Volume of damage zone (105m3)

Sheardamage

Sheardamage

Resilience

Figure 6 Damage zone of 2 unit section

S1

minus05

minus75

minus7

minus65

minus6

minus55

minus5

minus45

minus4

minus35

minus3

minus25

minus2

minus15

minus1

0

(a)

minus2

minus36

minus34

minus32

minus30

minus28

minus26

minus24

minus22

minus20

minus18

minus16

minus14

minus12

minus10

minus8

minus6

minus4

S3

(b)

Figure 7 First and third principal stress of 2 unit section (MPa)

10 Mathematical Problems in Engineering

D

45

40

35

30

25

20

15

10

5

Figure 8 Displacement of 2 unit section (mm)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] Z-Y Ye L Hong X-L Liu and T-B Yin ldquoConstitutive modelof rock based onmicrostructures simulationrdquo Journal of CentralSouth University of Technology vol 15 no 2 pp 230ndash236 2008

[2] M R Salari S Saeb K J Willam S J Patchet and R C Car-rasco ldquoA coupled elastoplastic damage model for geomaterialsrdquoComputer Methods in Applied Mechanics and Engineering vol193 no 27-29 pp 2625ndash2643 2004

[3] G D Nguyen and A M Korsunsky ldquoDevelopment of anapproach to constitutive modelling of concrete isotropic dam-age coupled with plasticityrdquo International Journal of Solids andStructures vol 45 no 20 pp 5483ndash5501 2008

[4] X LiW-G Cao andY-H Su ldquoA statistical damage constitutivemodel for softening behavior of rocksrdquoEngineeringGeology vol143-144 pp 1ndash17 2012

[5] M Xiao ldquoAnalysis on stability of surrounding rock and damagefracture of concrete lining for high pressure branch pipesrdquo Jour-nal of Wuhan University of Hydraulic and Electric Engineeringno 6 pp 594ndash599 1995

[6] G Frantziskonis and C S Desai ldquoConstitutive model withstrain softeningrdquo International Journal of Solids and Structuresvol 23 no 6 pp 733ndash750 1987

[7] D Krajcinovic and S Mastilovic ldquoSome fundamental issues ofdamage mechanicsrdquo Mechanics of Materials vol 21 no 3 pp217ndash230 1995

[8] L Resende ldquoA Damage mechanics constitutive theory for theinelastic behaviour of concreterdquo Computer Methods in AppliedMechanics and Engineering vol 60 no 1 pp 57ndash93 1987

[9] M Jirasek ldquoNonlocal models for damage and fracture com-parison of approachesrdquo International Journal of Solids andStructures vol 35 no 31-32 pp 4133ndash4145 1998

[10] M Jirasek and S Marfia ldquoNon-local damage model based ondisplacement averagingrdquo International Journal for NumericalMethods in Engineering vol 63 no 1 pp 77ndash102 2005

[11] J Lee andG L Fenves ldquoPlastic-damagemodel for cyclic loadingof concrete structuresrdquo Journal of Engineering Mechanics vol124 no 8 pp 892ndash900 1998

[12] J Li and J Y Wu ldquoElastoplacstic damage constitutive modelfor concrete based on damage energy release rates part I basicformulationsrdquo China Civil Engineering Journal no 9 pp 14ndash202005

[13] J Y Wu and J Li ldquoElastoplastic damage constitutive modelfor concrete based on damage energy release rates part IInumerical algorithm and verificationsrdquo China Civil EngineeringJournal no 9 pp 21ndash27 2005

[14] J W Ju ldquoOn energy-based coupled elastoplastic damagetheories constitutive modeling and computational aspectsrdquoInternational Journal of Solids and Structures vol 25 no 7 pp803ndash833 1989

[15] Y Q Ying ldquoOn rock plasticity damage and their constitutiveformulationrdquo Scientia Geologica Sinica no 1 pp 63ndash70 1995

[16] J C Simo and J W Ju ldquoStrain- and stress-based continuumdamage models-I Formulationrdquo International Journal of Solidsand Structures vol 23 no 7 pp 821ndash840 1987

[17] M H Yu YW ZanW Fan J Zhao and Z Z Dong ldquoAnvancesin strength theory of rock in 20 centurymdash100 years in memoryof theMohr-Coulomb Strehgth theoryrdquo Chinese Journal of RockMechanics and Engineering no 5 pp 545ndash550 2000

[18] O Zienkiewicz and G Pande ldquoSome useful forms of isotropicyield surfaces for soil and rockmechanicsrdquo in Finite Elements inGeomechanics pp 179ndash198 Wiley Cambridge UK 1977

[19] M Xiao Study on numerical analysis method of stability andsupporting for underground caverns [PhD thesis] Wuhan Uni-versity Wuhan China 2002

[20] Y Liu A M Maniatty and H Antes ldquoInvestigation of aZienkiewiczndashPande yield surface and an elasticndashviscoplasticboundary element formulationrdquo Engineering Analysis withBoundary Elements vol 24 no 2 pp 207ndash211 2000

[21] Y-J Li D-L Zhang and B-G Liu ldquoDevelopment and ver-ification of strain-softening model considering deformationmodulus degradation in FLAC3Drdquo Rock and Soil Mechanicsvol 32 no 2 pp 647ndash652 2011

[22] M S Diederichs P K Kaiser and E Eberhardt ldquoDamageinitiation and propagation in hard rock during tunnelling andthe influence of near-face stress rotationrdquo International Journalof Rock Mechanics and Mining Sciences vol 41 no 5 pp 785ndash812 2004

[23] J Lee and G L Fenves ldquoA return-mapping algorithm forplastic-damage models 3-D and plane stress formulationrdquoInternational Journal for Numerical Methods in Engineering vol50 no 2 pp 487ndash506 2001

[24] J Huang Q Peng and M-X Chen ldquoAn improved return-map stress update algorithm for finite deformation analysisof general isotropic elastoplastic geomaterialsrdquo InternationalJournal for Numerical and Analytical Methods in Geomechanicsvol 38 no 6 pp 636ndash660 2014

[25] J Zhou X Yang W Fu et al ldquoExperimental test and fracturedamage mechanical characteristics of brittle rock under uniax-ial cyclic loading and unloading conditionsrdquo Chinese Journal ofRock Mechanics and Engineering vol 29 no 6 pp 1172ndash11832010

Submit your manuscripts athttpwwwhindawicom

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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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Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

tension damage this model divides stress into mean valuepart and deviatoric part and considers the damage criteriaand evolution laws respectively Jirasek [9 10] studied non-local constitutive models for damage and fracture processesof quasibrittle materials and explored a formulation withaveraging of the displacement field Lee and Fenves [11]establish a plastic damage model for concrete subjected tocyclic loading using the concepts of fracture-energy-baseddamage and stiffness degradation Li and Wu [12 13] followthe approach of stress splitting and derived a damage consti-tutive model for concrete in effective stress space based onenergy principal Ju [14] established an energy-based coupledelastoplastic damage theory by considering Helmholtz freeenergy featuring strain-split formulation which leads tomorerobust algorithm

Many works on damage model are dedicated towards thenotion of effective quantities making the establishment ofdamage state relatively indirect In this paper we follow theapproach of taking damage internal variable as a two-orderthree-dimension tensor but adopt the viewpoint raised byYing [15] that plastic deformation and damage of rock canbe treated uniformly as a macroscopic representation of theemergence development and accumulation of microscopicfaults within rock In this way a revised general form ofelastoplastic damage model for rock containing damagevariable of tensor form is derived By utilizing the assumptionof strain separation the tensor form damage variable isrelated to different damage forms in an intuitive way and theevolution laws of strength and stiffness for rock materials inthese damage states are described respectively

2 A Revised General Form ofElastoplastic Model with Damage Variablesof Tensor Form Considered

Continuum damage models are usually established by intro-ducing particular damage variables and evolution laws wherethe damage variables show to what extent damage developsNowadays what is wildly used in a large amount of literatureis the concept of effective stress in which take the scalardamage variable119863 for example119863 takes value 0 or 1 denotingno damage or full damage respectively and 1 minus 119863 can beseen as the damage coefficient a model containing damagevariable119863 can be established through the principal of equiva-lence (stress equivalence strain equivalence or energy equiv-alence) However when considered based on the intrinsicmechanism damage can together with plastic deformationbe seen as the result of emergence and development ofmicro-scopic faults that only differs from plasticity by the effectson strength and stiffness [15] In this section the results inreference [15] are generalized and derivation similar to plasticinternal variables is made about damage internal variables

As exposed in previous section rock damage has severalbasic types each affecting the strength and stiffness in dif-ferent way Similar to plastic internal variables introduce inconstitutive relation a damage internal variable Ω

119898119899(119898 119899 =

1 2 3) of tensor form which leads to

120590119894119895= 120590119894119895(120598119896119897 120585120573 Ω119898119899) (1)

where120590119894119895and 120598119896119897denote stress and strain components respec-

tively and 120585120573denotes plastic internal variable Differentiate

(1) with respect to time stress rate can be expressed as

119894119895= 119890

119894119895+ 119901

119894119895+ 119889

119894119895 (2)

where 119890119894119895= (120597120590

119894119895120597120598119896119897) 120598119896119897= 119863119894119895119896119897120598119896119897 119901119894119895= (120597120590

119894119895120597120585120573) 120585120573 and

119889

119894119895= (120597120590

119894119895120597Ω119898119899)Ω119898119899

each denotes elastic stress rate plasticstress rate and damage stress rate (Einsteinrsquos summationconvention is always assumed in this paper) here 119863

119894119895119896119897

denotes elastic stiffness For static problem there is no needto differentiate with respect to real time Similarly strain ratecan be expressed as

120598119894119895= 120598119890

119894119895+ 120598119901

119894119895+ 120598119889

119894119895 (3)

where 120598119890

119894119895= (120597120598119894119895120597120590119896119897)119896119897= 119862119894119895119896119897119896119897 120598119901119894119895= (120597120598119894119895120597120585120573) 120585120573 120598119889119894119895=

(120597120598119894119895120597Ω119898119899)Ω119898119899 and 119862

119894119895119896119897is the elastic flexibility

Assume 119891(120590119894119895 120585120573 Ω119898119899) and 119892(120598

119894119895 120585120573 Ω119898119899) are the bound

for plasticity and damage in stress and strain space In factshear damage criterion for rock material is often expressed asa form similar to plastic yield function (such as functions ofDrucker-Prager type) in application this justifies the unifiedtreatment for the judgment of plasticity and damage Formaterials satisfying Ilyushinrsquos postulate ∮120590

119894119895119889120598119894119895ge 0 similar

to the general theory for elastoplastic model a generalizedassociated flow law can be obtained by constructing a circularpath in strain space

119901

119894119895+ 119889

119894119895= minus

120597119892

120597120598119894119895

(120582 ge 0) (4)

Equation (4) is also suitable for softening material Similarly(4) can be extended to nonassociated materials Still treatdamage and plasticity in a unified way and it can be assumedthat

119901

119894119895+ 119889

119894119895= minus

120597119866

120597120598119894119895

(120582 ge 0) (5)

or

120598119901

119894119895+ 120598119889

119894119895=

120597119865

120597120598119894119895

(120582 ge 0) (6)

where 119865(120590119894119895 120585120573 Ω119898119899) and 119866(120598

119894119895 120585120573 Ω119898119899) denote plastic and

damage potential function in stress and strain space respec-tively

From the consistency condition in strain space that 119892 = 0it can be derived that 120598

119901

119894119895+ 120598119889

119894119895is proportional to 119892 when

loading namely 120598119901119894119895+ 120598119889

119894119895= 120598119894119895119892 where 120598

119894119895is the function of

strain and 119892 = (120597119892120597120598119898119899) 120598119898119899 More generally to take into

account situations when it is not loading introduceMacaulaybracket ⟨119909⟩ satisfying

⟨119909⟩ =

119909 (119909 ge 0)

0 (119909 lt 0)

(7)

it can be obtained that

120598119901

119894119895+ 120598119889

119894119895= 120598119894119895⟨119892⟩ (8)

Mathematical Problems in Engineering 3

Combine (6) and (8) it can be shown that 120598119901119894119895+ 120598119889

119894119895has direction

120597119865120597120598119894119895and is proportional to ⟨119892⟩ Thus it can be assumed

that

120598119901

119894119895+ 120598119889

119894119895= ]120597119865

120597120590119894119895

⟨119892⟩ (9)

and similarly for stress rate

119901

119894119895+ 119889

119894119895= minus]

120597119866

120597120590119896119897

⟨119892⟩ (10)

Let 119891 = (120597119891120597120590119901119902)119901119902

and 119892 = (120597119892120597120598119898119899) 120598119898119899 When loading

it can be obtained that 119892 = (120597119892120597120598119898119899)(119862119898119899119901119902

119901119902+ 120598119898119899119892) or

119891 = (1minus(120597119892120597120598119898119899)120598119898119899)119892 Suppose119891 and 119892 satisfy the relation

119891 = 120601119892 it can be derived that 120601 = 1 minus (120597119892120597120598119898119899)120598119898119899= 1 minus

(120597119892120597120598119894119895)](120597119865120597120590

119894119895) = 1minus]119867 where119867 = (120597119865120597120598

119894119895)(120597119892120597120598

119894119895) =

(120597119865120597120590119894119895)119863119894119895119896119897(120597119891120597120590

119896119897) When ] gt 0 we have 1] = 120601]+119867

Let ℎ = 120601] it can be obtained that ] = 1(119867+ℎ) and togetherwith (9) and (10) we get

119894119895= (119863

119894119895119896119897minus

1

119867 + ℎ

120597119866

120597120598119894119895

120597119892

120597120598119894119895

) 120598119896119897 (11)

120598119894119895= (119862119894119895119896119897+1

ℎ

120597119865

120597120590119894119895

120597119891

120597120590119896119897

) 119896119897

(ℎ = 0) (12)

Equations (11) and (12) have the same looking withgeneral elastoplastic relation but implicitly contain the effectof damage variable Ω

119898119899on rockrsquos strength and stiffness To

clarify it the specific form of ℎ is derived Introduce twohypothesizes raised in [15]

(1) Rockrsquos stiffness matrix degenerates as damage devel-ops Damage strain is induced by the degenera-tion of elastic stiffness matrix coefficients namely120597120598119894119895120597Ω119898119899= (120597119862

119894119895119896119897120597Ω119898119899)120590119896119897 Thus damage strain

rate can be defined as

120598119889

119894119895=

120597119862119894119895119896119897

120597Ω119898119899

120590119896119897Ω119898119899 (13)

(2) Ω119898119899

can be expressed as a homogeneous linear formof 120598119901119901119902 that is

Ω119898119899= 119872119898119899119901119902

120598119901

119901119902 (14)

where 119872119898119899119901119902

is the function of plastic strain anddamage variable

Substituting (13) and (14) into (12) leads to

120598119901

119894119895+

120597119862119894119895119896119897

120597Ω119898119899

120590119896119897119872119898119899119901119902

120598119901

119901119902=1

ℎ

120597119865

120597120590119894119895

120597119891

120597120590119896119897

119896119897 (15)

The equation above can be regarded as a system of linearequations about plastic strain rates which can be solved as

120598119901

119894119895=1

ℎ119870119894119895119896119897

120597119865

120597120590119894119895

120597119891

120597120590119898119899

119898119899 (16)

where

119870119894119895119896119897=1

2(120575119894119896120575119895119897+ 120575119894119897120575119895119896)

minus

(120597119862119894119895119898119899120597Ω119901119902) 120590119898119899119872119901119902119896119897

1 + (120597119862119904119905119898119899120597Ω119901119902) 120590119898119899119872119901119902119904119905

(17)

Substitute (16) into the consistency condition in stressspace

0 = 119891 =120597119891

120597120590119894119895

120590119894119895+120597119891

120597120585120573

120585120573+120597119891

Ω119896119897

Ω119896119897 (18)

Also consider that plastic internal variables 120585120573are usually

assumed as functions of plastic strain 120598119901119896119897 It can be derived

that

ℎ = minus119870119894119895119896119897

120597119865

120590119898119899

(120597119891

120585120573

120597120585120573

120597120598119901

119896119897

+120597119891

120597Ω119894119895

119872119894119895119896119897) (19)

Specifically suppose plastic internal variable satisfies 120585 =

radic(23) 120598119896119897120598119896119897 it can be obtained that

ℎ = minus120597119891

120597120585(2

3119861119894119895119861119894119895)

12

minus120597119891

120597Ω119894119895

119872119894119895119896119897119877119896119897 (20)

where 119861119894119895= 119870119894119895119896119897(120597119865120597120590

119896119897) In conclusion hypothesize (1)

implies the effect of damage on materialrsquos stiffness while(19) implies the effects on materialrsquos strength (plasticity anddamage bound surface)

Consider the Clausius-Duhem inequality

minus120588 ( + 120578 120579) + 120590119894119895120598119894119895minus1

120579119902119894

120597120579

120597119909119894

ge 0 (21)

where 120588 denotes density 119909 denotes spatial coordinates 120579denotes temperature 119902 denotes heat flow and120595 and 120578 denotefree energy and entropy per unit mass Similar to stress120595 canbe expressed as 120595 = 120595(120579 120598

119894119895 120585120573 Ω119898119899) and thus we have

=120597120595

120597120579

120579 +120597120595

120597120598119894119895

120598119894119895+120597120595

120597120585120573

120585120573+120597120595

120597Ω119898119899

Ω119898119899 (22)

Substituting (22) into (21) leads to

(120590119894119895minus 120588120597120595

120597120598119894119895

) 120598119894119895+120597120595

120597120585120573

120585120573+120597120595

120597Ω119898119899

Ω119898119899

minus 120588(120597120595

120597120579+ 120578) 120579 minus

1

120579119902119894

120597120579

120597119909119894

ge 0

(23)

Suppose the Coleman relation 120578 = minus120597120595120597120579 is satisfiedinequality (23) is valid if the inequalities

(120590119894119895minus 120588120597120595

120597120598119894119895

) 120598119894119895+120597120595

120597120585120573

120585120573+120597120595

120597Ω119898119899

Ω119898119899ge 0

1

120579119902119894

120597120579

120597119909119894

le 0

(24)

are satisfied which represent mechanical and thermal dissi-pation respectively

4 Mathematical Problems in Engineering

3 The Form of Damage Variables Based onSeparation of Tension and Shear

Tension and shear are the main types of damage for rockSimo and Ju [14 16] separate stress (strain) tensor into twodifferent parts and consider the damage effect respectivelyaccording to the assumption of separation of tension andshear Follow this approach and omit the effect of crushdamage variableΩ

119894119895can be reduced to a simpler form relating

to tension and shear independently Let119872119894119895119896119897

in (14) be

119872119894119895119896119897= 1205721119875119894119895119896119897+ 1205722119876119894119895119896119897 (25)

where

119875119894119895119896119897=

3

sum

120574=1

1198671(120598119901

120574) 119901120574

119894119901120574

119895119901120574

119896119901120574

119897

119876119894119895119896119897=

3

sum

120574=1

1198672(120598119901

120574) 119901120574

119894119901120574

119895119901120574

119896119901120574

119897

1198671(119909) =

1 119909 ge 0

0 119909 lt 0

1198672(119909) =

0 119909 ge 0

1 119909 lt 0

(26)

Here119901120574119894denotes cosine of the angle between the 120574th principal

direction of plastic strain and each coordinate axis and 1205721

1205722are dimensionless function about plastic strain invariables

Equation (25) can be seen as a separation of damage variableinto the tension and shear part In fact the principal form of120598119901

119894119895can be obtained when applied with 119872

119894119895119896119897which deviates

according to the sign of its principal value Here 1198671and

1198672in (26) can be seen as cut-off functions that reserve

only positive and negative part of strain respectively thusseparating damage variable into the tension and shear parteach part casts back to the component form

Substitute (25) into (14) and let 119867(119909) = 12057211198671(119909) +

12057221198672(119909) it can be obtained that

Ω119894119895=

3

sum

120574=1

119867(120598119901

120574) 119901120574

119894119901120574

119895119901120574

119896119901120574

119897120598119901

119896119897 (27)

Let

Ω119894119895=

3

sum

120574=1

120596120574119901120574

119894119901120574

119895 (28)

To simplify derivation omit derivatives of principal directionwith respect to time we have

120574= 119867(120598

119901

120574) 119901120574

119896119901120574

119897120598119901

119896119897 (29)

Damage internal variable Ω119894119895is thus reduced to three vari-

ables 120596120574(120574 = 1 2 3) each relating to the three plastic

principal strains respectively It is now possible to replace thedamage variableΩ

119894119895with its simplified form120596

120574 By using (28)

we have 120597119891120597120596120574= (120597119891120597Ω

119894119895)(120597Ω119894119895120597120596120574) = (120597119891120597Ω

119894119895)119901120574

119894119901120574

119895

The damage part in (19) can thus be expressed as

120597119891

120597Ω119894119895

119872119894119895119896119897=

3

sum

120574=1

120597119891

120597120596120574

119867(120598119901

120574) 119901120574

119896119901120574

119897 (30)

4 Multiparameter Damage Model Based onZienkiewicz-Pande Criterion

To expose the model further in this section Zienkiewicz-Pande criterion with tension limit is taken as an exampleand the specific form of the multiparameter damage model isderivedThe criteria currently used very often in research andengineering include Drucker-Prager Mohr-Coulomb Hoek-Brown and unified strength theory suggested by Yu et al[17] of which theMohr-Coulomb criterion is widely adoptedThe existence of vertex singularity in bound surface will bringsome difficulties in iteration Zienkiewicz-Pande criterion is asmooth version of Mohr-Coulomb criterion [18] which usesa quadratic curve to approximate straight line in 120587-plane soas to eliminate singular vertex and benefit numerical imple-mentation also it takes intermediate principal stress intoaccount to some extentThe Zienkiewicz-Pande criterion hasbeen used in many engineering projects for the simulationlike excavation of underground caverns and shows a goodagreement with monitoring results [19]

41 The Specific Form of Multiparameter Damage Model Indamage judgment plastic yield surface is often used as thebound for shear damage and stress or strain limit as the boundfor tensionThus in this section Zienkiewicz-Pande criterionwith tension limit is assumed as the bound for plasticity anddamage namely

119891119904= 120590119898sin120601 minus 119888 cos120601

+ radic1

21198882

1

(119903120590

119892120601(120579120590))

2

+ 1198862sin2120601(31a)

119891119905= 1205901minus 120590119905 (31b)

Here the hyperbolic type of approximation in [20] is adoptedthat is

1198882

1=

2radic3

3 minus sin120601

119892120601(120579120590) =

2119896120601

(1 + 119896120601) minus (1 minus 119896

120601) sin 3120579

120590

119896120601=3 minus sin1206013 + sin120601

(32)

where 119888 is the cohesion 120601 is the angle of internal friction120590119905is the tension limit 120590

119898is the mean stress 120590

1198961198963 120579120590is

Mathematical Problems in Engineering 5

lode angle 119903120590is lode radius namely radic2119869

2 11988821is the revising

coefficient introduced in [20] and 119886 controls to what extentapproximation can be made to the Mohr-Coulomb yieldsurface Here the damage type is determined by the firstbound the state reaches when loading Let potential functionbe

119865119904= 120590119898sin120595 minus 119888 cos120595

+ radic1

21198882

1

(119903120590

119892120595(120579120590))

2

+ 1198862sin2120595(33a)

119865119905= 119891119905 (33b)

where

119892120595(120579120590) =

2119896120595

(1 + 119896120595) minus (1 minus 119896

120595) sin 3120579

120590

119896120595=3 minus sin1205953 + sin120595

(34)

Here120595 denotes the angle of dilatancy Consider that 1199032120590= 21198692

sin3120579120590= minus(3radic32)(119869

311986932

2) 1205901= 120590119898+ (2radic33)119869

12

2sin(120579120590+

(23)120587) and 120597120590119898120597120590119894119895= (13)120575

119894119895 1205971198692120597120590119894119895= 119904119894119895 and

1205971198693120597120590119894119895= 119904119894119896119904119896119895minus (23)119869

2120575119894119895 it can be derived that

120597119891119904

120597120590119894119895

= 1198601120575119894119895+ 1198602119904119894119895+ 1198603119904119894119896119904119896119895 (35a)

120597119891119905

120597120590119894119895

= 1198604120575119894119895+ 1198605119904119894119895+ 1198606119904119894119896119904119896119895 (35b)

where

1198601=1

3sin120601 minus

radic3

1198882

11198923

120601(120579120590)11986912

211988711198872

1198602=

1

21198882

11198922

120601(120579120590)1198871minus

9radic3

41198882

11198923

120601(120579120590)

1198693

11986932

2

11988711198872

1198603=

3radic3

21198882

11198923

120601(120579120590)

1

11986912

2

11988711198872

1198604=1

3(1 + 2119887

3)

1198605=radic3

3

1

11986912

2

1198874+3

2

1198693

1198692

2

1198873

1198606= minus

1

1198692

1198873

(36a)

1198871= (

1

1198882

11198922 (120579120590)1198692+ 1198862sin2120601)

minus12

1198872=

2119896120601(1 minus 119896

120601)

((1 + 119896120601) minus (1 minus 119896

120601) sin 3120579

120590)2

1198873=cos (120579

120590+ (23) 120587)

1003816100381610038161003816cos 31205791205901003816100381610038161003816

1198874= sin(120579

120590+2

3120587)

(36b)

Consider that 120597119892120597120598119894119895= (120597119891120597120590

119896119897)(120590119896119897120598119894119895) = 119863

119894119895119896119897(120597119891120597120590

119896119897)

it is easy to get 120597119892120597120598119894119895accordingly Similarly 120597119865120597120590

119894119895and

120597119866120597120598119894119895can be obtained

Let elastic flexibility be an isotropic tensor

119862119894119895119896119897= minus120583

119864120575119894119895120575119896119897+1 + 120583

2119864(120575119894119896120575119895119897+ 120575119894119897120575119895119896) (37)

where 119864 denotes Youngrsquos modulus and 120583 denotes Poissonrsquosratio Suppose only Youngrsquos modulus is affected by damageinternal variables then similar to (30) we can get

120597119862119894119895119898119899

120597Ω119901119902

120590119898119899119872119901119902119896119897= 119870(1)

119894119895119870(2)

119896119897 (38a)

120597119862119904119905119898119899

120597Ω119901119902

120590119898119899119872119901119902119904119905= 119870(1)

119904119905119870(2)

119904119905 (38b)

where

119870(1)

119894119895=3120583

1198642120590119898120575119894119895minus2 (1 + 120583)

1198642120590119894119895

119870(2)

119896119897=

3

sum

120574=1

120597119864

120597120596120574

119867(120598119901

120574) 119901120574

119896119901120574

119897

(39)

Substitute (38a) and (38b) into (17) we can get 119870119894119895119896119897

whichtogether with (20) and (35a) and (35b) (also 120597119892120597120598

119894119895 120597119865120597120590

119894119895

and 120597119866120597120598119894119895which can be obtained in a similar approach)

leads to a constitutive relation in the form as (11) and (12)

42 The Damage Evolution of Material Strength and StiffnessMaterial strength and stiffness degenerate as damage evolvesHere those strength and stiffness parameters are treated asfunctions of the reduced form of damage variables 120596

120573and

their specific forms are examined Consider that damagevariable has a degenerating effect on cohesion and angle ofinternal friction it can be derived that

120597119891119904

120597120596120574

=120597119891119904

120597119888

120597119888

120597120596120574

+120597119891119904

120597120601

120597120601

120597120596120574

(for shearing) (40a)

Similarly consider the degenerating effect damage variablehas on tension limit we have

120597119891119905

120597120596120574

= minus120597120590119905

120597120596120574

(for tension) (40b)

6 Mathematical Problems in Engineering

Since Zienkiewicz-Pande yield surface is a smooth approx-imation to the Mohr-Coulomb yield surface and the Mohr-Coulomb criterion has the form

119888 =120590119888

2

1 minus sin120601cos120601 (41)

during uniaxial compression it can be assumed that thecohesion 119888 in Zienkiewicz-Pande criterion is proportional touniaxial compressive strength 120590

119888 Instead of plastic hardening

effect damage makes materialrsquos strength degenerate To sim-plify implementation omit the effect damage variable has onangle of internal friction and also consider the function typeadopted in [21] for the degeneration of materialrsquos parameterslet 119888 and 120590

119905be linear functions for damage variables

119888 =1 minus sin1206012 cos120601

(1205901198880minus 119864119904ℎ (120596120574)) (42a)

120590119905= 1205901199050minus 119864119905ℎ (120596120574) (42b)

where 1205901198880

and 1205901199050

denote the original compression andtension strength and 119864

119904and 119864

119905are damagemodulus for shear

and tension respectivelyAs internal microscopic cracks develop very fast after

reaching damage [22] exponential function is adopted

119864 = 1198640119890minus119877ℎ(120596120574) (43)

where1198640is the original Youngrsquosmodulus119877 ismaterialrsquos dam-

age coefficient and ℎ(120596120574) is taken as the linear combination

of damage variables 120596120574 Specifically let

ℎ (120596120574) =

max (120596120574) minusmin (120596

120574) (shear)

max (120596120574) (tension)

(44)

Given the irreversibility of damage during the damageevolution of materialrsquos parameters the minima of materialparameters along the evolution path are used

5 Numerical Example

The algorithm for solving group of nonlinear equationsincludes Newton-Raphson method modified Newton-Raphson method and quasi-Newton-Raphson method ofwhich Newton-Raphson method and modified Newton-Raphson method can effectively utilize the sparsity ofstiffness matrix Consider that the model raised in this paperhas many factors interacting with each other and thus highlynonlinearized a framework for nonlinear finite elementanalysis by using Newton-Raphson method is implementedwhich calls specific constitutive model to handle iterativesolving

Typical implementation ofmaterialmodel can be reducedto the following problems

(1) The establishment of tangent stiffness matrix whichcan be implemented using (11)

0

20

40

60

80

100

120

140

160

Stre

ss

0001 0002 0003 0004 0005 00060000Strain

Figure 1 Stress-strain curve under uniaxial compression (MPa)

(2) State update namely update of stress strain andmaterial parameters of each quadrature point whenstrain increment is known Typical implementationincludes explicit method (eg Runge-Kutta method)and implicit method (eg return map method [2324]) To simplify implementation explicit method isadopted which takes the incremental form of (11) andperform update and revision for state and materialparameters at the end of each increment step

To validate the effectiveness of the model establishedconsider the cylindrical sample in [25] Let 119864 = 3143GPa120583 = 023 119888 = 2622MPa and 120601 = 447∘ Apply uniaxialcompression to the sample model The stress-strain curve isdepicted in Figure 1 Comparison with the experiment resultof uniaxial compression in [25] shows that the establishedmodel can simulate this process well

Consider a cubic finite element model that each side haslength 1m Let 119864 = 317GPa 120583 = 026 119888 = 30MPa120601 = 464

∘ and 120590119905= 35MPa Apply cyclic compression

and tension uniaxial load The stress-strain curves are shownin Figure 2 When strain reaches 00018 and 00028 forcompression or when strain reaches 000016 and 00022 fortension unload stress to zero and then apply again Thetangent stiffness during unloading is smaller than elasticstiffness as the introduction of damage The model presentedin this paper can simulate damage behavior under cycliccompression and tension effectively

6 Engineering Application

In this section the established model is used for the excava-tion simulation of Zhenrsquoan underground power station

61 Project Overview and Numerical Modeling Zhenrsquoanpumped storage power station is located in Shanxi Provincein China with installed capacity of 1400MW (4 times 350MW)The underground powerhouse is made up of the main pow-erhouse main transform house tailrace surge chamber and

Mathematical Problems in Engineering 7

Table 1 Mechanical parameters for rock

Density (gcm3) Youngrsquos modulus (GPa) Poissonrsquos ratio Cohesion (MPa) Internal friction angle (∘) Tension limit (MPa)267 130 027 10 477 045

000

00

000

05

000

10

000

15

000

20

000

25

000

30

000

45

000

40

000

35

Strain

30

25

20

15

10

5

0

Stre

ss

(a) Compression

000

000

000

005

000

010

000

015

000

020

000

025

000

030

000

040

000

035

Strain

Stre

ss

40

35

30

25

20

15

10

05

00

(b) Tension

Figure 2 Stress-strain curve under cyclic uniaxial compression and tension (MPa)

several other tunnels connecting each cavern Mechanicalparameters for rock are shown in Table 1 Finite elementmodel shown in Figure 3 contains 480608 isoparametric 8-node elements with 109011 elements to be excavated dividedinto 10 stages 119909-axis of the model is perpendicular to lon-gitudinal axis of main powerhouse with a range of 31864m119910-axis is along the longitudinal axis with a range of 3896m119911-axis is along the vertical axis with a range of 5686m Initialstress is shown in Figure 4 It is obtained by inverse analysisbased on 4 gauging points near the carven and the resultshows that it is generated mainly by gravity The establishedmodel is used for excavation simulation of the undergroundpowerhouse

62 Result Analysis

621 Damage Zone Distribution To simulate the process ofexcavation divide the computation into 10 steps each witha layer shown in Figure 3 removed and let the iterationreach convergence from the unbalanced state Figure 5 givesthe development of damage zone where the third and sixthstage are the excavation of lateral caverns resulting insome finite elements returning from damage bound Whenexcavation is complete damage zone distribution of 2 unitsectionrsquos surrounding rocks is shown in Figure 6 Depth ofdamage zone reaches a maximum of 21m near the arch roofDepth of damage zone near side wall ranges from 184m to201m and ranges from 70m to 90m near main transformhouse Tension damage zone is mostly distributed near

the intersection of tunnels and caverns where release ofinitial stress along multiple direction makes it easy to crack

622 Stress Distribution After excavation stress distribu-tions of surrounding rock of each unit section are roughlysimilar The first and third principal stress of 2 unit sectionrsquossurrounding rocks are shown in Figure 7 the direction of thethird principal stress is almost vertical which is the same asthe direction of gravity and first principal stress is along theradial direction Near the intersection of each tunnel with themain powerhouse the third principal stress concentrates andreaches a maximum of minus20MPa First principal stress in themiddle part of 2 unit sectionrsquos side wall reaches a peak ofminus237MPa and third principal stress is around minus20MPa

623 Displacement Distribution What is shown in Figure 8is the displacement of 2 unit section Release of initialstress makes the displacement directing towards the cavernsDisplacement near the intersection of each cavern is relativelylarger than that of other areas In other areas displacementfield is distributed relatively uniformly After excavation 2unit sectionrsquos arch roof rsquos displacement resiles to 30mmMaximum displacement of main powerhousersquos upstream anddownstream side wall is 421mm and 355mm respectivelyThe values are 138mm and 196mm for main transformhouse

From the analysis above it is easy to know that thedamage zone stress and displacement distribution obtained

8 Mathematical Problems in Engineering

(a) (b)

Figure 3 Computation model (a) and excavation elements (b)S1

minus05

minus9

minus85

minus8

minus75

minus7

minus65

minus6

minus55

minus5

minus45

minus4

minus35

minus3

minus25

minus2

minus15

minus1

(a)

S3

minus1

minus2

minus3

minus4

minus5

minus6

minus7

minus8

minus9

minus10

minus11

minus12

minus13

(b)

Figure 4 Initial first (a) and third (b) principal stress (MPa)

using the model established in this paper are similar tothose using Zienkiewicz-Pande or Mohr-Coulomb criterionbecause of the form of shear damage bound adopted Theintroduction of damage mechanismmakes it more natural tofigure out the shear and tension damage area

7 Conclusion

This paper approaches based on the microscopic mechanismof damage for rock material and establishes in an intuitiveway a multiparameter elastoplastic damage model for rockthat is applicable to engineering With the assumption thatdamage comes into existence as the materialrsquos strength and

stiffness degenerate and that damage is interconnected withplastic deformation a revised general form for elastoplasticdamage model containing damage variable of tensor formis established By considering plastic strain separation theexpression of damage variable reflecting the damage mech-anism for shear and tension simultaneously is derived Byadopting Zienkiewicz-Pande criterion with tension limit asthe bound for plasticity and damage the specific form for thedamage model is derived and implemented

The model established in this paper is physically intuitiveand has relativelywell-based theoretical backgroundNumer-ical experiments and engineering application show that thismodel can reflect the damage behavior of rock effectively

Mathematical Problems in Engineering 9

1 2 3 4 5 6 7 8 9 10Stage

25

20

15

10

5

0

Volu

me o

f dam

age z

one

Shear damageTension damage

Figure 5 Volume of damage zone (105m3)

Sheardamage

Sheardamage

Resilience

Figure 6 Damage zone of 2 unit section

S1

minus05

minus75

minus7

minus65

minus6

minus55

minus5

minus45

minus4

minus35

minus3

minus25

minus2

minus15

minus1

0

(a)

minus2

minus36

minus34

minus32

minus30

minus28

minus26

minus24

minus22

minus20

minus18

minus16

minus14

minus12

minus10

minus8

minus6

minus4

S3

(b)

Figure 7 First and third principal stress of 2 unit section (MPa)

10 Mathematical Problems in Engineering

D

45

40

35

30

25

20

15

10

5

Figure 8 Displacement of 2 unit section (mm)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] Z-Y Ye L Hong X-L Liu and T-B Yin ldquoConstitutive modelof rock based onmicrostructures simulationrdquo Journal of CentralSouth University of Technology vol 15 no 2 pp 230ndash236 2008

[2] M R Salari S Saeb K J Willam S J Patchet and R C Car-rasco ldquoA coupled elastoplastic damage model for geomaterialsrdquoComputer Methods in Applied Mechanics and Engineering vol193 no 27-29 pp 2625ndash2643 2004

[3] G D Nguyen and A M Korsunsky ldquoDevelopment of anapproach to constitutive modelling of concrete isotropic dam-age coupled with plasticityrdquo International Journal of Solids andStructures vol 45 no 20 pp 5483ndash5501 2008

[4] X LiW-G Cao andY-H Su ldquoA statistical damage constitutivemodel for softening behavior of rocksrdquoEngineeringGeology vol143-144 pp 1ndash17 2012

[5] M Xiao ldquoAnalysis on stability of surrounding rock and damagefracture of concrete lining for high pressure branch pipesrdquo Jour-nal of Wuhan University of Hydraulic and Electric Engineeringno 6 pp 594ndash599 1995

[6] G Frantziskonis and C S Desai ldquoConstitutive model withstrain softeningrdquo International Journal of Solids and Structuresvol 23 no 6 pp 733ndash750 1987

[7] D Krajcinovic and S Mastilovic ldquoSome fundamental issues ofdamage mechanicsrdquo Mechanics of Materials vol 21 no 3 pp217ndash230 1995

[8] L Resende ldquoA Damage mechanics constitutive theory for theinelastic behaviour of concreterdquo Computer Methods in AppliedMechanics and Engineering vol 60 no 1 pp 57ndash93 1987

[9] M Jirasek ldquoNonlocal models for damage and fracture com-parison of approachesrdquo International Journal of Solids andStructures vol 35 no 31-32 pp 4133ndash4145 1998

[10] M Jirasek and S Marfia ldquoNon-local damage model based ondisplacement averagingrdquo International Journal for NumericalMethods in Engineering vol 63 no 1 pp 77ndash102 2005

[11] J Lee andG L Fenves ldquoPlastic-damagemodel for cyclic loadingof concrete structuresrdquo Journal of Engineering Mechanics vol124 no 8 pp 892ndash900 1998

[12] J Li and J Y Wu ldquoElastoplacstic damage constitutive modelfor concrete based on damage energy release rates part I basicformulationsrdquo China Civil Engineering Journal no 9 pp 14ndash202005

[13] J Y Wu and J Li ldquoElastoplastic damage constitutive modelfor concrete based on damage energy release rates part IInumerical algorithm and verificationsrdquo China Civil EngineeringJournal no 9 pp 21ndash27 2005

[14] J W Ju ldquoOn energy-based coupled elastoplastic damagetheories constitutive modeling and computational aspectsrdquoInternational Journal of Solids and Structures vol 25 no 7 pp803ndash833 1989

[15] Y Q Ying ldquoOn rock plasticity damage and their constitutiveformulationrdquo Scientia Geologica Sinica no 1 pp 63ndash70 1995

[16] J C Simo and J W Ju ldquoStrain- and stress-based continuumdamage models-I Formulationrdquo International Journal of Solidsand Structures vol 23 no 7 pp 821ndash840 1987

[17] M H Yu YW ZanW Fan J Zhao and Z Z Dong ldquoAnvancesin strength theory of rock in 20 centurymdash100 years in memoryof theMohr-Coulomb Strehgth theoryrdquo Chinese Journal of RockMechanics and Engineering no 5 pp 545ndash550 2000

[18] O Zienkiewicz and G Pande ldquoSome useful forms of isotropicyield surfaces for soil and rockmechanicsrdquo in Finite Elements inGeomechanics pp 179ndash198 Wiley Cambridge UK 1977

[19] M Xiao Study on numerical analysis method of stability andsupporting for underground caverns [PhD thesis] Wuhan Uni-versity Wuhan China 2002

[20] Y Liu A M Maniatty and H Antes ldquoInvestigation of aZienkiewiczndashPande yield surface and an elasticndashviscoplasticboundary element formulationrdquo Engineering Analysis withBoundary Elements vol 24 no 2 pp 207ndash211 2000

[21] Y-J Li D-L Zhang and B-G Liu ldquoDevelopment and ver-ification of strain-softening model considering deformationmodulus degradation in FLAC3Drdquo Rock and Soil Mechanicsvol 32 no 2 pp 647ndash652 2011

[22] M S Diederichs P K Kaiser and E Eberhardt ldquoDamageinitiation and propagation in hard rock during tunnelling andthe influence of near-face stress rotationrdquo International Journalof Rock Mechanics and Mining Sciences vol 41 no 5 pp 785ndash812 2004

[23] J Lee and G L Fenves ldquoA return-mapping algorithm forplastic-damage models 3-D and plane stress formulationrdquoInternational Journal for Numerical Methods in Engineering vol50 no 2 pp 487ndash506 2001

[24] J Huang Q Peng and M-X Chen ldquoAn improved return-map stress update algorithm for finite deformation analysisof general isotropic elastoplastic geomaterialsrdquo InternationalJournal for Numerical and Analytical Methods in Geomechanicsvol 38 no 6 pp 636ndash660 2014

[25] J Zhou X Yang W Fu et al ldquoExperimental test and fracturedamage mechanical characteristics of brittle rock under uniax-ial cyclic loading and unloading conditionsrdquo Chinese Journal ofRock Mechanics and Engineering vol 29 no 6 pp 1172ndash11832010

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

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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 3

Combine (6) and (8) it can be shown that 120598119901119894119895+ 120598119889

119894119895has direction

120597119865120597120598119894119895and is proportional to ⟨119892⟩ Thus it can be assumed

that

120598119901

119894119895+ 120598119889

119894119895= ]120597119865

120597120590119894119895

⟨119892⟩ (9)

and similarly for stress rate

119901

119894119895+ 119889

119894119895= minus]

120597119866

120597120590119896119897

⟨119892⟩ (10)

Let 119891 = (120597119891120597120590119901119902)119901119902

and 119892 = (120597119892120597120598119898119899) 120598119898119899 When loading

it can be obtained that 119892 = (120597119892120597120598119898119899)(119862119898119899119901119902

119901119902+ 120598119898119899119892) or

119891 = (1minus(120597119892120597120598119898119899)120598119898119899)119892 Suppose119891 and 119892 satisfy the relation

119891 = 120601119892 it can be derived that 120601 = 1 minus (120597119892120597120598119898119899)120598119898119899= 1 minus

(120597119892120597120598119894119895)](120597119865120597120590

119894119895) = 1minus]119867 where119867 = (120597119865120597120598

119894119895)(120597119892120597120598

119894119895) =

(120597119865120597120590119894119895)119863119894119895119896119897(120597119891120597120590

119896119897) When ] gt 0 we have 1] = 120601]+119867

Let ℎ = 120601] it can be obtained that ] = 1(119867+ℎ) and togetherwith (9) and (10) we get

119894119895= (119863

119894119895119896119897minus

1

119867 + ℎ

120597119866

120597120598119894119895

120597119892

120597120598119894119895

) 120598119896119897 (11)

120598119894119895= (119862119894119895119896119897+1

ℎ

120597119865

120597120590119894119895

120597119891

120597120590119896119897

) 119896119897

(ℎ = 0) (12)

Equations (11) and (12) have the same looking withgeneral elastoplastic relation but implicitly contain the effectof damage variable Ω

119898119899on rockrsquos strength and stiffness To

clarify it the specific form of ℎ is derived Introduce twohypothesizes raised in [15]

(1) Rockrsquos stiffness matrix degenerates as damage devel-ops Damage strain is induced by the degenera-tion of elastic stiffness matrix coefficients namely120597120598119894119895120597Ω119898119899= (120597119862

119894119895119896119897120597Ω119898119899)120590119896119897 Thus damage strain

rate can be defined as

120598119889

119894119895=

120597119862119894119895119896119897

120597Ω119898119899

120590119896119897Ω119898119899 (13)

(2) Ω119898119899

can be expressed as a homogeneous linear formof 120598119901119901119902 that is

Ω119898119899= 119872119898119899119901119902

120598119901

119901119902 (14)

where 119872119898119899119901119902

is the function of plastic strain anddamage variable

Substituting (13) and (14) into (12) leads to

120598119901

119894119895+

120597119862119894119895119896119897

120597Ω119898119899

120590119896119897119872119898119899119901119902

120598119901

119901119902=1

ℎ

120597119865

120597120590119894119895

120597119891

120597120590119896119897

119896119897 (15)

The equation above can be regarded as a system of linearequations about plastic strain rates which can be solved as

120598119901

119894119895=1

ℎ119870119894119895119896119897

120597119865

120597120590119894119895

120597119891

120597120590119898119899

119898119899 (16)

where

119870119894119895119896119897=1

2(120575119894119896120575119895119897+ 120575119894119897120575119895119896)

minus

(120597119862119894119895119898119899120597Ω119901119902) 120590119898119899119872119901119902119896119897

1 + (120597119862119904119905119898119899120597Ω119901119902) 120590119898119899119872119901119902119904119905

(17)

Substitute (16) into the consistency condition in stressspace

0 = 119891 =120597119891

120597120590119894119895

120590119894119895+120597119891

120597120585120573

120585120573+120597119891

Ω119896119897

Ω119896119897 (18)

Also consider that plastic internal variables 120585120573are usually

assumed as functions of plastic strain 120598119901119896119897 It can be derived

that

ℎ = minus119870119894119895119896119897

120597119865

120590119898119899

(120597119891

120585120573

120597120585120573

120597120598119901

119896119897

+120597119891

120597Ω119894119895

119872119894119895119896119897) (19)

Specifically suppose plastic internal variable satisfies 120585 =

radic(23) 120598119896119897120598119896119897 it can be obtained that

ℎ = minus120597119891

120597120585(2

3119861119894119895119861119894119895)

12

minus120597119891

120597Ω119894119895

119872119894119895119896119897119877119896119897 (20)

where 119861119894119895= 119870119894119895119896119897(120597119865120597120590

119896119897) In conclusion hypothesize (1)

implies the effect of damage on materialrsquos stiffness while(19) implies the effects on materialrsquos strength (plasticity anddamage bound surface)

Consider the Clausius-Duhem inequality

minus120588 ( + 120578 120579) + 120590119894119895120598119894119895minus1

120579119902119894

120597120579

120597119909119894

ge 0 (21)

where 120588 denotes density 119909 denotes spatial coordinates 120579denotes temperature 119902 denotes heat flow and120595 and 120578 denotefree energy and entropy per unit mass Similar to stress120595 canbe expressed as 120595 = 120595(120579 120598

119894119895 120585120573 Ω119898119899) and thus we have

=120597120595

120597120579

120579 +120597120595

120597120598119894119895

120598119894119895+120597120595

120597120585120573

120585120573+120597120595

120597Ω119898119899

Ω119898119899 (22)

Substituting (22) into (21) leads to

(120590119894119895minus 120588120597120595

120597120598119894119895

) 120598119894119895+120597120595

120597120585120573

120585120573+120597120595

120597Ω119898119899

Ω119898119899

minus 120588(120597120595

120597120579+ 120578) 120579 minus

1

120579119902119894

120597120579

120597119909119894

ge 0

(23)

Suppose the Coleman relation 120578 = minus120597120595120597120579 is satisfiedinequality (23) is valid if the inequalities

(120590119894119895minus 120588120597120595

120597120598119894119895

) 120598119894119895+120597120595

120597120585120573

120585120573+120597120595

120597Ω119898119899

Ω119898119899ge 0

1

120579119902119894

120597120579

120597119909119894

le 0

(24)

are satisfied which represent mechanical and thermal dissi-pation respectively

4 Mathematical Problems in Engineering

3 The Form of Damage Variables Based onSeparation of Tension and Shear

Tension and shear are the main types of damage for rockSimo and Ju [14 16] separate stress (strain) tensor into twodifferent parts and consider the damage effect respectivelyaccording to the assumption of separation of tension andshear Follow this approach and omit the effect of crushdamage variableΩ

119894119895can be reduced to a simpler form relating

to tension and shear independently Let119872119894119895119896119897

in (14) be

119872119894119895119896119897= 1205721119875119894119895119896119897+ 1205722119876119894119895119896119897 (25)

where

119875119894119895119896119897=

3

sum

120574=1

1198671(120598119901

120574) 119901120574

119894119901120574

119895119901120574

119896119901120574

119897

119876119894119895119896119897=

3

sum

120574=1

1198672(120598119901

120574) 119901120574

119894119901120574

119895119901120574

119896119901120574

119897

1198671(119909) =

1 119909 ge 0

0 119909 lt 0

1198672(119909) =

0 119909 ge 0

1 119909 lt 0

(26)

Here119901120574119894denotes cosine of the angle between the 120574th principal

direction of plastic strain and each coordinate axis and 1205721

1205722are dimensionless function about plastic strain invariables

Equation (25) can be seen as a separation of damage variableinto the tension and shear part In fact the principal form of120598119901

119894119895can be obtained when applied with 119872

119894119895119896119897which deviates

according to the sign of its principal value Here 1198671and

1198672in (26) can be seen as cut-off functions that reserve

only positive and negative part of strain respectively thusseparating damage variable into the tension and shear parteach part casts back to the component form

Substitute (25) into (14) and let 119867(119909) = 12057211198671(119909) +

12057221198672(119909) it can be obtained that

Ω119894119895=

3

sum

120574=1

119867(120598119901

120574) 119901120574

119894119901120574

119895119901120574

119896119901120574

119897120598119901

119896119897 (27)

Let

Ω119894119895=

3

sum

120574=1

120596120574119901120574

119894119901120574

119895 (28)

To simplify derivation omit derivatives of principal directionwith respect to time we have

120574= 119867(120598

119901

120574) 119901120574

119896119901120574

119897120598119901

119896119897 (29)

Damage internal variable Ω119894119895is thus reduced to three vari-

ables 120596120574(120574 = 1 2 3) each relating to the three plastic

principal strains respectively It is now possible to replace thedamage variableΩ

119894119895with its simplified form120596

120574 By using (28)

we have 120597119891120597120596120574= (120597119891120597Ω

119894119895)(120597Ω119894119895120597120596120574) = (120597119891120597Ω

119894119895)119901120574

119894119901120574

119895

The damage part in (19) can thus be expressed as

120597119891

120597Ω119894119895

119872119894119895119896119897=

3

sum

120574=1

120597119891

120597120596120574

119867(120598119901

120574) 119901120574

119896119901120574

119897 (30)

4 Multiparameter Damage Model Based onZienkiewicz-Pande Criterion

To expose the model further in this section Zienkiewicz-Pande criterion with tension limit is taken as an exampleand the specific form of the multiparameter damage model isderivedThe criteria currently used very often in research andengineering include Drucker-Prager Mohr-Coulomb Hoek-Brown and unified strength theory suggested by Yu et al[17] of which theMohr-Coulomb criterion is widely adoptedThe existence of vertex singularity in bound surface will bringsome difficulties in iteration Zienkiewicz-Pande criterion is asmooth version of Mohr-Coulomb criterion [18] which usesa quadratic curve to approximate straight line in 120587-plane soas to eliminate singular vertex and benefit numerical imple-mentation also it takes intermediate principal stress intoaccount to some extentThe Zienkiewicz-Pande criterion hasbeen used in many engineering projects for the simulationlike excavation of underground caverns and shows a goodagreement with monitoring results [19]

41 The Specific Form of Multiparameter Damage Model Indamage judgment plastic yield surface is often used as thebound for shear damage and stress or strain limit as the boundfor tensionThus in this section Zienkiewicz-Pande criterionwith tension limit is assumed as the bound for plasticity anddamage namely

119891119904= 120590119898sin120601 minus 119888 cos120601

+ radic1

21198882

1

(119903120590

119892120601(120579120590))

2

+ 1198862sin2120601(31a)

119891119905= 1205901minus 120590119905 (31b)

Here the hyperbolic type of approximation in [20] is adoptedthat is

1198882

1=

2radic3

3 minus sin120601

119892120601(120579120590) =

2119896120601

(1 + 119896120601) minus (1 minus 119896

120601) sin 3120579

120590

119896120601=3 minus sin1206013 + sin120601

(32)

where 119888 is the cohesion 120601 is the angle of internal friction120590119905is the tension limit 120590

119898is the mean stress 120590

1198961198963 120579120590is

Mathematical Problems in Engineering 5

lode angle 119903120590is lode radius namely radic2119869

2 11988821is the revising

coefficient introduced in [20] and 119886 controls to what extentapproximation can be made to the Mohr-Coulomb yieldsurface Here the damage type is determined by the firstbound the state reaches when loading Let potential functionbe

119865119904= 120590119898sin120595 minus 119888 cos120595

+ radic1

21198882

1

(119903120590

119892120595(120579120590))

2

+ 1198862sin2120595(33a)

119865119905= 119891119905 (33b)

where

119892120595(120579120590) =

2119896120595

(1 + 119896120595) minus (1 minus 119896

120595) sin 3120579

120590

119896120595=3 minus sin1205953 + sin120595

(34)

Here120595 denotes the angle of dilatancy Consider that 1199032120590= 21198692

sin3120579120590= minus(3radic32)(119869

311986932

2) 1205901= 120590119898+ (2radic33)119869

12

2sin(120579120590+

(23)120587) and 120597120590119898120597120590119894119895= (13)120575

119894119895 1205971198692120597120590119894119895= 119904119894119895 and

1205971198693120597120590119894119895= 119904119894119896119904119896119895minus (23)119869

2120575119894119895 it can be derived that

120597119891119904

120597120590119894119895

= 1198601120575119894119895+ 1198602119904119894119895+ 1198603119904119894119896119904119896119895 (35a)

120597119891119905

120597120590119894119895

= 1198604120575119894119895+ 1198605119904119894119895+ 1198606119904119894119896119904119896119895 (35b)

where

1198601=1

3sin120601 minus

radic3

1198882

11198923

120601(120579120590)11986912

211988711198872

1198602=

1

21198882

11198922

120601(120579120590)1198871minus

9radic3

41198882

11198923

120601(120579120590)

1198693

11986932

2

11988711198872

1198603=

3radic3

21198882

11198923

120601(120579120590)

1

11986912

2

11988711198872

1198604=1

3(1 + 2119887

3)

1198605=radic3

3

1

11986912

2

1198874+3

2

1198693

1198692

2

1198873

1198606= minus

1

1198692

1198873

(36a)

1198871= (

1

1198882

11198922 (120579120590)1198692+ 1198862sin2120601)

minus12

1198872=

2119896120601(1 minus 119896

120601)

((1 + 119896120601) minus (1 minus 119896

120601) sin 3120579

120590)2

1198873=cos (120579

120590+ (23) 120587)

1003816100381610038161003816cos 31205791205901003816100381610038161003816

1198874= sin(120579

120590+2

3120587)

(36b)

Consider that 120597119892120597120598119894119895= (120597119891120597120590

119896119897)(120590119896119897120598119894119895) = 119863

119894119895119896119897(120597119891120597120590

119896119897)

it is easy to get 120597119892120597120598119894119895accordingly Similarly 120597119865120597120590

119894119895and

120597119866120597120598119894119895can be obtained

Let elastic flexibility be an isotropic tensor

119862119894119895119896119897= minus120583

119864120575119894119895120575119896119897+1 + 120583

2119864(120575119894119896120575119895119897+ 120575119894119897120575119895119896) (37)

where 119864 denotes Youngrsquos modulus and 120583 denotes Poissonrsquosratio Suppose only Youngrsquos modulus is affected by damageinternal variables then similar to (30) we can get

120597119862119894119895119898119899

120597Ω119901119902

120590119898119899119872119901119902119896119897= 119870(1)

119894119895119870(2)

119896119897 (38a)

120597119862119904119905119898119899

120597Ω119901119902

120590119898119899119872119901119902119904119905= 119870(1)

119904119905119870(2)

119904119905 (38b)

where

119870(1)

119894119895=3120583

1198642120590119898120575119894119895minus2 (1 + 120583)

1198642120590119894119895

119870(2)

119896119897=

3

sum

120574=1

120597119864

120597120596120574

119867(120598119901

120574) 119901120574

119896119901120574

119897

(39)

Substitute (38a) and (38b) into (17) we can get 119870119894119895119896119897

whichtogether with (20) and (35a) and (35b) (also 120597119892120597120598

119894119895 120597119865120597120590

119894119895

and 120597119866120597120598119894119895which can be obtained in a similar approach)

leads to a constitutive relation in the form as (11) and (12)

42 The Damage Evolution of Material Strength and StiffnessMaterial strength and stiffness degenerate as damage evolvesHere those strength and stiffness parameters are treated asfunctions of the reduced form of damage variables 120596

120573and

their specific forms are examined Consider that damagevariable has a degenerating effect on cohesion and angle ofinternal friction it can be derived that

120597119891119904

120597120596120574

=120597119891119904

120597119888

120597119888

120597120596120574

+120597119891119904

120597120601

120597120601

120597120596120574

(for shearing) (40a)

Similarly consider the degenerating effect damage variablehas on tension limit we have

120597119891119905

120597120596120574

= minus120597120590119905

120597120596120574

(for tension) (40b)

6 Mathematical Problems in Engineering

Since Zienkiewicz-Pande yield surface is a smooth approx-imation to the Mohr-Coulomb yield surface and the Mohr-Coulomb criterion has the form

119888 =120590119888

2

1 minus sin120601cos120601 (41)

during uniaxial compression it can be assumed that thecohesion 119888 in Zienkiewicz-Pande criterion is proportional touniaxial compressive strength 120590

119888 Instead of plastic hardening

effect damage makes materialrsquos strength degenerate To sim-plify implementation omit the effect damage variable has onangle of internal friction and also consider the function typeadopted in [21] for the degeneration of materialrsquos parameterslet 119888 and 120590

119905be linear functions for damage variables

119888 =1 minus sin1206012 cos120601

(1205901198880minus 119864119904ℎ (120596120574)) (42a)

120590119905= 1205901199050minus 119864119905ℎ (120596120574) (42b)

where 1205901198880

and 1205901199050

denote the original compression andtension strength and 119864

119904and 119864

119905are damagemodulus for shear

and tension respectivelyAs internal microscopic cracks develop very fast after

reaching damage [22] exponential function is adopted

119864 = 1198640119890minus119877ℎ(120596120574) (43)

where1198640is the original Youngrsquosmodulus119877 ismaterialrsquos dam-

age coefficient and ℎ(120596120574) is taken as the linear combination

of damage variables 120596120574 Specifically let

ℎ (120596120574) =

max (120596120574) minusmin (120596

120574) (shear)

max (120596120574) (tension)

(44)

Given the irreversibility of damage during the damageevolution of materialrsquos parameters the minima of materialparameters along the evolution path are used

5 Numerical Example

The algorithm for solving group of nonlinear equationsincludes Newton-Raphson method modified Newton-Raphson method and quasi-Newton-Raphson method ofwhich Newton-Raphson method and modified Newton-Raphson method can effectively utilize the sparsity ofstiffness matrix Consider that the model raised in this paperhas many factors interacting with each other and thus highlynonlinearized a framework for nonlinear finite elementanalysis by using Newton-Raphson method is implementedwhich calls specific constitutive model to handle iterativesolving

Typical implementation ofmaterialmodel can be reducedto the following problems

(1) The establishment of tangent stiffness matrix whichcan be implemented using (11)

0

20

40

60

80

100

120

140

160

Stre

ss

0001 0002 0003 0004 0005 00060000Strain

Figure 1 Stress-strain curve under uniaxial compression (MPa)

(2) State update namely update of stress strain andmaterial parameters of each quadrature point whenstrain increment is known Typical implementationincludes explicit method (eg Runge-Kutta method)and implicit method (eg return map method [2324]) To simplify implementation explicit method isadopted which takes the incremental form of (11) andperform update and revision for state and materialparameters at the end of each increment step

To validate the effectiveness of the model establishedconsider the cylindrical sample in [25] Let 119864 = 3143GPa120583 = 023 119888 = 2622MPa and 120601 = 447∘ Apply uniaxialcompression to the sample model The stress-strain curve isdepicted in Figure 1 Comparison with the experiment resultof uniaxial compression in [25] shows that the establishedmodel can simulate this process well

Consider a cubic finite element model that each side haslength 1m Let 119864 = 317GPa 120583 = 026 119888 = 30MPa120601 = 464

∘ and 120590119905= 35MPa Apply cyclic compression

and tension uniaxial load The stress-strain curves are shownin Figure 2 When strain reaches 00018 and 00028 forcompression or when strain reaches 000016 and 00022 fortension unload stress to zero and then apply again Thetangent stiffness during unloading is smaller than elasticstiffness as the introduction of damage The model presentedin this paper can simulate damage behavior under cycliccompression and tension effectively

6 Engineering Application

In this section the established model is used for the excava-tion simulation of Zhenrsquoan underground power station

61 Project Overview and Numerical Modeling Zhenrsquoanpumped storage power station is located in Shanxi Provincein China with installed capacity of 1400MW (4 times 350MW)The underground powerhouse is made up of the main pow-erhouse main transform house tailrace surge chamber and

Mathematical Problems in Engineering 7

Table 1 Mechanical parameters for rock

Density (gcm3) Youngrsquos modulus (GPa) Poissonrsquos ratio Cohesion (MPa) Internal friction angle (∘) Tension limit (MPa)267 130 027 10 477 045

000

00

000

05

000

10

000

15

000

20

000

25

000

30

000

45

000

40

000

35

Strain

30

25

20

15

10

5

0

Stre

ss

(a) Compression

000

000

000

005

000

010

000

015

000

020

000

025

000

030

000

040

000

035

Strain

Stre

ss

40

35

30

25

20

15

10

05

00

(b) Tension

Figure 2 Stress-strain curve under cyclic uniaxial compression and tension (MPa)

several other tunnels connecting each cavern Mechanicalparameters for rock are shown in Table 1 Finite elementmodel shown in Figure 3 contains 480608 isoparametric 8-node elements with 109011 elements to be excavated dividedinto 10 stages 119909-axis of the model is perpendicular to lon-gitudinal axis of main powerhouse with a range of 31864m119910-axis is along the longitudinal axis with a range of 3896m119911-axis is along the vertical axis with a range of 5686m Initialstress is shown in Figure 4 It is obtained by inverse analysisbased on 4 gauging points near the carven and the resultshows that it is generated mainly by gravity The establishedmodel is used for excavation simulation of the undergroundpowerhouse

62 Result Analysis

621 Damage Zone Distribution To simulate the process ofexcavation divide the computation into 10 steps each witha layer shown in Figure 3 removed and let the iterationreach convergence from the unbalanced state Figure 5 givesthe development of damage zone where the third and sixthstage are the excavation of lateral caverns resulting insome finite elements returning from damage bound Whenexcavation is complete damage zone distribution of 2 unitsectionrsquos surrounding rocks is shown in Figure 6 Depth ofdamage zone reaches a maximum of 21m near the arch roofDepth of damage zone near side wall ranges from 184m to201m and ranges from 70m to 90m near main transformhouse Tension damage zone is mostly distributed near

the intersection of tunnels and caverns where release ofinitial stress along multiple direction makes it easy to crack

622 Stress Distribution After excavation stress distribu-tions of surrounding rock of each unit section are roughlysimilar The first and third principal stress of 2 unit sectionrsquossurrounding rocks are shown in Figure 7 the direction of thethird principal stress is almost vertical which is the same asthe direction of gravity and first principal stress is along theradial direction Near the intersection of each tunnel with themain powerhouse the third principal stress concentrates andreaches a maximum of minus20MPa First principal stress in themiddle part of 2 unit sectionrsquos side wall reaches a peak ofminus237MPa and third principal stress is around minus20MPa

623 Displacement Distribution What is shown in Figure 8is the displacement of 2 unit section Release of initialstress makes the displacement directing towards the cavernsDisplacement near the intersection of each cavern is relativelylarger than that of other areas In other areas displacementfield is distributed relatively uniformly After excavation 2unit sectionrsquos arch roof rsquos displacement resiles to 30mmMaximum displacement of main powerhousersquos upstream anddownstream side wall is 421mm and 355mm respectivelyThe values are 138mm and 196mm for main transformhouse

From the analysis above it is easy to know that thedamage zone stress and displacement distribution obtained

8 Mathematical Problems in Engineering

(a) (b)

Figure 3 Computation model (a) and excavation elements (b)S1

minus05

minus9

minus85

minus8

minus75

minus7

minus65

minus6

minus55

minus5

minus45

minus4

minus35

minus3

minus25

minus2

minus15

minus1

(a)

S3

minus1

minus2

minus3

minus4

minus5

minus6

minus7

minus8

minus9

minus10

minus11

minus12

minus13

(b)

Figure 4 Initial first (a) and third (b) principal stress (MPa)

using the model established in this paper are similar tothose using Zienkiewicz-Pande or Mohr-Coulomb criterionbecause of the form of shear damage bound adopted Theintroduction of damage mechanismmakes it more natural tofigure out the shear and tension damage area

7 Conclusion

This paper approaches based on the microscopic mechanismof damage for rock material and establishes in an intuitiveway a multiparameter elastoplastic damage model for rockthat is applicable to engineering With the assumption thatdamage comes into existence as the materialrsquos strength and

stiffness degenerate and that damage is interconnected withplastic deformation a revised general form for elastoplasticdamage model containing damage variable of tensor formis established By considering plastic strain separation theexpression of damage variable reflecting the damage mech-anism for shear and tension simultaneously is derived Byadopting Zienkiewicz-Pande criterion with tension limit asthe bound for plasticity and damage the specific form for thedamage model is derived and implemented

The model established in this paper is physically intuitiveand has relativelywell-based theoretical backgroundNumer-ical experiments and engineering application show that thismodel can reflect the damage behavior of rock effectively

Mathematical Problems in Engineering 9

1 2 3 4 5 6 7 8 9 10Stage

25

20

15

10

5

0

Volu

me o

f dam

age z

one

Shear damageTension damage

Figure 5 Volume of damage zone (105m3)

Sheardamage

Sheardamage

Resilience

Figure 6 Damage zone of 2 unit section

S1

minus05

minus75

minus7

minus65

minus6

minus55

minus5

minus45

minus4

minus35

minus3

minus25

minus2

minus15

minus1

0

(a)

minus2

minus36

minus34

minus32

minus30

minus28

minus26

minus24

minus22

minus20

minus18

minus16

minus14

minus12

minus10

minus8

minus6

minus4

S3

(b)

Figure 7 First and third principal stress of 2 unit section (MPa)

10 Mathematical Problems in Engineering

D

45

40

35

30

25

20

15

10

5

Figure 8 Displacement of 2 unit section (mm)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] Z-Y Ye L Hong X-L Liu and T-B Yin ldquoConstitutive modelof rock based onmicrostructures simulationrdquo Journal of CentralSouth University of Technology vol 15 no 2 pp 230ndash236 2008

[2] M R Salari S Saeb K J Willam S J Patchet and R C Car-rasco ldquoA coupled elastoplastic damage model for geomaterialsrdquoComputer Methods in Applied Mechanics and Engineering vol193 no 27-29 pp 2625ndash2643 2004

[3] G D Nguyen and A M Korsunsky ldquoDevelopment of anapproach to constitutive modelling of concrete isotropic dam-age coupled with plasticityrdquo International Journal of Solids andStructures vol 45 no 20 pp 5483ndash5501 2008

[4] X LiW-G Cao andY-H Su ldquoA statistical damage constitutivemodel for softening behavior of rocksrdquoEngineeringGeology vol143-144 pp 1ndash17 2012

[5] M Xiao ldquoAnalysis on stability of surrounding rock and damagefracture of concrete lining for high pressure branch pipesrdquo Jour-nal of Wuhan University of Hydraulic and Electric Engineeringno 6 pp 594ndash599 1995

[6] G Frantziskonis and C S Desai ldquoConstitutive model withstrain softeningrdquo International Journal of Solids and Structuresvol 23 no 6 pp 733ndash750 1987

[7] D Krajcinovic and S Mastilovic ldquoSome fundamental issues ofdamage mechanicsrdquo Mechanics of Materials vol 21 no 3 pp217ndash230 1995

[8] L Resende ldquoA Damage mechanics constitutive theory for theinelastic behaviour of concreterdquo Computer Methods in AppliedMechanics and Engineering vol 60 no 1 pp 57ndash93 1987

[9] M Jirasek ldquoNonlocal models for damage and fracture com-parison of approachesrdquo International Journal of Solids andStructures vol 35 no 31-32 pp 4133ndash4145 1998

[10] M Jirasek and S Marfia ldquoNon-local damage model based ondisplacement averagingrdquo International Journal for NumericalMethods in Engineering vol 63 no 1 pp 77ndash102 2005

[11] J Lee andG L Fenves ldquoPlastic-damagemodel for cyclic loadingof concrete structuresrdquo Journal of Engineering Mechanics vol124 no 8 pp 892ndash900 1998

[12] J Li and J Y Wu ldquoElastoplacstic damage constitutive modelfor concrete based on damage energy release rates part I basicformulationsrdquo China Civil Engineering Journal no 9 pp 14ndash202005

[13] J Y Wu and J Li ldquoElastoplastic damage constitutive modelfor concrete based on damage energy release rates part IInumerical algorithm and verificationsrdquo China Civil EngineeringJournal no 9 pp 21ndash27 2005

[14] J W Ju ldquoOn energy-based coupled elastoplastic damagetheories constitutive modeling and computational aspectsrdquoInternational Journal of Solids and Structures vol 25 no 7 pp803ndash833 1989

[15] Y Q Ying ldquoOn rock plasticity damage and their constitutiveformulationrdquo Scientia Geologica Sinica no 1 pp 63ndash70 1995

[16] J C Simo and J W Ju ldquoStrain- and stress-based continuumdamage models-I Formulationrdquo International Journal of Solidsand Structures vol 23 no 7 pp 821ndash840 1987

[17] M H Yu YW ZanW Fan J Zhao and Z Z Dong ldquoAnvancesin strength theory of rock in 20 centurymdash100 years in memoryof theMohr-Coulomb Strehgth theoryrdquo Chinese Journal of RockMechanics and Engineering no 5 pp 545ndash550 2000

[18] O Zienkiewicz and G Pande ldquoSome useful forms of isotropicyield surfaces for soil and rockmechanicsrdquo in Finite Elements inGeomechanics pp 179ndash198 Wiley Cambridge UK 1977

[19] M Xiao Study on numerical analysis method of stability andsupporting for underground caverns [PhD thesis] Wuhan Uni-versity Wuhan China 2002

[20] Y Liu A M Maniatty and H Antes ldquoInvestigation of aZienkiewiczndashPande yield surface and an elasticndashviscoplasticboundary element formulationrdquo Engineering Analysis withBoundary Elements vol 24 no 2 pp 207ndash211 2000

[21] Y-J Li D-L Zhang and B-G Liu ldquoDevelopment and ver-ification of strain-softening model considering deformationmodulus degradation in FLAC3Drdquo Rock and Soil Mechanicsvol 32 no 2 pp 647ndash652 2011

[22] M S Diederichs P K Kaiser and E Eberhardt ldquoDamageinitiation and propagation in hard rock during tunnelling andthe influence of near-face stress rotationrdquo International Journalof Rock Mechanics and Mining Sciences vol 41 no 5 pp 785ndash812 2004

[23] J Lee and G L Fenves ldquoA return-mapping algorithm forplastic-damage models 3-D and plane stress formulationrdquoInternational Journal for Numerical Methods in Engineering vol50 no 2 pp 487ndash506 2001

[24] J Huang Q Peng and M-X Chen ldquoAn improved return-map stress update algorithm for finite deformation analysisof general isotropic elastoplastic geomaterialsrdquo InternationalJournal for Numerical and Analytical Methods in Geomechanicsvol 38 no 6 pp 636ndash660 2014

[25] J Zhou X Yang W Fu et al ldquoExperimental test and fracturedamage mechanical characteristics of brittle rock under uniax-ial cyclic loading and unloading conditionsrdquo Chinese Journal ofRock Mechanics and Engineering vol 29 no 6 pp 1172ndash11832010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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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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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

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

3 The Form of Damage Variables Based onSeparation of Tension and Shear

Tension and shear are the main types of damage for rockSimo and Ju [14 16] separate stress (strain) tensor into twodifferent parts and consider the damage effect respectivelyaccording to the assumption of separation of tension andshear Follow this approach and omit the effect of crushdamage variableΩ

119894119895can be reduced to a simpler form relating

to tension and shear independently Let119872119894119895119896119897

in (14) be

119872119894119895119896119897= 1205721119875119894119895119896119897+ 1205722119876119894119895119896119897 (25)

where

119875119894119895119896119897=

3

sum

120574=1

1198671(120598119901

120574) 119901120574

119894119901120574

119895119901120574

119896119901120574

119897

119876119894119895119896119897=

3

sum

120574=1

1198672(120598119901

120574) 119901120574

119894119901120574

119895119901120574

119896119901120574

119897

1198671(119909) =

1 119909 ge 0

0 119909 lt 0

1198672(119909) =

0 119909 ge 0

1 119909 lt 0

(26)

Here119901120574119894denotes cosine of the angle between the 120574th principal

direction of plastic strain and each coordinate axis and 1205721

1205722are dimensionless function about plastic strain invariables

Equation (25) can be seen as a separation of damage variableinto the tension and shear part In fact the principal form of120598119901

119894119895can be obtained when applied with 119872

119894119895119896119897which deviates

according to the sign of its principal value Here 1198671and

1198672in (26) can be seen as cut-off functions that reserve

only positive and negative part of strain respectively thusseparating damage variable into the tension and shear parteach part casts back to the component form

Substitute (25) into (14) and let 119867(119909) = 12057211198671(119909) +

12057221198672(119909) it can be obtained that

Ω119894119895=

3

sum

120574=1

119867(120598119901

120574) 119901120574

119894119901120574

119895119901120574

119896119901120574

119897120598119901

119896119897 (27)

Let

Ω119894119895=

3

sum

120574=1

120596120574119901120574

119894119901120574

119895 (28)

To simplify derivation omit derivatives of principal directionwith respect to time we have

120574= 119867(120598

119901

120574) 119901120574

119896119901120574

119897120598119901

119896119897 (29)

Damage internal variable Ω119894119895is thus reduced to three vari-

ables 120596120574(120574 = 1 2 3) each relating to the three plastic

principal strains respectively It is now possible to replace thedamage variableΩ

119894119895with its simplified form120596

120574 By using (28)

we have 120597119891120597120596120574= (120597119891120597Ω

119894119895)(120597Ω119894119895120597120596120574) = (120597119891120597Ω

119894119895)119901120574

119894119901120574

119895

The damage part in (19) can thus be expressed as

120597119891

120597Ω119894119895

119872119894119895119896119897=

3

sum

120574=1

120597119891

120597120596120574

119867(120598119901

120574) 119901120574

119896119901120574

119897 (30)

4 Multiparameter Damage Model Based onZienkiewicz-Pande Criterion

To expose the model further in this section Zienkiewicz-Pande criterion with tension limit is taken as an exampleand the specific form of the multiparameter damage model isderivedThe criteria currently used very often in research andengineering include Drucker-Prager Mohr-Coulomb Hoek-Brown and unified strength theory suggested by Yu et al[17] of which theMohr-Coulomb criterion is widely adoptedThe existence of vertex singularity in bound surface will bringsome difficulties in iteration Zienkiewicz-Pande criterion is asmooth version of Mohr-Coulomb criterion [18] which usesa quadratic curve to approximate straight line in 120587-plane soas to eliminate singular vertex and benefit numerical imple-mentation also it takes intermediate principal stress intoaccount to some extentThe Zienkiewicz-Pande criterion hasbeen used in many engineering projects for the simulationlike excavation of underground caverns and shows a goodagreement with monitoring results [19]

41 The Specific Form of Multiparameter Damage Model Indamage judgment plastic yield surface is often used as thebound for shear damage and stress or strain limit as the boundfor tensionThus in this section Zienkiewicz-Pande criterionwith tension limit is assumed as the bound for plasticity anddamage namely

119891119904= 120590119898sin120601 minus 119888 cos120601

+ radic1

21198882

1

(119903120590

119892120601(120579120590))

2

+ 1198862sin2120601(31a)

119891119905= 1205901minus 120590119905 (31b)

Here the hyperbolic type of approximation in [20] is adoptedthat is

1198882

1=

2radic3

3 minus sin120601

119892120601(120579120590) =

2119896120601

(1 + 119896120601) minus (1 minus 119896

120601) sin 3120579

120590

119896120601=3 minus sin1206013 + sin120601

(32)

where 119888 is the cohesion 120601 is the angle of internal friction120590119905is the tension limit 120590

119898is the mean stress 120590

1198961198963 120579120590is

Mathematical Problems in Engineering 5

lode angle 119903120590is lode radius namely radic2119869

2 11988821is the revising

coefficient introduced in [20] and 119886 controls to what extentapproximation can be made to the Mohr-Coulomb yieldsurface Here the damage type is determined by the firstbound the state reaches when loading Let potential functionbe

119865119904= 120590119898sin120595 minus 119888 cos120595

+ radic1

21198882

1

(119903120590

119892120595(120579120590))

2

+ 1198862sin2120595(33a)

119865119905= 119891119905 (33b)

where

119892120595(120579120590) =

2119896120595

(1 + 119896120595) minus (1 minus 119896

120595) sin 3120579

120590

119896120595=3 minus sin1205953 + sin120595

(34)

Here120595 denotes the angle of dilatancy Consider that 1199032120590= 21198692

sin3120579120590= minus(3radic32)(119869

311986932

2) 1205901= 120590119898+ (2radic33)119869

12

2sin(120579120590+

(23)120587) and 120597120590119898120597120590119894119895= (13)120575

119894119895 1205971198692120597120590119894119895= 119904119894119895 and

1205971198693120597120590119894119895= 119904119894119896119904119896119895minus (23)119869

2120575119894119895 it can be derived that

120597119891119904

120597120590119894119895

= 1198601120575119894119895+ 1198602119904119894119895+ 1198603119904119894119896119904119896119895 (35a)

120597119891119905

120597120590119894119895

= 1198604120575119894119895+ 1198605119904119894119895+ 1198606119904119894119896119904119896119895 (35b)

where

1198601=1

3sin120601 minus

radic3

1198882

11198923

120601(120579120590)11986912

211988711198872

1198602=

1

21198882

11198922

120601(120579120590)1198871minus

9radic3

41198882

11198923

120601(120579120590)

1198693

11986932

2

11988711198872

1198603=

3radic3

21198882

11198923

120601(120579120590)

1

11986912

2

11988711198872

1198604=1

3(1 + 2119887

3)

1198605=radic3

3

1

11986912

2

1198874+3

2

1198693

1198692

2

1198873

1198606= minus

1

1198692

1198873

(36a)

1198871= (

1

1198882

11198922 (120579120590)1198692+ 1198862sin2120601)

minus12

1198872=

2119896120601(1 minus 119896

120601)

((1 + 119896120601) minus (1 minus 119896

120601) sin 3120579

120590)2

1198873=cos (120579

120590+ (23) 120587)

1003816100381610038161003816cos 31205791205901003816100381610038161003816

1198874= sin(120579

120590+2

3120587)

(36b)

Consider that 120597119892120597120598119894119895= (120597119891120597120590

119896119897)(120590119896119897120598119894119895) = 119863

119894119895119896119897(120597119891120597120590

119896119897)

it is easy to get 120597119892120597120598119894119895accordingly Similarly 120597119865120597120590

119894119895and

120597119866120597120598119894119895can be obtained

Let elastic flexibility be an isotropic tensor

119862119894119895119896119897= minus120583

119864120575119894119895120575119896119897+1 + 120583

2119864(120575119894119896120575119895119897+ 120575119894119897120575119895119896) (37)

where 119864 denotes Youngrsquos modulus and 120583 denotes Poissonrsquosratio Suppose only Youngrsquos modulus is affected by damageinternal variables then similar to (30) we can get

120597119862119894119895119898119899

120597Ω119901119902

120590119898119899119872119901119902119896119897= 119870(1)

119894119895119870(2)

119896119897 (38a)

120597119862119904119905119898119899

120597Ω119901119902

120590119898119899119872119901119902119904119905= 119870(1)

119904119905119870(2)

119904119905 (38b)

where

119870(1)

119894119895=3120583

1198642120590119898120575119894119895minus2 (1 + 120583)

1198642120590119894119895

119870(2)

119896119897=

3

sum

120574=1

120597119864

120597120596120574

119867(120598119901

120574) 119901120574

119896119901120574

119897

(39)

Substitute (38a) and (38b) into (17) we can get 119870119894119895119896119897

whichtogether with (20) and (35a) and (35b) (also 120597119892120597120598

119894119895 120597119865120597120590

119894119895

and 120597119866120597120598119894119895which can be obtained in a similar approach)

leads to a constitutive relation in the form as (11) and (12)

42 The Damage Evolution of Material Strength and StiffnessMaterial strength and stiffness degenerate as damage evolvesHere those strength and stiffness parameters are treated asfunctions of the reduced form of damage variables 120596

120573and

their specific forms are examined Consider that damagevariable has a degenerating effect on cohesion and angle ofinternal friction it can be derived that

120597119891119904

120597120596120574

=120597119891119904

120597119888

120597119888

120597120596120574

+120597119891119904

120597120601

120597120601

120597120596120574

(for shearing) (40a)

Similarly consider the degenerating effect damage variablehas on tension limit we have

120597119891119905

120597120596120574

= minus120597120590119905

120597120596120574

(for tension) (40b)

6 Mathematical Problems in Engineering

Since Zienkiewicz-Pande yield surface is a smooth approx-imation to the Mohr-Coulomb yield surface and the Mohr-Coulomb criterion has the form

119888 =120590119888

2

1 minus sin120601cos120601 (41)

during uniaxial compression it can be assumed that thecohesion 119888 in Zienkiewicz-Pande criterion is proportional touniaxial compressive strength 120590

119888 Instead of plastic hardening

effect damage makes materialrsquos strength degenerate To sim-plify implementation omit the effect damage variable has onangle of internal friction and also consider the function typeadopted in [21] for the degeneration of materialrsquos parameterslet 119888 and 120590

119905be linear functions for damage variables

119888 =1 minus sin1206012 cos120601

(1205901198880minus 119864119904ℎ (120596120574)) (42a)

120590119905= 1205901199050minus 119864119905ℎ (120596120574) (42b)

where 1205901198880

and 1205901199050

denote the original compression andtension strength and 119864

119904and 119864

119905are damagemodulus for shear

and tension respectivelyAs internal microscopic cracks develop very fast after

reaching damage [22] exponential function is adopted

119864 = 1198640119890minus119877ℎ(120596120574) (43)

where1198640is the original Youngrsquosmodulus119877 ismaterialrsquos dam-

age coefficient and ℎ(120596120574) is taken as the linear combination

of damage variables 120596120574 Specifically let

ℎ (120596120574) =

max (120596120574) minusmin (120596

120574) (shear)

max (120596120574) (tension)

(44)

Given the irreversibility of damage during the damageevolution of materialrsquos parameters the minima of materialparameters along the evolution path are used

5 Numerical Example

The algorithm for solving group of nonlinear equationsincludes Newton-Raphson method modified Newton-Raphson method and quasi-Newton-Raphson method ofwhich Newton-Raphson method and modified Newton-Raphson method can effectively utilize the sparsity ofstiffness matrix Consider that the model raised in this paperhas many factors interacting with each other and thus highlynonlinearized a framework for nonlinear finite elementanalysis by using Newton-Raphson method is implementedwhich calls specific constitutive model to handle iterativesolving

Typical implementation ofmaterialmodel can be reducedto the following problems

(1) The establishment of tangent stiffness matrix whichcan be implemented using (11)

0

20

40

60

80

100

120

140

160

Stre

ss

0001 0002 0003 0004 0005 00060000Strain

Figure 1 Stress-strain curve under uniaxial compression (MPa)

(2) State update namely update of stress strain andmaterial parameters of each quadrature point whenstrain increment is known Typical implementationincludes explicit method (eg Runge-Kutta method)and implicit method (eg return map method [2324]) To simplify implementation explicit method isadopted which takes the incremental form of (11) andperform update and revision for state and materialparameters at the end of each increment step

To validate the effectiveness of the model establishedconsider the cylindrical sample in [25] Let 119864 = 3143GPa120583 = 023 119888 = 2622MPa and 120601 = 447∘ Apply uniaxialcompression to the sample model The stress-strain curve isdepicted in Figure 1 Comparison with the experiment resultof uniaxial compression in [25] shows that the establishedmodel can simulate this process well

Consider a cubic finite element model that each side haslength 1m Let 119864 = 317GPa 120583 = 026 119888 = 30MPa120601 = 464

∘ and 120590119905= 35MPa Apply cyclic compression

and tension uniaxial load The stress-strain curves are shownin Figure 2 When strain reaches 00018 and 00028 forcompression or when strain reaches 000016 and 00022 fortension unload stress to zero and then apply again Thetangent stiffness during unloading is smaller than elasticstiffness as the introduction of damage The model presentedin this paper can simulate damage behavior under cycliccompression and tension effectively

6 Engineering Application

In this section the established model is used for the excava-tion simulation of Zhenrsquoan underground power station

61 Project Overview and Numerical Modeling Zhenrsquoanpumped storage power station is located in Shanxi Provincein China with installed capacity of 1400MW (4 times 350MW)The underground powerhouse is made up of the main pow-erhouse main transform house tailrace surge chamber and

Mathematical Problems in Engineering 7

Table 1 Mechanical parameters for rock

Density (gcm3) Youngrsquos modulus (GPa) Poissonrsquos ratio Cohesion (MPa) Internal friction angle (∘) Tension limit (MPa)267 130 027 10 477 045

000

00

000

05

000

10

000

15

000

20

000

25

000

30

000

45

000

40

000

35

Strain

30

25

20

15

10

5

0

Stre

ss

(a) Compression

000

000

000

005

000

010

000

015

000

020

000

025

000

030

000

040

000

035

Strain

Stre

ss

40

35

30

25

20

15

10

05

00

(b) Tension

Figure 2 Stress-strain curve under cyclic uniaxial compression and tension (MPa)

several other tunnels connecting each cavern Mechanicalparameters for rock are shown in Table 1 Finite elementmodel shown in Figure 3 contains 480608 isoparametric 8-node elements with 109011 elements to be excavated dividedinto 10 stages 119909-axis of the model is perpendicular to lon-gitudinal axis of main powerhouse with a range of 31864m119910-axis is along the longitudinal axis with a range of 3896m119911-axis is along the vertical axis with a range of 5686m Initialstress is shown in Figure 4 It is obtained by inverse analysisbased on 4 gauging points near the carven and the resultshows that it is generated mainly by gravity The establishedmodel is used for excavation simulation of the undergroundpowerhouse

62 Result Analysis

621 Damage Zone Distribution To simulate the process ofexcavation divide the computation into 10 steps each witha layer shown in Figure 3 removed and let the iterationreach convergence from the unbalanced state Figure 5 givesthe development of damage zone where the third and sixthstage are the excavation of lateral caverns resulting insome finite elements returning from damage bound Whenexcavation is complete damage zone distribution of 2 unitsectionrsquos surrounding rocks is shown in Figure 6 Depth ofdamage zone reaches a maximum of 21m near the arch roofDepth of damage zone near side wall ranges from 184m to201m and ranges from 70m to 90m near main transformhouse Tension damage zone is mostly distributed near

the intersection of tunnels and caverns where release ofinitial stress along multiple direction makes it easy to crack

622 Stress Distribution After excavation stress distribu-tions of surrounding rock of each unit section are roughlysimilar The first and third principal stress of 2 unit sectionrsquossurrounding rocks are shown in Figure 7 the direction of thethird principal stress is almost vertical which is the same asthe direction of gravity and first principal stress is along theradial direction Near the intersection of each tunnel with themain powerhouse the third principal stress concentrates andreaches a maximum of minus20MPa First principal stress in themiddle part of 2 unit sectionrsquos side wall reaches a peak ofminus237MPa and third principal stress is around minus20MPa

623 Displacement Distribution What is shown in Figure 8is the displacement of 2 unit section Release of initialstress makes the displacement directing towards the cavernsDisplacement near the intersection of each cavern is relativelylarger than that of other areas In other areas displacementfield is distributed relatively uniformly After excavation 2unit sectionrsquos arch roof rsquos displacement resiles to 30mmMaximum displacement of main powerhousersquos upstream anddownstream side wall is 421mm and 355mm respectivelyThe values are 138mm and 196mm for main transformhouse

From the analysis above it is easy to know that thedamage zone stress and displacement distribution obtained

8 Mathematical Problems in Engineering

(a) (b)

Figure 3 Computation model (a) and excavation elements (b)S1

minus05

minus9

minus85

minus8

minus75

minus7

minus65

minus6

minus55

minus5

minus45

minus4

minus35

minus3

minus25

minus2

minus15

minus1

(a)

S3

minus1

minus2

minus3

minus4

minus5

minus6

minus7

minus8

minus9

minus10

minus11

minus12

minus13

(b)

Figure 4 Initial first (a) and third (b) principal stress (MPa)

using the model established in this paper are similar tothose using Zienkiewicz-Pande or Mohr-Coulomb criterionbecause of the form of shear damage bound adopted Theintroduction of damage mechanismmakes it more natural tofigure out the shear and tension damage area

7 Conclusion

This paper approaches based on the microscopic mechanismof damage for rock material and establishes in an intuitiveway a multiparameter elastoplastic damage model for rockthat is applicable to engineering With the assumption thatdamage comes into existence as the materialrsquos strength and

stiffness degenerate and that damage is interconnected withplastic deformation a revised general form for elastoplasticdamage model containing damage variable of tensor formis established By considering plastic strain separation theexpression of damage variable reflecting the damage mech-anism for shear and tension simultaneously is derived Byadopting Zienkiewicz-Pande criterion with tension limit asthe bound for plasticity and damage the specific form for thedamage model is derived and implemented

The model established in this paper is physically intuitiveand has relativelywell-based theoretical backgroundNumer-ical experiments and engineering application show that thismodel can reflect the damage behavior of rock effectively

Mathematical Problems in Engineering 9

1 2 3 4 5 6 7 8 9 10Stage

25

20

15

10

5

0

Volu

me o

f dam

age z

one

Shear damageTension damage

Figure 5 Volume of damage zone (105m3)

Sheardamage

Sheardamage

Resilience

Figure 6 Damage zone of 2 unit section

S1

minus05

minus75

minus7

minus65

minus6

minus55

minus5

minus45

minus4

minus35

minus3

minus25

minus2

minus15

minus1

0

(a)

minus2

minus36

minus34

minus32

minus30

minus28

minus26

minus24

minus22

minus20

minus18

minus16

minus14

minus12

minus10

minus8

minus6

minus4

S3

(b)

Figure 7 First and third principal stress of 2 unit section (MPa)

10 Mathematical Problems in Engineering

D

45

40

35

30

25

20

15

10

5

Figure 8 Displacement of 2 unit section (mm)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] Z-Y Ye L Hong X-L Liu and T-B Yin ldquoConstitutive modelof rock based onmicrostructures simulationrdquo Journal of CentralSouth University of Technology vol 15 no 2 pp 230ndash236 2008

[2] M R Salari S Saeb K J Willam S J Patchet and R C Car-rasco ldquoA coupled elastoplastic damage model for geomaterialsrdquoComputer Methods in Applied Mechanics and Engineering vol193 no 27-29 pp 2625ndash2643 2004

[3] G D Nguyen and A M Korsunsky ldquoDevelopment of anapproach to constitutive modelling of concrete isotropic dam-age coupled with plasticityrdquo International Journal of Solids andStructures vol 45 no 20 pp 5483ndash5501 2008

[4] X LiW-G Cao andY-H Su ldquoA statistical damage constitutivemodel for softening behavior of rocksrdquoEngineeringGeology vol143-144 pp 1ndash17 2012

[5] M Xiao ldquoAnalysis on stability of surrounding rock and damagefracture of concrete lining for high pressure branch pipesrdquo Jour-nal of Wuhan University of Hydraulic and Electric Engineeringno 6 pp 594ndash599 1995

[6] G Frantziskonis and C S Desai ldquoConstitutive model withstrain softeningrdquo International Journal of Solids and Structuresvol 23 no 6 pp 733ndash750 1987

[7] D Krajcinovic and S Mastilovic ldquoSome fundamental issues ofdamage mechanicsrdquo Mechanics of Materials vol 21 no 3 pp217ndash230 1995

[8] L Resende ldquoA Damage mechanics constitutive theory for theinelastic behaviour of concreterdquo Computer Methods in AppliedMechanics and Engineering vol 60 no 1 pp 57ndash93 1987

[9] M Jirasek ldquoNonlocal models for damage and fracture com-parison of approachesrdquo International Journal of Solids andStructures vol 35 no 31-32 pp 4133ndash4145 1998

[10] M Jirasek and S Marfia ldquoNon-local damage model based ondisplacement averagingrdquo International Journal for NumericalMethods in Engineering vol 63 no 1 pp 77ndash102 2005

[11] J Lee andG L Fenves ldquoPlastic-damagemodel for cyclic loadingof concrete structuresrdquo Journal of Engineering Mechanics vol124 no 8 pp 892ndash900 1998

[12] J Li and J Y Wu ldquoElastoplacstic damage constitutive modelfor concrete based on damage energy release rates part I basicformulationsrdquo China Civil Engineering Journal no 9 pp 14ndash202005

[13] J Y Wu and J Li ldquoElastoplastic damage constitutive modelfor concrete based on damage energy release rates part IInumerical algorithm and verificationsrdquo China Civil EngineeringJournal no 9 pp 21ndash27 2005

[14] J W Ju ldquoOn energy-based coupled elastoplastic damagetheories constitutive modeling and computational aspectsrdquoInternational Journal of Solids and Structures vol 25 no 7 pp803ndash833 1989

[15] Y Q Ying ldquoOn rock plasticity damage and their constitutiveformulationrdquo Scientia Geologica Sinica no 1 pp 63ndash70 1995

[16] J C Simo and J W Ju ldquoStrain- and stress-based continuumdamage models-I Formulationrdquo International Journal of Solidsand Structures vol 23 no 7 pp 821ndash840 1987

[17] M H Yu YW ZanW Fan J Zhao and Z Z Dong ldquoAnvancesin strength theory of rock in 20 centurymdash100 years in memoryof theMohr-Coulomb Strehgth theoryrdquo Chinese Journal of RockMechanics and Engineering no 5 pp 545ndash550 2000

[18] O Zienkiewicz and G Pande ldquoSome useful forms of isotropicyield surfaces for soil and rockmechanicsrdquo in Finite Elements inGeomechanics pp 179ndash198 Wiley Cambridge UK 1977

[19] M Xiao Study on numerical analysis method of stability andsupporting for underground caverns [PhD thesis] Wuhan Uni-versity Wuhan China 2002

[20] Y Liu A M Maniatty and H Antes ldquoInvestigation of aZienkiewiczndashPande yield surface and an elasticndashviscoplasticboundary element formulationrdquo Engineering Analysis withBoundary Elements vol 24 no 2 pp 207ndash211 2000

[21] Y-J Li D-L Zhang and B-G Liu ldquoDevelopment and ver-ification of strain-softening model considering deformationmodulus degradation in FLAC3Drdquo Rock and Soil Mechanicsvol 32 no 2 pp 647ndash652 2011

[22] M S Diederichs P K Kaiser and E Eberhardt ldquoDamageinitiation and propagation in hard rock during tunnelling andthe influence of near-face stress rotationrdquo International Journalof Rock Mechanics and Mining Sciences vol 41 no 5 pp 785ndash812 2004

[23] J Lee and G L Fenves ldquoA return-mapping algorithm forplastic-damage models 3-D and plane stress formulationrdquoInternational Journal for Numerical Methods in Engineering vol50 no 2 pp 487ndash506 2001

[24] J Huang Q Peng and M-X Chen ldquoAn improved return-map stress update algorithm for finite deformation analysisof general isotropic elastoplastic geomaterialsrdquo InternationalJournal for Numerical and Analytical Methods in Geomechanicsvol 38 no 6 pp 636ndash660 2014

[25] J Zhou X Yang W Fu et al ldquoExperimental test and fracturedamage mechanical characteristics of brittle rock under uniax-ial cyclic loading and unloading conditionsrdquo Chinese Journal ofRock Mechanics and Engineering vol 29 no 6 pp 1172ndash11832010

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 5

lode angle 119903120590is lode radius namely radic2119869

2 11988821is the revising

coefficient introduced in [20] and 119886 controls to what extentapproximation can be made to the Mohr-Coulomb yieldsurface Here the damage type is determined by the firstbound the state reaches when loading Let potential functionbe

119865119904= 120590119898sin120595 minus 119888 cos120595

+ radic1

21198882

1

(119903120590

119892120595(120579120590))

2

+ 1198862sin2120595(33a)

119865119905= 119891119905 (33b)

where

119892120595(120579120590) =

2119896120595

(1 + 119896120595) minus (1 minus 119896

120595) sin 3120579

120590

119896120595=3 minus sin1205953 + sin120595

(34)

Here120595 denotes the angle of dilatancy Consider that 1199032120590= 21198692

sin3120579120590= minus(3radic32)(119869

311986932

2) 1205901= 120590119898+ (2radic33)119869

12

2sin(120579120590+

(23)120587) and 120597120590119898120597120590119894119895= (13)120575

119894119895 1205971198692120597120590119894119895= 119904119894119895 and

1205971198693120597120590119894119895= 119904119894119896119904119896119895minus (23)119869

2120575119894119895 it can be derived that

120597119891119904

120597120590119894119895

= 1198601120575119894119895+ 1198602119904119894119895+ 1198603119904119894119896119904119896119895 (35a)

120597119891119905

120597120590119894119895

= 1198604120575119894119895+ 1198605119904119894119895+ 1198606119904119894119896119904119896119895 (35b)

where

1198601=1

3sin120601 minus

radic3

1198882

11198923

120601(120579120590)11986912

211988711198872

1198602=

1

21198882

11198922

120601(120579120590)1198871minus

9radic3

41198882

11198923

120601(120579120590)

1198693

11986932

2

11988711198872

1198603=

3radic3

21198882

11198923

120601(120579120590)

1

11986912

2

11988711198872

1198604=1

3(1 + 2119887

3)

1198605=radic3

3

1

11986912

2

1198874+3

2

1198693

1198692

2

1198873

1198606= minus

1

1198692

1198873

(36a)

1198871= (

1

1198882

11198922 (120579120590)1198692+ 1198862sin2120601)

minus12

1198872=

2119896120601(1 minus 119896

120601)

((1 + 119896120601) minus (1 minus 119896

120601) sin 3120579

120590)2

1198873=cos (120579

120590+ (23) 120587)

1003816100381610038161003816cos 31205791205901003816100381610038161003816

1198874= sin(120579

120590+2

3120587)

(36b)

Consider that 120597119892120597120598119894119895= (120597119891120597120590

119896119897)(120590119896119897120598119894119895) = 119863

119894119895119896119897(120597119891120597120590

119896119897)

it is easy to get 120597119892120597120598119894119895accordingly Similarly 120597119865120597120590

119894119895and

120597119866120597120598119894119895can be obtained

Let elastic flexibility be an isotropic tensor

119862119894119895119896119897= minus120583

119864120575119894119895120575119896119897+1 + 120583

2119864(120575119894119896120575119895119897+ 120575119894119897120575119895119896) (37)

where 119864 denotes Youngrsquos modulus and 120583 denotes Poissonrsquosratio Suppose only Youngrsquos modulus is affected by damageinternal variables then similar to (30) we can get

120597119862119894119895119898119899

120597Ω119901119902

120590119898119899119872119901119902119896119897= 119870(1)

119894119895119870(2)

119896119897 (38a)

120597119862119904119905119898119899

120597Ω119901119902

120590119898119899119872119901119902119904119905= 119870(1)

119904119905119870(2)

119904119905 (38b)

where

119870(1)

119894119895=3120583

1198642120590119898120575119894119895minus2 (1 + 120583)

1198642120590119894119895

119870(2)

119896119897=

3

sum

120574=1

120597119864

120597120596120574

119867(120598119901

120574) 119901120574

119896119901120574

119897

(39)

Substitute (38a) and (38b) into (17) we can get 119870119894119895119896119897

whichtogether with (20) and (35a) and (35b) (also 120597119892120597120598

119894119895 120597119865120597120590

119894119895

and 120597119866120597120598119894119895which can be obtained in a similar approach)

leads to a constitutive relation in the form as (11) and (12)

42 The Damage Evolution of Material Strength and StiffnessMaterial strength and stiffness degenerate as damage evolvesHere those strength and stiffness parameters are treated asfunctions of the reduced form of damage variables 120596

120573and

their specific forms are examined Consider that damagevariable has a degenerating effect on cohesion and angle ofinternal friction it can be derived that

120597119891119904

120597120596120574

=120597119891119904

120597119888

120597119888

120597120596120574

+120597119891119904

120597120601

120597120601

120597120596120574

(for shearing) (40a)

Similarly consider the degenerating effect damage variablehas on tension limit we have

120597119891119905

120597120596120574

= minus120597120590119905

120597120596120574

(for tension) (40b)

6 Mathematical Problems in Engineering

Since Zienkiewicz-Pande yield surface is a smooth approx-imation to the Mohr-Coulomb yield surface and the Mohr-Coulomb criterion has the form

119888 =120590119888

2

1 minus sin120601cos120601 (41)

during uniaxial compression it can be assumed that thecohesion 119888 in Zienkiewicz-Pande criterion is proportional touniaxial compressive strength 120590

119888 Instead of plastic hardening

effect damage makes materialrsquos strength degenerate To sim-plify implementation omit the effect damage variable has onangle of internal friction and also consider the function typeadopted in [21] for the degeneration of materialrsquos parameterslet 119888 and 120590

119905be linear functions for damage variables

119888 =1 minus sin1206012 cos120601

(1205901198880minus 119864119904ℎ (120596120574)) (42a)

120590119905= 1205901199050minus 119864119905ℎ (120596120574) (42b)

where 1205901198880

and 1205901199050

denote the original compression andtension strength and 119864

119904and 119864

119905are damagemodulus for shear

and tension respectivelyAs internal microscopic cracks develop very fast after

reaching damage [22] exponential function is adopted

119864 = 1198640119890minus119877ℎ(120596120574) (43)

where1198640is the original Youngrsquosmodulus119877 ismaterialrsquos dam-

age coefficient and ℎ(120596120574) is taken as the linear combination

of damage variables 120596120574 Specifically let

ℎ (120596120574) =

max (120596120574) minusmin (120596

120574) (shear)

max (120596120574) (tension)

(44)

Given the irreversibility of damage during the damageevolution of materialrsquos parameters the minima of materialparameters along the evolution path are used

5 Numerical Example

The algorithm for solving group of nonlinear equationsincludes Newton-Raphson method modified Newton-Raphson method and quasi-Newton-Raphson method ofwhich Newton-Raphson method and modified Newton-Raphson method can effectively utilize the sparsity ofstiffness matrix Consider that the model raised in this paperhas many factors interacting with each other and thus highlynonlinearized a framework for nonlinear finite elementanalysis by using Newton-Raphson method is implementedwhich calls specific constitutive model to handle iterativesolving

Typical implementation ofmaterialmodel can be reducedto the following problems

(1) The establishment of tangent stiffness matrix whichcan be implemented using (11)

0

20

40

60

80

100

120

140

160

Stre

ss

0001 0002 0003 0004 0005 00060000Strain

Figure 1 Stress-strain curve under uniaxial compression (MPa)

(2) State update namely update of stress strain andmaterial parameters of each quadrature point whenstrain increment is known Typical implementationincludes explicit method (eg Runge-Kutta method)and implicit method (eg return map method [2324]) To simplify implementation explicit method isadopted which takes the incremental form of (11) andperform update and revision for state and materialparameters at the end of each increment step

To validate the effectiveness of the model establishedconsider the cylindrical sample in [25] Let 119864 = 3143GPa120583 = 023 119888 = 2622MPa and 120601 = 447∘ Apply uniaxialcompression to the sample model The stress-strain curve isdepicted in Figure 1 Comparison with the experiment resultof uniaxial compression in [25] shows that the establishedmodel can simulate this process well

Consider a cubic finite element model that each side haslength 1m Let 119864 = 317GPa 120583 = 026 119888 = 30MPa120601 = 464

∘ and 120590119905= 35MPa Apply cyclic compression

and tension uniaxial load The stress-strain curves are shownin Figure 2 When strain reaches 00018 and 00028 forcompression or when strain reaches 000016 and 00022 fortension unload stress to zero and then apply again Thetangent stiffness during unloading is smaller than elasticstiffness as the introduction of damage The model presentedin this paper can simulate damage behavior under cycliccompression and tension effectively

6 Engineering Application

In this section the established model is used for the excava-tion simulation of Zhenrsquoan underground power station

61 Project Overview and Numerical Modeling Zhenrsquoanpumped storage power station is located in Shanxi Provincein China with installed capacity of 1400MW (4 times 350MW)The underground powerhouse is made up of the main pow-erhouse main transform house tailrace surge chamber and

Mathematical Problems in Engineering 7

Table 1 Mechanical parameters for rock

Density (gcm3) Youngrsquos modulus (GPa) Poissonrsquos ratio Cohesion (MPa) Internal friction angle (∘) Tension limit (MPa)267 130 027 10 477 045

000

00

000

05

000

10

000

15

000

20

000

25

000

30

000

45

000

40

000

35

Strain

30

25

20

15

10

5

0

Stre

ss

(a) Compression

000

000

000

005

000

010

000

015

000

020

000

025

000

030

000

040

000

035

Strain

Stre

ss

40

35

30

25

20

15

10

05

00

(b) Tension

Figure 2 Stress-strain curve under cyclic uniaxial compression and tension (MPa)

several other tunnels connecting each cavern Mechanicalparameters for rock are shown in Table 1 Finite elementmodel shown in Figure 3 contains 480608 isoparametric 8-node elements with 109011 elements to be excavated dividedinto 10 stages 119909-axis of the model is perpendicular to lon-gitudinal axis of main powerhouse with a range of 31864m119910-axis is along the longitudinal axis with a range of 3896m119911-axis is along the vertical axis with a range of 5686m Initialstress is shown in Figure 4 It is obtained by inverse analysisbased on 4 gauging points near the carven and the resultshows that it is generated mainly by gravity The establishedmodel is used for excavation simulation of the undergroundpowerhouse

62 Result Analysis

621 Damage Zone Distribution To simulate the process ofexcavation divide the computation into 10 steps each witha layer shown in Figure 3 removed and let the iterationreach convergence from the unbalanced state Figure 5 givesthe development of damage zone where the third and sixthstage are the excavation of lateral caverns resulting insome finite elements returning from damage bound Whenexcavation is complete damage zone distribution of 2 unitsectionrsquos surrounding rocks is shown in Figure 6 Depth ofdamage zone reaches a maximum of 21m near the arch roofDepth of damage zone near side wall ranges from 184m to201m and ranges from 70m to 90m near main transformhouse Tension damage zone is mostly distributed near

the intersection of tunnels and caverns where release ofinitial stress along multiple direction makes it easy to crack

622 Stress Distribution After excavation stress distribu-tions of surrounding rock of each unit section are roughlysimilar The first and third principal stress of 2 unit sectionrsquossurrounding rocks are shown in Figure 7 the direction of thethird principal stress is almost vertical which is the same asthe direction of gravity and first principal stress is along theradial direction Near the intersection of each tunnel with themain powerhouse the third principal stress concentrates andreaches a maximum of minus20MPa First principal stress in themiddle part of 2 unit sectionrsquos side wall reaches a peak ofminus237MPa and third principal stress is around minus20MPa

623 Displacement Distribution What is shown in Figure 8is the displacement of 2 unit section Release of initialstress makes the displacement directing towards the cavernsDisplacement near the intersection of each cavern is relativelylarger than that of other areas In other areas displacementfield is distributed relatively uniformly After excavation 2unit sectionrsquos arch roof rsquos displacement resiles to 30mmMaximum displacement of main powerhousersquos upstream anddownstream side wall is 421mm and 355mm respectivelyThe values are 138mm and 196mm for main transformhouse

From the analysis above it is easy to know that thedamage zone stress and displacement distribution obtained

8 Mathematical Problems in Engineering

(a) (b)

Figure 3 Computation model (a) and excavation elements (b)S1

minus05

minus9

minus85

minus8

minus75

minus7

minus65

minus6

minus55

minus5

minus45

minus4

minus35

minus3

minus25

minus2

minus15

minus1

(a)

S3

minus1

minus2

minus3

minus4

minus5

minus6

minus7

minus8

minus9

minus10

minus11

minus12

minus13

(b)

Figure 4 Initial first (a) and third (b) principal stress (MPa)

using the model established in this paper are similar tothose using Zienkiewicz-Pande or Mohr-Coulomb criterionbecause of the form of shear damage bound adopted Theintroduction of damage mechanismmakes it more natural tofigure out the shear and tension damage area

7 Conclusion

This paper approaches based on the microscopic mechanismof damage for rock material and establishes in an intuitiveway a multiparameter elastoplastic damage model for rockthat is applicable to engineering With the assumption thatdamage comes into existence as the materialrsquos strength and

stiffness degenerate and that damage is interconnected withplastic deformation a revised general form for elastoplasticdamage model containing damage variable of tensor formis established By considering plastic strain separation theexpression of damage variable reflecting the damage mech-anism for shear and tension simultaneously is derived Byadopting Zienkiewicz-Pande criterion with tension limit asthe bound for plasticity and damage the specific form for thedamage model is derived and implemented

The model established in this paper is physically intuitiveand has relativelywell-based theoretical backgroundNumer-ical experiments and engineering application show that thismodel can reflect the damage behavior of rock effectively

Mathematical Problems in Engineering 9

1 2 3 4 5 6 7 8 9 10Stage

25

20

15

10

5

0

Volu

me o

f dam

age z

one

Shear damageTension damage

Figure 5 Volume of damage zone (105m3)

Sheardamage

Sheardamage

Resilience

Figure 6 Damage zone of 2 unit section

S1

minus05

minus75

minus7

minus65

minus6

minus55

minus5

minus45

minus4

minus35

minus3

minus25

minus2

minus15

minus1

0

(a)

minus2

minus36

minus34

minus32

minus30

minus28

minus26

minus24

minus22

minus20

minus18

minus16

minus14

minus12

minus10

minus8

minus6

minus4

S3

(b)

Figure 7 First and third principal stress of 2 unit section (MPa)

10 Mathematical Problems in Engineering

D

45

40

35

30

25

20

15

10

5

Figure 8 Displacement of 2 unit section (mm)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] Z-Y Ye L Hong X-L Liu and T-B Yin ldquoConstitutive modelof rock based onmicrostructures simulationrdquo Journal of CentralSouth University of Technology vol 15 no 2 pp 230ndash236 2008

[2] M R Salari S Saeb K J Willam S J Patchet and R C Car-rasco ldquoA coupled elastoplastic damage model for geomaterialsrdquoComputer Methods in Applied Mechanics and Engineering vol193 no 27-29 pp 2625ndash2643 2004

[3] G D Nguyen and A M Korsunsky ldquoDevelopment of anapproach to constitutive modelling of concrete isotropic dam-age coupled with plasticityrdquo International Journal of Solids andStructures vol 45 no 20 pp 5483ndash5501 2008

[4] X LiW-G Cao andY-H Su ldquoA statistical damage constitutivemodel for softening behavior of rocksrdquoEngineeringGeology vol143-144 pp 1ndash17 2012

[5] M Xiao ldquoAnalysis on stability of surrounding rock and damagefracture of concrete lining for high pressure branch pipesrdquo Jour-nal of Wuhan University of Hydraulic and Electric Engineeringno 6 pp 594ndash599 1995

[6] G Frantziskonis and C S Desai ldquoConstitutive model withstrain softeningrdquo International Journal of Solids and Structuresvol 23 no 6 pp 733ndash750 1987

[7] D Krajcinovic and S Mastilovic ldquoSome fundamental issues ofdamage mechanicsrdquo Mechanics of Materials vol 21 no 3 pp217ndash230 1995

[8] L Resende ldquoA Damage mechanics constitutive theory for theinelastic behaviour of concreterdquo Computer Methods in AppliedMechanics and Engineering vol 60 no 1 pp 57ndash93 1987

[9] M Jirasek ldquoNonlocal models for damage and fracture com-parison of approachesrdquo International Journal of Solids andStructures vol 35 no 31-32 pp 4133ndash4145 1998

[10] M Jirasek and S Marfia ldquoNon-local damage model based ondisplacement averagingrdquo International Journal for NumericalMethods in Engineering vol 63 no 1 pp 77ndash102 2005

[11] J Lee andG L Fenves ldquoPlastic-damagemodel for cyclic loadingof concrete structuresrdquo Journal of Engineering Mechanics vol124 no 8 pp 892ndash900 1998

[12] J Li and J Y Wu ldquoElastoplacstic damage constitutive modelfor concrete based on damage energy release rates part I basicformulationsrdquo China Civil Engineering Journal no 9 pp 14ndash202005

[13] J Y Wu and J Li ldquoElastoplastic damage constitutive modelfor concrete based on damage energy release rates part IInumerical algorithm and verificationsrdquo China Civil EngineeringJournal no 9 pp 21ndash27 2005

[14] J W Ju ldquoOn energy-based coupled elastoplastic damagetheories constitutive modeling and computational aspectsrdquoInternational Journal of Solids and Structures vol 25 no 7 pp803ndash833 1989

[15] Y Q Ying ldquoOn rock plasticity damage and their constitutiveformulationrdquo Scientia Geologica Sinica no 1 pp 63ndash70 1995

[16] J C Simo and J W Ju ldquoStrain- and stress-based continuumdamage models-I Formulationrdquo International Journal of Solidsand Structures vol 23 no 7 pp 821ndash840 1987

[17] M H Yu YW ZanW Fan J Zhao and Z Z Dong ldquoAnvancesin strength theory of rock in 20 centurymdash100 years in memoryof theMohr-Coulomb Strehgth theoryrdquo Chinese Journal of RockMechanics and Engineering no 5 pp 545ndash550 2000

[18] O Zienkiewicz and G Pande ldquoSome useful forms of isotropicyield surfaces for soil and rockmechanicsrdquo in Finite Elements inGeomechanics pp 179ndash198 Wiley Cambridge UK 1977

[19] M Xiao Study on numerical analysis method of stability andsupporting for underground caverns [PhD thesis] Wuhan Uni-versity Wuhan China 2002

[20] Y Liu A M Maniatty and H Antes ldquoInvestigation of aZienkiewiczndashPande yield surface and an elasticndashviscoplasticboundary element formulationrdquo Engineering Analysis withBoundary Elements vol 24 no 2 pp 207ndash211 2000

[21] Y-J Li D-L Zhang and B-G Liu ldquoDevelopment and ver-ification of strain-softening model considering deformationmodulus degradation in FLAC3Drdquo Rock and Soil Mechanicsvol 32 no 2 pp 647ndash652 2011

[22] M S Diederichs P K Kaiser and E Eberhardt ldquoDamageinitiation and propagation in hard rock during tunnelling andthe influence of near-face stress rotationrdquo International Journalof Rock Mechanics and Mining Sciences vol 41 no 5 pp 785ndash812 2004

[23] J Lee and G L Fenves ldquoA return-mapping algorithm forplastic-damage models 3-D and plane stress formulationrdquoInternational Journal for Numerical Methods in Engineering vol50 no 2 pp 487ndash506 2001

[24] J Huang Q Peng and M-X Chen ldquoAn improved return-map stress update algorithm for finite deformation analysisof general isotropic elastoplastic geomaterialsrdquo InternationalJournal for Numerical and Analytical Methods in Geomechanicsvol 38 no 6 pp 636ndash660 2014

[25] J Zhou X Yang W Fu et al ldquoExperimental test and fracturedamage mechanical characteristics of brittle rock under uniax-ial cyclic loading and unloading conditionsrdquo Chinese Journal ofRock Mechanics and Engineering vol 29 no 6 pp 1172ndash11832010

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

6 Mathematical Problems in Engineering

Since Zienkiewicz-Pande yield surface is a smooth approx-imation to the Mohr-Coulomb yield surface and the Mohr-Coulomb criterion has the form

119888 =120590119888

2

1 minus sin120601cos120601 (41)

during uniaxial compression it can be assumed that thecohesion 119888 in Zienkiewicz-Pande criterion is proportional touniaxial compressive strength 120590

119888 Instead of plastic hardening

effect damage makes materialrsquos strength degenerate To sim-plify implementation omit the effect damage variable has onangle of internal friction and also consider the function typeadopted in [21] for the degeneration of materialrsquos parameterslet 119888 and 120590

119905be linear functions for damage variables

119888 =1 minus sin1206012 cos120601

(1205901198880minus 119864119904ℎ (120596120574)) (42a)

120590119905= 1205901199050minus 119864119905ℎ (120596120574) (42b)

where 1205901198880

and 1205901199050

denote the original compression andtension strength and 119864

119904and 119864

119905are damagemodulus for shear

and tension respectivelyAs internal microscopic cracks develop very fast after

reaching damage [22] exponential function is adopted

119864 = 1198640119890minus119877ℎ(120596120574) (43)

where1198640is the original Youngrsquosmodulus119877 ismaterialrsquos dam-

age coefficient and ℎ(120596120574) is taken as the linear combination

of damage variables 120596120574 Specifically let

ℎ (120596120574) =

max (120596120574) minusmin (120596

120574) (shear)

max (120596120574) (tension)

(44)

Given the irreversibility of damage during the damageevolution of materialrsquos parameters the minima of materialparameters along the evolution path are used

5 Numerical Example

The algorithm for solving group of nonlinear equationsincludes Newton-Raphson method modified Newton-Raphson method and quasi-Newton-Raphson method ofwhich Newton-Raphson method and modified Newton-Raphson method can effectively utilize the sparsity ofstiffness matrix Consider that the model raised in this paperhas many factors interacting with each other and thus highlynonlinearized a framework for nonlinear finite elementanalysis by using Newton-Raphson method is implementedwhich calls specific constitutive model to handle iterativesolving

Typical implementation ofmaterialmodel can be reducedto the following problems

(1) The establishment of tangent stiffness matrix whichcan be implemented using (11)

0

20

40

60

80

100

120

140

160

Stre

ss

0001 0002 0003 0004 0005 00060000Strain

Figure 1 Stress-strain curve under uniaxial compression (MPa)

(2) State update namely update of stress strain andmaterial parameters of each quadrature point whenstrain increment is known Typical implementationincludes explicit method (eg Runge-Kutta method)and implicit method (eg return map method [2324]) To simplify implementation explicit method isadopted which takes the incremental form of (11) andperform update and revision for state and materialparameters at the end of each increment step

To validate the effectiveness of the model establishedconsider the cylindrical sample in [25] Let 119864 = 3143GPa120583 = 023 119888 = 2622MPa and 120601 = 447∘ Apply uniaxialcompression to the sample model The stress-strain curve isdepicted in Figure 1 Comparison with the experiment resultof uniaxial compression in [25] shows that the establishedmodel can simulate this process well

Consider a cubic finite element model that each side haslength 1m Let 119864 = 317GPa 120583 = 026 119888 = 30MPa120601 = 464

∘ and 120590119905= 35MPa Apply cyclic compression

and tension uniaxial load The stress-strain curves are shownin Figure 2 When strain reaches 00018 and 00028 forcompression or when strain reaches 000016 and 00022 fortension unload stress to zero and then apply again Thetangent stiffness during unloading is smaller than elasticstiffness as the introduction of damage The model presentedin this paper can simulate damage behavior under cycliccompression and tension effectively

6 Engineering Application

In this section the established model is used for the excava-tion simulation of Zhenrsquoan underground power station

61 Project Overview and Numerical Modeling Zhenrsquoanpumped storage power station is located in Shanxi Provincein China with installed capacity of 1400MW (4 times 350MW)The underground powerhouse is made up of the main pow-erhouse main transform house tailrace surge chamber and

Mathematical Problems in Engineering 7

Table 1 Mechanical parameters for rock

Density (gcm3) Youngrsquos modulus (GPa) Poissonrsquos ratio Cohesion (MPa) Internal friction angle (∘) Tension limit (MPa)267 130 027 10 477 045

000

00

000

05

000

10

000

15

000

20

000

25

000

30

000

45

000

40

000

35

Strain

30

25

20

15

10

5

0

Stre

ss

(a) Compression

000

000

000

005

000

010

000

015

000

020

000

025

000

030

000

040

000

035

Strain

Stre

ss

40

35

30

25

20

15

10

05

00

(b) Tension

Figure 2 Stress-strain curve under cyclic uniaxial compression and tension (MPa)

several other tunnels connecting each cavern Mechanicalparameters for rock are shown in Table 1 Finite elementmodel shown in Figure 3 contains 480608 isoparametric 8-node elements with 109011 elements to be excavated dividedinto 10 stages 119909-axis of the model is perpendicular to lon-gitudinal axis of main powerhouse with a range of 31864m119910-axis is along the longitudinal axis with a range of 3896m119911-axis is along the vertical axis with a range of 5686m Initialstress is shown in Figure 4 It is obtained by inverse analysisbased on 4 gauging points near the carven and the resultshows that it is generated mainly by gravity The establishedmodel is used for excavation simulation of the undergroundpowerhouse

62 Result Analysis

621 Damage Zone Distribution To simulate the process ofexcavation divide the computation into 10 steps each witha layer shown in Figure 3 removed and let the iterationreach convergence from the unbalanced state Figure 5 givesthe development of damage zone where the third and sixthstage are the excavation of lateral caverns resulting insome finite elements returning from damage bound Whenexcavation is complete damage zone distribution of 2 unitsectionrsquos surrounding rocks is shown in Figure 6 Depth ofdamage zone reaches a maximum of 21m near the arch roofDepth of damage zone near side wall ranges from 184m to201m and ranges from 70m to 90m near main transformhouse Tension damage zone is mostly distributed near

the intersection of tunnels and caverns where release ofinitial stress along multiple direction makes it easy to crack

622 Stress Distribution After excavation stress distribu-tions of surrounding rock of each unit section are roughlysimilar The first and third principal stress of 2 unit sectionrsquossurrounding rocks are shown in Figure 7 the direction of thethird principal stress is almost vertical which is the same asthe direction of gravity and first principal stress is along theradial direction Near the intersection of each tunnel with themain powerhouse the third principal stress concentrates andreaches a maximum of minus20MPa First principal stress in themiddle part of 2 unit sectionrsquos side wall reaches a peak ofminus237MPa and third principal stress is around minus20MPa

623 Displacement Distribution What is shown in Figure 8is the displacement of 2 unit section Release of initialstress makes the displacement directing towards the cavernsDisplacement near the intersection of each cavern is relativelylarger than that of other areas In other areas displacementfield is distributed relatively uniformly After excavation 2unit sectionrsquos arch roof rsquos displacement resiles to 30mmMaximum displacement of main powerhousersquos upstream anddownstream side wall is 421mm and 355mm respectivelyThe values are 138mm and 196mm for main transformhouse

From the analysis above it is easy to know that thedamage zone stress and displacement distribution obtained

8 Mathematical Problems in Engineering

(a) (b)

Figure 3 Computation model (a) and excavation elements (b)S1

minus05

minus9

minus85

minus8

minus75

minus7

minus65

minus6

minus55

minus5

minus45

minus4

minus35

minus3

minus25

minus2

minus15

minus1

(a)

S3

minus1

minus2

minus3

minus4

minus5

minus6

minus7

minus8

minus9

minus10

minus11

minus12

minus13

(b)

Figure 4 Initial first (a) and third (b) principal stress (MPa)

using the model established in this paper are similar tothose using Zienkiewicz-Pande or Mohr-Coulomb criterionbecause of the form of shear damage bound adopted Theintroduction of damage mechanismmakes it more natural tofigure out the shear and tension damage area

7 Conclusion

This paper approaches based on the microscopic mechanismof damage for rock material and establishes in an intuitiveway a multiparameter elastoplastic damage model for rockthat is applicable to engineering With the assumption thatdamage comes into existence as the materialrsquos strength and

stiffness degenerate and that damage is interconnected withplastic deformation a revised general form for elastoplasticdamage model containing damage variable of tensor formis established By considering plastic strain separation theexpression of damage variable reflecting the damage mech-anism for shear and tension simultaneously is derived Byadopting Zienkiewicz-Pande criterion with tension limit asthe bound for plasticity and damage the specific form for thedamage model is derived and implemented

The model established in this paper is physically intuitiveand has relativelywell-based theoretical backgroundNumer-ical experiments and engineering application show that thismodel can reflect the damage behavior of rock effectively

Mathematical Problems in Engineering 9

1 2 3 4 5 6 7 8 9 10Stage

25

20

15

10

5

0

Volu

me o

f dam

age z

one

Shear damageTension damage

Figure 5 Volume of damage zone (105m3)

Sheardamage

Sheardamage

Resilience

Figure 6 Damage zone of 2 unit section

S1

minus05

minus75

minus7

minus65

minus6

minus55

minus5

minus45

minus4

minus35

minus3

minus25

minus2

minus15

minus1

0

(a)

minus2

minus36

minus34

minus32

minus30

minus28

minus26

minus24

minus22

minus20

minus18

minus16

minus14

minus12

minus10

minus8

minus6

minus4

S3

(b)

Figure 7 First and third principal stress of 2 unit section (MPa)

10 Mathematical Problems in Engineering

D

45

40

35

30

25

20

15

10

5

Figure 8 Displacement of 2 unit section (mm)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] Z-Y Ye L Hong X-L Liu and T-B Yin ldquoConstitutive modelof rock based onmicrostructures simulationrdquo Journal of CentralSouth University of Technology vol 15 no 2 pp 230ndash236 2008

[2] M R Salari S Saeb K J Willam S J Patchet and R C Car-rasco ldquoA coupled elastoplastic damage model for geomaterialsrdquoComputer Methods in Applied Mechanics and Engineering vol193 no 27-29 pp 2625ndash2643 2004

[3] G D Nguyen and A M Korsunsky ldquoDevelopment of anapproach to constitutive modelling of concrete isotropic dam-age coupled with plasticityrdquo International Journal of Solids andStructures vol 45 no 20 pp 5483ndash5501 2008

[4] X LiW-G Cao andY-H Su ldquoA statistical damage constitutivemodel for softening behavior of rocksrdquoEngineeringGeology vol143-144 pp 1ndash17 2012

[5] M Xiao ldquoAnalysis on stability of surrounding rock and damagefracture of concrete lining for high pressure branch pipesrdquo Jour-nal of Wuhan University of Hydraulic and Electric Engineeringno 6 pp 594ndash599 1995

[6] G Frantziskonis and C S Desai ldquoConstitutive model withstrain softeningrdquo International Journal of Solids and Structuresvol 23 no 6 pp 733ndash750 1987

[7] D Krajcinovic and S Mastilovic ldquoSome fundamental issues ofdamage mechanicsrdquo Mechanics of Materials vol 21 no 3 pp217ndash230 1995

[8] L Resende ldquoA Damage mechanics constitutive theory for theinelastic behaviour of concreterdquo Computer Methods in AppliedMechanics and Engineering vol 60 no 1 pp 57ndash93 1987

[9] M Jirasek ldquoNonlocal models for damage and fracture com-parison of approachesrdquo International Journal of Solids andStructures vol 35 no 31-32 pp 4133ndash4145 1998

[10] M Jirasek and S Marfia ldquoNon-local damage model based ondisplacement averagingrdquo International Journal for NumericalMethods in Engineering vol 63 no 1 pp 77ndash102 2005

[11] J Lee andG L Fenves ldquoPlastic-damagemodel for cyclic loadingof concrete structuresrdquo Journal of Engineering Mechanics vol124 no 8 pp 892ndash900 1998

[12] J Li and J Y Wu ldquoElastoplacstic damage constitutive modelfor concrete based on damage energy release rates part I basicformulationsrdquo China Civil Engineering Journal no 9 pp 14ndash202005

[13] J Y Wu and J Li ldquoElastoplastic damage constitutive modelfor concrete based on damage energy release rates part IInumerical algorithm and verificationsrdquo China Civil EngineeringJournal no 9 pp 21ndash27 2005

[14] J W Ju ldquoOn energy-based coupled elastoplastic damagetheories constitutive modeling and computational aspectsrdquoInternational Journal of Solids and Structures vol 25 no 7 pp803ndash833 1989

[15] Y Q Ying ldquoOn rock plasticity damage and their constitutiveformulationrdquo Scientia Geologica Sinica no 1 pp 63ndash70 1995

[16] J C Simo and J W Ju ldquoStrain- and stress-based continuumdamage models-I Formulationrdquo International Journal of Solidsand Structures vol 23 no 7 pp 821ndash840 1987

[17] M H Yu YW ZanW Fan J Zhao and Z Z Dong ldquoAnvancesin strength theory of rock in 20 centurymdash100 years in memoryof theMohr-Coulomb Strehgth theoryrdquo Chinese Journal of RockMechanics and Engineering no 5 pp 545ndash550 2000

[18] O Zienkiewicz and G Pande ldquoSome useful forms of isotropicyield surfaces for soil and rockmechanicsrdquo in Finite Elements inGeomechanics pp 179ndash198 Wiley Cambridge UK 1977

[19] M Xiao Study on numerical analysis method of stability andsupporting for underground caverns [PhD thesis] Wuhan Uni-versity Wuhan China 2002

[20] Y Liu A M Maniatty and H Antes ldquoInvestigation of aZienkiewiczndashPande yield surface and an elasticndashviscoplasticboundary element formulationrdquo Engineering Analysis withBoundary Elements vol 24 no 2 pp 207ndash211 2000

[21] Y-J Li D-L Zhang and B-G Liu ldquoDevelopment and ver-ification of strain-softening model considering deformationmodulus degradation in FLAC3Drdquo Rock and Soil Mechanicsvol 32 no 2 pp 647ndash652 2011

[22] M S Diederichs P K Kaiser and E Eberhardt ldquoDamageinitiation and propagation in hard rock during tunnelling andthe influence of near-face stress rotationrdquo International Journalof Rock Mechanics and Mining Sciences vol 41 no 5 pp 785ndash812 2004

[23] J Lee and G L Fenves ldquoA return-mapping algorithm forplastic-damage models 3-D and plane stress formulationrdquoInternational Journal for Numerical Methods in Engineering vol50 no 2 pp 487ndash506 2001

[24] J Huang Q Peng and M-X Chen ldquoAn improved return-map stress update algorithm for finite deformation analysisof general isotropic elastoplastic geomaterialsrdquo InternationalJournal for Numerical and Analytical Methods in Geomechanicsvol 38 no 6 pp 636ndash660 2014

[25] J Zhou X Yang W Fu et al ldquoExperimental test and fracturedamage mechanical characteristics of brittle rock under uniax-ial cyclic loading and unloading conditionsrdquo Chinese Journal ofRock Mechanics and Engineering vol 29 no 6 pp 1172ndash11832010

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 7

Table 1 Mechanical parameters for rock

Density (gcm3) Youngrsquos modulus (GPa) Poissonrsquos ratio Cohesion (MPa) Internal friction angle (∘) Tension limit (MPa)267 130 027 10 477 045

000

00

000

05

000

10

000

15

000

20

000

25

000

30

000

45

000

40

000

35

Strain

30

25

20

15

10

5

0

Stre

ss

(a) Compression

000

000

000

005

000

010

000

015

000

020

000

025

000

030

000

040

000

035

Strain

Stre

ss

40

35

30

25

20

15

10

05

00

(b) Tension

Figure 2 Stress-strain curve under cyclic uniaxial compression and tension (MPa)

several other tunnels connecting each cavern Mechanicalparameters for rock are shown in Table 1 Finite elementmodel shown in Figure 3 contains 480608 isoparametric 8-node elements with 109011 elements to be excavated dividedinto 10 stages 119909-axis of the model is perpendicular to lon-gitudinal axis of main powerhouse with a range of 31864m119910-axis is along the longitudinal axis with a range of 3896m119911-axis is along the vertical axis with a range of 5686m Initialstress is shown in Figure 4 It is obtained by inverse analysisbased on 4 gauging points near the carven and the resultshows that it is generated mainly by gravity The establishedmodel is used for excavation simulation of the undergroundpowerhouse

62 Result Analysis

621 Damage Zone Distribution To simulate the process ofexcavation divide the computation into 10 steps each witha layer shown in Figure 3 removed and let the iterationreach convergence from the unbalanced state Figure 5 givesthe development of damage zone where the third and sixthstage are the excavation of lateral caverns resulting insome finite elements returning from damage bound Whenexcavation is complete damage zone distribution of 2 unitsectionrsquos surrounding rocks is shown in Figure 6 Depth ofdamage zone reaches a maximum of 21m near the arch roofDepth of damage zone near side wall ranges from 184m to201m and ranges from 70m to 90m near main transformhouse Tension damage zone is mostly distributed near

the intersection of tunnels and caverns where release ofinitial stress along multiple direction makes it easy to crack

622 Stress Distribution After excavation stress distribu-tions of surrounding rock of each unit section are roughlysimilar The first and third principal stress of 2 unit sectionrsquossurrounding rocks are shown in Figure 7 the direction of thethird principal stress is almost vertical which is the same asthe direction of gravity and first principal stress is along theradial direction Near the intersection of each tunnel with themain powerhouse the third principal stress concentrates andreaches a maximum of minus20MPa First principal stress in themiddle part of 2 unit sectionrsquos side wall reaches a peak ofminus237MPa and third principal stress is around minus20MPa

623 Displacement Distribution What is shown in Figure 8is the displacement of 2 unit section Release of initialstress makes the displacement directing towards the cavernsDisplacement near the intersection of each cavern is relativelylarger than that of other areas In other areas displacementfield is distributed relatively uniformly After excavation 2unit sectionrsquos arch roof rsquos displacement resiles to 30mmMaximum displacement of main powerhousersquos upstream anddownstream side wall is 421mm and 355mm respectivelyThe values are 138mm and 196mm for main transformhouse

From the analysis above it is easy to know that thedamage zone stress and displacement distribution obtained

8 Mathematical Problems in Engineering

(a) (b)

Figure 3 Computation model (a) and excavation elements (b)S1

minus05

minus9

minus85

minus8

minus75

minus7

minus65

minus6

minus55

minus5

minus45

minus4

minus35

minus3

minus25

minus2

minus15

minus1

(a)

S3

minus1

minus2

minus3

minus4

minus5

minus6

minus7

minus8

minus9

minus10

minus11

minus12

minus13

(b)

Figure 4 Initial first (a) and third (b) principal stress (MPa)

using the model established in this paper are similar tothose using Zienkiewicz-Pande or Mohr-Coulomb criterionbecause of the form of shear damage bound adopted Theintroduction of damage mechanismmakes it more natural tofigure out the shear and tension damage area

7 Conclusion

This paper approaches based on the microscopic mechanismof damage for rock material and establishes in an intuitiveway a multiparameter elastoplastic damage model for rockthat is applicable to engineering With the assumption thatdamage comes into existence as the materialrsquos strength and

stiffness degenerate and that damage is interconnected withplastic deformation a revised general form for elastoplasticdamage model containing damage variable of tensor formis established By considering plastic strain separation theexpression of damage variable reflecting the damage mech-anism for shear and tension simultaneously is derived Byadopting Zienkiewicz-Pande criterion with tension limit asthe bound for plasticity and damage the specific form for thedamage model is derived and implemented

The model established in this paper is physically intuitiveand has relativelywell-based theoretical backgroundNumer-ical experiments and engineering application show that thismodel can reflect the damage behavior of rock effectively

Mathematical Problems in Engineering 9

1 2 3 4 5 6 7 8 9 10Stage

25

20

15

10

5

0

Volu

me o

f dam

age z

one

Shear damageTension damage

Figure 5 Volume of damage zone (105m3)

Sheardamage

Sheardamage

Resilience

Figure 6 Damage zone of 2 unit section

S1

minus05

minus75

minus7

minus65

minus6

minus55

minus5

minus45

minus4

minus35

minus3

minus25

minus2

minus15

minus1

0

(a)

minus2

minus36

minus34

minus32

minus30

minus28

minus26

minus24

minus22

minus20

minus18

minus16

minus14

minus12

minus10

minus8

minus6

minus4

S3

(b)

Figure 7 First and third principal stress of 2 unit section (MPa)

10 Mathematical Problems in Engineering

D

45

40

35

30

25

20

15

10

5

Figure 8 Displacement of 2 unit section (mm)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] Z-Y Ye L Hong X-L Liu and T-B Yin ldquoConstitutive modelof rock based onmicrostructures simulationrdquo Journal of CentralSouth University of Technology vol 15 no 2 pp 230ndash236 2008

[2] M R Salari S Saeb K J Willam S J Patchet and R C Car-rasco ldquoA coupled elastoplastic damage model for geomaterialsrdquoComputer Methods in Applied Mechanics and Engineering vol193 no 27-29 pp 2625ndash2643 2004

[3] G D Nguyen and A M Korsunsky ldquoDevelopment of anapproach to constitutive modelling of concrete isotropic dam-age coupled with plasticityrdquo International Journal of Solids andStructures vol 45 no 20 pp 5483ndash5501 2008

[4] X LiW-G Cao andY-H Su ldquoA statistical damage constitutivemodel for softening behavior of rocksrdquoEngineeringGeology vol143-144 pp 1ndash17 2012

[5] M Xiao ldquoAnalysis on stability of surrounding rock and damagefracture of concrete lining for high pressure branch pipesrdquo Jour-nal of Wuhan University of Hydraulic and Electric Engineeringno 6 pp 594ndash599 1995

[6] G Frantziskonis and C S Desai ldquoConstitutive model withstrain softeningrdquo International Journal of Solids and Structuresvol 23 no 6 pp 733ndash750 1987

[7] D Krajcinovic and S Mastilovic ldquoSome fundamental issues ofdamage mechanicsrdquo Mechanics of Materials vol 21 no 3 pp217ndash230 1995

[8] L Resende ldquoA Damage mechanics constitutive theory for theinelastic behaviour of concreterdquo Computer Methods in AppliedMechanics and Engineering vol 60 no 1 pp 57ndash93 1987

[9] M Jirasek ldquoNonlocal models for damage and fracture com-parison of approachesrdquo International Journal of Solids andStructures vol 35 no 31-32 pp 4133ndash4145 1998

[10] M Jirasek and S Marfia ldquoNon-local damage model based ondisplacement averagingrdquo International Journal for NumericalMethods in Engineering vol 63 no 1 pp 77ndash102 2005

[11] J Lee andG L Fenves ldquoPlastic-damagemodel for cyclic loadingof concrete structuresrdquo Journal of Engineering Mechanics vol124 no 8 pp 892ndash900 1998

[12] J Li and J Y Wu ldquoElastoplacstic damage constitutive modelfor concrete based on damage energy release rates part I basicformulationsrdquo China Civil Engineering Journal no 9 pp 14ndash202005

[13] J Y Wu and J Li ldquoElastoplastic damage constitutive modelfor concrete based on damage energy release rates part IInumerical algorithm and verificationsrdquo China Civil EngineeringJournal no 9 pp 21ndash27 2005

[14] J W Ju ldquoOn energy-based coupled elastoplastic damagetheories constitutive modeling and computational aspectsrdquoInternational Journal of Solids and Structures vol 25 no 7 pp803ndash833 1989

[15] Y Q Ying ldquoOn rock plasticity damage and their constitutiveformulationrdquo Scientia Geologica Sinica no 1 pp 63ndash70 1995

[16] J C Simo and J W Ju ldquoStrain- and stress-based continuumdamage models-I Formulationrdquo International Journal of Solidsand Structures vol 23 no 7 pp 821ndash840 1987

[17] M H Yu YW ZanW Fan J Zhao and Z Z Dong ldquoAnvancesin strength theory of rock in 20 centurymdash100 years in memoryof theMohr-Coulomb Strehgth theoryrdquo Chinese Journal of RockMechanics and Engineering no 5 pp 545ndash550 2000

[18] O Zienkiewicz and G Pande ldquoSome useful forms of isotropicyield surfaces for soil and rockmechanicsrdquo in Finite Elements inGeomechanics pp 179ndash198 Wiley Cambridge UK 1977

[19] M Xiao Study on numerical analysis method of stability andsupporting for underground caverns [PhD thesis] Wuhan Uni-versity Wuhan China 2002

[20] Y Liu A M Maniatty and H Antes ldquoInvestigation of aZienkiewiczndashPande yield surface and an elasticndashviscoplasticboundary element formulationrdquo Engineering Analysis withBoundary Elements vol 24 no 2 pp 207ndash211 2000

[21] Y-J Li D-L Zhang and B-G Liu ldquoDevelopment and ver-ification of strain-softening model considering deformationmodulus degradation in FLAC3Drdquo Rock and Soil Mechanicsvol 32 no 2 pp 647ndash652 2011

[22] M S Diederichs P K Kaiser and E Eberhardt ldquoDamageinitiation and propagation in hard rock during tunnelling andthe influence of near-face stress rotationrdquo International Journalof Rock Mechanics and Mining Sciences vol 41 no 5 pp 785ndash812 2004

[23] J Lee and G L Fenves ldquoA return-mapping algorithm forplastic-damage models 3-D and plane stress formulationrdquoInternational Journal for Numerical Methods in Engineering vol50 no 2 pp 487ndash506 2001

[24] J Huang Q Peng and M-X Chen ldquoAn improved return-map stress update algorithm for finite deformation analysisof general isotropic elastoplastic geomaterialsrdquo InternationalJournal for Numerical and Analytical Methods in Geomechanicsvol 38 no 6 pp 636ndash660 2014

[25] J Zhou X Yang W Fu et al ldquoExperimental test and fracturedamage mechanical characteristics of brittle rock under uniax-ial cyclic loading and unloading conditionsrdquo Chinese Journal ofRock Mechanics and Engineering vol 29 no 6 pp 1172ndash11832010

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

(a) (b)

Figure 3 Computation model (a) and excavation elements (b)S1

minus05

minus9

minus85

minus8

minus75

minus7

minus65

minus6

minus55

minus5

minus45

minus4

minus35

minus3

minus25

minus2

minus15

minus1

(a)

S3

minus1

minus2

minus3

minus4

minus5

minus6

minus7

minus8

minus9

minus10

minus11

minus12

minus13

(b)

Figure 4 Initial first (a) and third (b) principal stress (MPa)

using the model established in this paper are similar tothose using Zienkiewicz-Pande or Mohr-Coulomb criterionbecause of the form of shear damage bound adopted Theintroduction of damage mechanismmakes it more natural tofigure out the shear and tension damage area

7 Conclusion

This paper approaches based on the microscopic mechanismof damage for rock material and establishes in an intuitiveway a multiparameter elastoplastic damage model for rockthat is applicable to engineering With the assumption thatdamage comes into existence as the materialrsquos strength and

stiffness degenerate and that damage is interconnected withplastic deformation a revised general form for elastoplasticdamage model containing damage variable of tensor formis established By considering plastic strain separation theexpression of damage variable reflecting the damage mech-anism for shear and tension simultaneously is derived Byadopting Zienkiewicz-Pande criterion with tension limit asthe bound for plasticity and damage the specific form for thedamage model is derived and implemented

The model established in this paper is physically intuitiveand has relativelywell-based theoretical backgroundNumer-ical experiments and engineering application show that thismodel can reflect the damage behavior of rock effectively

Mathematical Problems in Engineering 9

1 2 3 4 5 6 7 8 9 10Stage

25

20

15

10

5

0

Volu

me o

f dam

age z

one

Shear damageTension damage

Figure 5 Volume of damage zone (105m3)

Sheardamage

Sheardamage

Resilience

Figure 6 Damage zone of 2 unit section

S1

minus05

minus75

minus7

minus65

minus6

minus55

minus5

minus45

minus4

minus35

minus3

minus25

minus2

minus15

minus1

0

(a)

minus2

minus36

minus34

minus32

minus30

minus28

minus26

minus24

minus22

minus20

minus18

minus16

minus14

minus12

minus10

minus8

minus6

minus4

S3

(b)

Figure 7 First and third principal stress of 2 unit section (MPa)

10 Mathematical Problems in Engineering

D

45

40

35

30

25

20

15

10

5

Figure 8 Displacement of 2 unit section (mm)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] Z-Y Ye L Hong X-L Liu and T-B Yin ldquoConstitutive modelof rock based onmicrostructures simulationrdquo Journal of CentralSouth University of Technology vol 15 no 2 pp 230ndash236 2008

[2] M R Salari S Saeb K J Willam S J Patchet and R C Car-rasco ldquoA coupled elastoplastic damage model for geomaterialsrdquoComputer Methods in Applied Mechanics and Engineering vol193 no 27-29 pp 2625ndash2643 2004

[3] G D Nguyen and A M Korsunsky ldquoDevelopment of anapproach to constitutive modelling of concrete isotropic dam-age coupled with plasticityrdquo International Journal of Solids andStructures vol 45 no 20 pp 5483ndash5501 2008

[4] X LiW-G Cao andY-H Su ldquoA statistical damage constitutivemodel for softening behavior of rocksrdquoEngineeringGeology vol143-144 pp 1ndash17 2012

[5] M Xiao ldquoAnalysis on stability of surrounding rock and damagefracture of concrete lining for high pressure branch pipesrdquo Jour-nal of Wuhan University of Hydraulic and Electric Engineeringno 6 pp 594ndash599 1995

[6] G Frantziskonis and C S Desai ldquoConstitutive model withstrain softeningrdquo International Journal of Solids and Structuresvol 23 no 6 pp 733ndash750 1987

[7] D Krajcinovic and S Mastilovic ldquoSome fundamental issues ofdamage mechanicsrdquo Mechanics of Materials vol 21 no 3 pp217ndash230 1995

[8] L Resende ldquoA Damage mechanics constitutive theory for theinelastic behaviour of concreterdquo Computer Methods in AppliedMechanics and Engineering vol 60 no 1 pp 57ndash93 1987

[9] M Jirasek ldquoNonlocal models for damage and fracture com-parison of approachesrdquo International Journal of Solids andStructures vol 35 no 31-32 pp 4133ndash4145 1998

[10] M Jirasek and S Marfia ldquoNon-local damage model based ondisplacement averagingrdquo International Journal for NumericalMethods in Engineering vol 63 no 1 pp 77ndash102 2005

[11] J Lee andG L Fenves ldquoPlastic-damagemodel for cyclic loadingof concrete structuresrdquo Journal of Engineering Mechanics vol124 no 8 pp 892ndash900 1998

[12] J Li and J Y Wu ldquoElastoplacstic damage constitutive modelfor concrete based on damage energy release rates part I basicformulationsrdquo China Civil Engineering Journal no 9 pp 14ndash202005

[13] J Y Wu and J Li ldquoElastoplastic damage constitutive modelfor concrete based on damage energy release rates part IInumerical algorithm and verificationsrdquo China Civil EngineeringJournal no 9 pp 21ndash27 2005

[14] J W Ju ldquoOn energy-based coupled elastoplastic damagetheories constitutive modeling and computational aspectsrdquoInternational Journal of Solids and Structures vol 25 no 7 pp803ndash833 1989

[15] Y Q Ying ldquoOn rock plasticity damage and their constitutiveformulationrdquo Scientia Geologica Sinica no 1 pp 63ndash70 1995

[16] J C Simo and J W Ju ldquoStrain- and stress-based continuumdamage models-I Formulationrdquo International Journal of Solidsand Structures vol 23 no 7 pp 821ndash840 1987

[17] M H Yu YW ZanW Fan J Zhao and Z Z Dong ldquoAnvancesin strength theory of rock in 20 centurymdash100 years in memoryof theMohr-Coulomb Strehgth theoryrdquo Chinese Journal of RockMechanics and Engineering no 5 pp 545ndash550 2000

[18] O Zienkiewicz and G Pande ldquoSome useful forms of isotropicyield surfaces for soil and rockmechanicsrdquo in Finite Elements inGeomechanics pp 179ndash198 Wiley Cambridge UK 1977

[19] M Xiao Study on numerical analysis method of stability andsupporting for underground caverns [PhD thesis] Wuhan Uni-versity Wuhan China 2002

[20] Y Liu A M Maniatty and H Antes ldquoInvestigation of aZienkiewiczndashPande yield surface and an elasticndashviscoplasticboundary element formulationrdquo Engineering Analysis withBoundary Elements vol 24 no 2 pp 207ndash211 2000

[21] Y-J Li D-L Zhang and B-G Liu ldquoDevelopment and ver-ification of strain-softening model considering deformationmodulus degradation in FLAC3Drdquo Rock and Soil Mechanicsvol 32 no 2 pp 647ndash652 2011

[22] M S Diederichs P K Kaiser and E Eberhardt ldquoDamageinitiation and propagation in hard rock during tunnelling andthe influence of near-face stress rotationrdquo International Journalof Rock Mechanics and Mining Sciences vol 41 no 5 pp 785ndash812 2004

[23] J Lee and G L Fenves ldquoA return-mapping algorithm forplastic-damage models 3-D and plane stress formulationrdquoInternational Journal for Numerical Methods in Engineering vol50 no 2 pp 487ndash506 2001

[24] J Huang Q Peng and M-X Chen ldquoAn improved return-map stress update algorithm for finite deformation analysisof general isotropic elastoplastic geomaterialsrdquo InternationalJournal for Numerical and Analytical Methods in Geomechanicsvol 38 no 6 pp 636ndash660 2014

[25] J Zhou X Yang W Fu et al ldquoExperimental test and fracturedamage mechanical characteristics of brittle rock under uniax-ial cyclic loading and unloading conditionsrdquo Chinese Journal ofRock Mechanics and Engineering vol 29 no 6 pp 1172ndash11832010

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

1 2 3 4 5 6 7 8 9 10Stage

25

20

15

10

5

0

Volu

me o

f dam

age z

one

Shear damageTension damage

Figure 5 Volume of damage zone (105m3)

Sheardamage

Sheardamage

Resilience

Figure 6 Damage zone of 2 unit section

S1

minus05

minus75

minus7

minus65

minus6

minus55

minus5

minus45

minus4

minus35

minus3

minus25

minus2

minus15

minus1

0

(a)

minus2

minus36

minus34

minus32

minus30

minus28

minus26

minus24

minus22

minus20

minus18

minus16

minus14

minus12

minus10

minus8

minus6

minus4

S3

(b)

Figure 7 First and third principal stress of 2 unit section (MPa)

10 Mathematical Problems in Engineering

D

45

40

35

30

25

20

15

10

5

Figure 8 Displacement of 2 unit section (mm)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] Z-Y Ye L Hong X-L Liu and T-B Yin ldquoConstitutive modelof rock based onmicrostructures simulationrdquo Journal of CentralSouth University of Technology vol 15 no 2 pp 230ndash236 2008

[2] M R Salari S Saeb K J Willam S J Patchet and R C Car-rasco ldquoA coupled elastoplastic damage model for geomaterialsrdquoComputer Methods in Applied Mechanics and Engineering vol193 no 27-29 pp 2625ndash2643 2004

[3] G D Nguyen and A M Korsunsky ldquoDevelopment of anapproach to constitutive modelling of concrete isotropic dam-age coupled with plasticityrdquo International Journal of Solids andStructures vol 45 no 20 pp 5483ndash5501 2008

[4] X LiW-G Cao andY-H Su ldquoA statistical damage constitutivemodel for softening behavior of rocksrdquoEngineeringGeology vol143-144 pp 1ndash17 2012

[5] M Xiao ldquoAnalysis on stability of surrounding rock and damagefracture of concrete lining for high pressure branch pipesrdquo Jour-nal of Wuhan University of Hydraulic and Electric Engineeringno 6 pp 594ndash599 1995

[6] G Frantziskonis and C S Desai ldquoConstitutive model withstrain softeningrdquo International Journal of Solids and Structuresvol 23 no 6 pp 733ndash750 1987

[7] D Krajcinovic and S Mastilovic ldquoSome fundamental issues ofdamage mechanicsrdquo Mechanics of Materials vol 21 no 3 pp217ndash230 1995

[8] L Resende ldquoA Damage mechanics constitutive theory for theinelastic behaviour of concreterdquo Computer Methods in AppliedMechanics and Engineering vol 60 no 1 pp 57ndash93 1987

[9] M Jirasek ldquoNonlocal models for damage and fracture com-parison of approachesrdquo International Journal of Solids andStructures vol 35 no 31-32 pp 4133ndash4145 1998

[10] M Jirasek and S Marfia ldquoNon-local damage model based ondisplacement averagingrdquo International Journal for NumericalMethods in Engineering vol 63 no 1 pp 77ndash102 2005

[11] J Lee andG L Fenves ldquoPlastic-damagemodel for cyclic loadingof concrete structuresrdquo Journal of Engineering Mechanics vol124 no 8 pp 892ndash900 1998

[12] J Li and J Y Wu ldquoElastoplacstic damage constitutive modelfor concrete based on damage energy release rates part I basicformulationsrdquo China Civil Engineering Journal no 9 pp 14ndash202005

[13] J Y Wu and J Li ldquoElastoplastic damage constitutive modelfor concrete based on damage energy release rates part IInumerical algorithm and verificationsrdquo China Civil EngineeringJournal no 9 pp 21ndash27 2005

[14] J W Ju ldquoOn energy-based coupled elastoplastic damagetheories constitutive modeling and computational aspectsrdquoInternational Journal of Solids and Structures vol 25 no 7 pp803ndash833 1989

[15] Y Q Ying ldquoOn rock plasticity damage and their constitutiveformulationrdquo Scientia Geologica Sinica no 1 pp 63ndash70 1995

[16] J C Simo and J W Ju ldquoStrain- and stress-based continuumdamage models-I Formulationrdquo International Journal of Solidsand Structures vol 23 no 7 pp 821ndash840 1987

[17] M H Yu YW ZanW Fan J Zhao and Z Z Dong ldquoAnvancesin strength theory of rock in 20 centurymdash100 years in memoryof theMohr-Coulomb Strehgth theoryrdquo Chinese Journal of RockMechanics and Engineering no 5 pp 545ndash550 2000

[18] O Zienkiewicz and G Pande ldquoSome useful forms of isotropicyield surfaces for soil and rockmechanicsrdquo in Finite Elements inGeomechanics pp 179ndash198 Wiley Cambridge UK 1977

[19] M Xiao Study on numerical analysis method of stability andsupporting for underground caverns [PhD thesis] Wuhan Uni-versity Wuhan China 2002

[20] Y Liu A M Maniatty and H Antes ldquoInvestigation of aZienkiewiczndashPande yield surface and an elasticndashviscoplasticboundary element formulationrdquo Engineering Analysis withBoundary Elements vol 24 no 2 pp 207ndash211 2000

[21] Y-J Li D-L Zhang and B-G Liu ldquoDevelopment and ver-ification of strain-softening model considering deformationmodulus degradation in FLAC3Drdquo Rock and Soil Mechanicsvol 32 no 2 pp 647ndash652 2011

[22] M S Diederichs P K Kaiser and E Eberhardt ldquoDamageinitiation and propagation in hard rock during tunnelling andthe influence of near-face stress rotationrdquo International Journalof Rock Mechanics and Mining Sciences vol 41 no 5 pp 785ndash812 2004

[23] J Lee and G L Fenves ldquoA return-mapping algorithm forplastic-damage models 3-D and plane stress formulationrdquoInternational Journal for Numerical Methods in Engineering vol50 no 2 pp 487ndash506 2001

[24] J Huang Q Peng and M-X Chen ldquoAn improved return-map stress update algorithm for finite deformation analysisof general isotropic elastoplastic geomaterialsrdquo InternationalJournal for Numerical and Analytical Methods in Geomechanicsvol 38 no 6 pp 636ndash660 2014

[25] J Zhou X Yang W Fu et al ldquoExperimental test and fracturedamage mechanical characteristics of brittle rock under uniax-ial cyclic loading and unloading conditionsrdquo Chinese Journal ofRock Mechanics and Engineering vol 29 no 6 pp 1172ndash11832010

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

D

45

40

35

30

25

20

15

10

5

Figure 8 Displacement of 2 unit section (mm)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] Z-Y Ye L Hong X-L Liu and T-B Yin ldquoConstitutive modelof rock based onmicrostructures simulationrdquo Journal of CentralSouth University of Technology vol 15 no 2 pp 230ndash236 2008

[2] M R Salari S Saeb K J Willam S J Patchet and R C Car-rasco ldquoA coupled elastoplastic damage model for geomaterialsrdquoComputer Methods in Applied Mechanics and Engineering vol193 no 27-29 pp 2625ndash2643 2004

[3] G D Nguyen and A M Korsunsky ldquoDevelopment of anapproach to constitutive modelling of concrete isotropic dam-age coupled with plasticityrdquo International Journal of Solids andStructures vol 45 no 20 pp 5483ndash5501 2008

[4] X LiW-G Cao andY-H Su ldquoA statistical damage constitutivemodel for softening behavior of rocksrdquoEngineeringGeology vol143-144 pp 1ndash17 2012

[5] M Xiao ldquoAnalysis on stability of surrounding rock and damagefracture of concrete lining for high pressure branch pipesrdquo Jour-nal of Wuhan University of Hydraulic and Electric Engineeringno 6 pp 594ndash599 1995

[6] G Frantziskonis and C S Desai ldquoConstitutive model withstrain softeningrdquo International Journal of Solids and Structuresvol 23 no 6 pp 733ndash750 1987

[7] D Krajcinovic and S Mastilovic ldquoSome fundamental issues ofdamage mechanicsrdquo Mechanics of Materials vol 21 no 3 pp217ndash230 1995

[8] L Resende ldquoA Damage mechanics constitutive theory for theinelastic behaviour of concreterdquo Computer Methods in AppliedMechanics and Engineering vol 60 no 1 pp 57ndash93 1987

[9] M Jirasek ldquoNonlocal models for damage and fracture com-parison of approachesrdquo International Journal of Solids andStructures vol 35 no 31-32 pp 4133ndash4145 1998

[10] M Jirasek and S Marfia ldquoNon-local damage model based ondisplacement averagingrdquo International Journal for NumericalMethods in Engineering vol 63 no 1 pp 77ndash102 2005

[11] J Lee andG L Fenves ldquoPlastic-damagemodel for cyclic loadingof concrete structuresrdquo Journal of Engineering Mechanics vol124 no 8 pp 892ndash900 1998

[12] J Li and J Y Wu ldquoElastoplacstic damage constitutive modelfor concrete based on damage energy release rates part I basicformulationsrdquo China Civil Engineering Journal no 9 pp 14ndash202005

[13] J Y Wu and J Li ldquoElastoplastic damage constitutive modelfor concrete based on damage energy release rates part IInumerical algorithm and verificationsrdquo China Civil EngineeringJournal no 9 pp 21ndash27 2005

[14] J W Ju ldquoOn energy-based coupled elastoplastic damagetheories constitutive modeling and computational aspectsrdquoInternational Journal of Solids and Structures vol 25 no 7 pp803ndash833 1989

[15] Y Q Ying ldquoOn rock plasticity damage and their constitutiveformulationrdquo Scientia Geologica Sinica no 1 pp 63ndash70 1995

[16] J C Simo and J W Ju ldquoStrain- and stress-based continuumdamage models-I Formulationrdquo International Journal of Solidsand Structures vol 23 no 7 pp 821ndash840 1987

[17] M H Yu YW ZanW Fan J Zhao and Z Z Dong ldquoAnvancesin strength theory of rock in 20 centurymdash100 years in memoryof theMohr-Coulomb Strehgth theoryrdquo Chinese Journal of RockMechanics and Engineering no 5 pp 545ndash550 2000

[18] O Zienkiewicz and G Pande ldquoSome useful forms of isotropicyield surfaces for soil and rockmechanicsrdquo in Finite Elements inGeomechanics pp 179ndash198 Wiley Cambridge UK 1977

[19] M Xiao Study on numerical analysis method of stability andsupporting for underground caverns [PhD thesis] Wuhan Uni-versity Wuhan China 2002

[20] Y Liu A M Maniatty and H Antes ldquoInvestigation of aZienkiewiczndashPande yield surface and an elasticndashviscoplasticboundary element formulationrdquo Engineering Analysis withBoundary Elements vol 24 no 2 pp 207ndash211 2000

[21] Y-J Li D-L Zhang and B-G Liu ldquoDevelopment and ver-ification of strain-softening model considering deformationmodulus degradation in FLAC3Drdquo Rock and Soil Mechanicsvol 32 no 2 pp 647ndash652 2011

[22] M S Diederichs P K Kaiser and E Eberhardt ldquoDamageinitiation and propagation in hard rock during tunnelling andthe influence of near-face stress rotationrdquo International Journalof Rock Mechanics and Mining Sciences vol 41 no 5 pp 785ndash812 2004

[23] J Lee and G L Fenves ldquoA return-mapping algorithm forplastic-damage models 3-D and plane stress formulationrdquoInternational Journal for Numerical Methods in Engineering vol50 no 2 pp 487ndash506 2001

[24] J Huang Q Peng and M-X Chen ldquoAn improved return-map stress update algorithm for finite deformation analysisof general isotropic elastoplastic geomaterialsrdquo InternationalJournal for Numerical and Analytical Methods in Geomechanicsvol 38 no 6 pp 636ndash660 2014

[25] J Zhou X Yang W Fu et al ldquoExperimental test and fracturedamage mechanical characteristics of brittle rock under uniax-ial cyclic loading and unloading conditionsrdquo Chinese Journal ofRock Mechanics and Engineering vol 29 no 6 pp 1172ndash11832010

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