Research Article Computational Modelling of Fracture ...downloads.hindawi.com/journals/mpe/2016/3231092.pdf · Research Article Computational Modelling of Fracture Propagation in

Research ArticleComputational Modelling of Fracture Propagation in RocksUsing a Coupled Elastic-Plasticity-Damage Model

Isa Kolo Rashid K Abu Al-Rub and Rita L Sousa

Institute Center for Energy Mechanical and Materials Engineering Department Masdar Institute of Science and TechnologyAbu Dhabi UAE

Correspondence should be addressed to Rashid K Abu Al-Rub rabualrubmasdaracae

Received 31 January 2016 Accepted 27 April 2016

Academic Editor Paolo Lonetti

Copyright copy 2016 Isa Kolo et alThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

A coupled elastic-plasticity-damage constitutive model AK Model is applied to predict fracture propagation in rocks The quasi-brittle material model captures anisotropic effects and the distinct behavior of rocks in tension and compression Calibration of theconstitutivemodel is realized using experimental data for CarraramarbleThrough theWeibull distribution function heterogeneityeffect is captured by spatially varying the elastic properties of the rock Favorable comparison between model predictions andexperiments for single-flawed specimens reveal that the AK Model is reliable and accurate for modelling fracture propagation inrocks

1 Introduction

An understanding of fracture initiation and propagation inrocks is important in reservoir (hydrocarbon and geother-mal)management and energy exploration activities includingenhanced geothermal systems oil and gas extraction andunderground water transport This has triggered numerousresearch efforts through experimental studies and numericalmodelling of rock fractures Numerical modelling could bevery challenging especially when other field observationslike effects of temperature and confining pressure are to becoupled with the fracturing process [1 2]

While some numerical models assume an elastic rockbehavior other models try to characterize rock behavior intothe plastic region Assumptions such as material isotropyare generally made for simplicity but relaxing such assump-tions is necessary to obtain more accurate results Somemodels rely on predefining crack locations [1] or extensionof preexisting cracks The coupled damage-plasticity modeloriginally developed by Cicekli et al [3] and later modifiedby Abu Al-Rub and Kim [4] referred to as AK Model inthis work captures crack initiation and propagation withoutthe need to predefine crack locations a priori [4] The modelis based on a continuum damage mechanics approach andgenerally applicable to quasi-brittle materials Many such

models exist with varying degrees in the extent to whichimportant physical phenomena are captured [5ndash8]

Another crucial consideration in carbonate reservoirs isthe heterogeneity of constituent rocks Studies have shownthat the carbonate reservoirs in the Middle-East are veryheterogeneous in terms of rock types [9] Reservoir hetero-geneity is also important in carbon capture and sequestration[10] Moreover the compressive strength of carbonates hasbeen shown to be a function of the grain size porosity andelasticmodulus [11]Hence for proper reservoir characteriza-tion and economic evaluation fracture propagation damageand heterogeneity should all be taken into account Hereinlies the uniqueness of the AK Model Even with materialmodels abound for describing rock behavior many do notcombine the effects of both damage and heterogeneity andsome that do are limited to isotropic andor elastic properties[12]Thiswork presents a computationalmodel that considers(1) the plastic deformation in addition to elastic deformationin rocks (2) damage localization (3) anisotropy in rockbehavior and (4) heterogeneity of rock sampleunit

This study focuses on qualitative validation of the modelfor predicting fracture propagation in rocks and the incorpo-ration of heterogeneity effects A brief look at previous workson rock fracture propagation is presented next (in Section 2)

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016 Article ID 3231092 15 pageshttpdxdoiorg10115520163231092

2 Mathematical Problems in Engineering

Subsequently (Section 3) the constitutive model is conciselydescribed without detailing its numerical implementationUsing Carrara marble as a representative rock material themodel is then calibrated (Section 4) and with experimentson single-flawed rock specimens the capability of the modelin predicting fracture patterns is demonstrated in Section 5(model validation) In Section 6 some methods of incorpo-rating the effects of heterogeneity in modelling geologicalmedia are highlighted before the spatial variation of elasticproperties is used to study heterogeneity effects through theWeibull distribution function

2 Fracture Prediction in Rocks

Crack initiation and propagation in rocks have been thesubject of various past and current studies The approachis to study propagation of cracks from precracked (flawed)specimens using experiments (eg [13ndash17]) and numeri-cal simulations (see [18ndash21]) Bobet and Einstein [21 22]used both experiments and numerical simulations to studycrack initiation propagation and coalescence in rock Theyobtained good simulation results with the upgraded codeFROCK which is based on the Displacement DiscontinuityMethod (DDM) A later work by Goncalves Da Silva andEinstein [23] improved the capabilities of the code by intro-ducing a new strain-based criterion as well as a normal stress-dependent criterion for crack development

AUTODYN is a nonlinear hydrodynamics code compat-ible with ANSYS Li and Wong [24] used it to study theinfluence of flaw inclination angle and loading condition oncrack propagation Tang [12] developed the 2D finite elementcode RFPA 2D The code incorporates the effect of bothdamage and heterogeneity and its extended 3D version hasbeen used successfully in predicting fracture of specimensin triaxial compression [19] In another study Zhang andWong [25] used the Bonded Particle Method (BPM) toinvestigate the effect of loading rate on crack behavior offlawed specimens Crack coalescence mode was observed tochange from that dominated by the tensile segment to thatdominated by the shear-band

FROCK is based on DDM which is a boundary elementmethod (BEM) and according to Khair et al [26] the FiniteElement Method (FEM) is superior to BEM in predictingsubsurface fractures While the AUTODYN-based modeladopted by [24] captures damage like the RFPA it relieson Drucker-Prager criterion which has been proven tooverestimate intact rock strength [27] Additionally RFPAdoes not consider anisotropy and plastic deformation inrocks This calls for an elastoplastic-damage model based onFEM The model by Abu Al-Rub and Kim [4] AK Modelfor plain concrete considers damage effects anisotropy andplastic deformation It also adopts the Lubliner yield criterionwhich is an improvement on the Drucker-Prager criterionThis model is implemented as a user material subroutine inAbaqus a commercial finite element code

It is noteworthy that Linear Elastic Fracture Mechanics(LEFM) has also been used to simulate crack propagation inrocks (eg see [29] which uses FRANC2D crack propagationsimulator) However LEFM is not able to predict fracture

initiation and when a crack is assumed the stress intensityand fracture toughness on which LEFM is based could bemeaningless for the assumed flaw size [30] Also the materialbehavior near the crack tip region (fracture process zone)could be inelastic and nonlinear [30] making LEFM onlyapplicable when the size of process zone (L) is significantlysmall with respect to the smallest critical dimension of thestructure (D) ndash DL gt 100 [31] This influenced the decisionof [32] in their elastic Abaqus analysis to take stress measure-ments at a distance from the flaw tip that is to avoid stressesin the process zone which make no physical sense Whilecohesive zone modelling has been introduced to addressthese issues in materials like concrete [33] increased numberor complexity of fractures might not be easily handled bycohesive zone modelling The continuum damage mechanics(CDM) approach adopted by the AK Model is generallymeritorious because (1) it accounts for localized damage(which actually happens in rock deformation) and micro-scopic initiation of cracks unlike Fracture Mechanics whichconsiders a clearly defined discrete macroscopic cracks (2)it is capable of analyzing complex fractures and networkswhich is not possible in elastic analysis (such as LinearElastic Fracture Mechanics (LEFM)) (3) there is no need forany special initial assumptions such as initial perturbationsand (4) it has no computational limitations on number offractures like the Extended Finite Element Method (XFEM)[34]

3 Elastoplastic-Damage Constitutive Model

A coupled elastoplasticity-damage model (AK Model) isadopted here It was developed by Abu Al-Rub and Kim[4] based on an earlier work by Cicekli et al [3] hencethe name AK For a full description of the model pleasesee [3 4] The model stands out because it presents acoupled anisotropic damage and plasticity constitutivemodelthat predicts rockrsquos distinct behavior in compression andtension with the following (1) a modified continuum damagemechanics framework to include quadratic isotropic andanisotropic variation of the effective (undamaged) stress interms of the nominal (damaged) stress The nominal andeffective configurations are explained in Section 31 and (2)two novel and different damage (power) evolution laws forboth tension and compression for a more accurate predictionof rock behavior after damage initiation

An overview of the modelrsquos main constitutive relationsis presented here to give a general idea without detailing itsnumerical implementation

31 Anisotropic Damage Model In damage mechanics thedamaged (nominal) configuration of a material is the normalstate of thematerial with imperfections like voids (pores) andcracks To analyze this an imaginary state of the materialcalled the effective (undamaged) configuration is assumed(see Figure 1)

The effective (undamaged) configuration of the materialconsiders is to be intact without any voids or cracks Strainis assumed to be constant in both configurations (ie strainequivalence hypothesis) and a damage internal state variable

Mathematical Problems in Engineering 3

AA

AD (area of cracks and voids)

(a) Damaged configuration (b) Effective configuration

120593 =A minus A

A=

AD

A

0 le 120593 le 1

120593 = 0rarrno damage120593 = 1rarr fracture

Figure 1 Illustration of nominal (damaged) and effective configuration

120593 is defined The damage variable which is a degradationvariable varies from 0 to 1 a value of zero indicates nodamage and one indicates full damage (ie fracture) Therelationship between the stresses in the damaged and effectiveconfigurations is given by

120590119894119895= (1 minus 120593)

2120590119894119895 (1)

where 120590119894119895and 120590

119894119895are the Cauchy stresses in the damaged and

effective configurations respectively Variables in the effectiveconfiguration are denoted by ( sdot )Thus the damaged elasticitytensor is given in terms of the effective elasticity tensor by

119864119894119895119896119897

= (1 minus 120593)2119864119894119895119896119897

(2)

Rock has a distinct behavior in compression and tensionThus the Cauchy stress tensor is decomposed into two partsusing spectral decomposition technique the positive part(tension) and the negative part (compression) The followingrelations show the spectral decomposition of the Cauchystress tensor for both damaged and effective configurations[4 35]

120590119894119895= 120590+

119894119895+ 120590minus

119894119895

120590119894119895= 120590+

119894119895+ 120590minus

119894119895

(3)

where (sdot)+ represents tensile components and (sdot)

minus representscompressive components

Fourth-order projection tensors and damage-effect ten-sors are introduced as well as the accompanying spectraldecomposition into tensile and compressive parts Withthis and further simplification the following relations areobtained [4 36]

120590119894119895= 119872119894119895119896119897

120590119896119897

119872119894119895119896119897

= (1 minus 120593+)2

119875+

119894119895119896119897+ (1 minus 120593

minus)2

119875minus

119894119895119896119897

119875+

119894119895119896119897=

3

sum

119896=1

119867(120590

(119896)

) 119899(119896)

119894119899(119896)

119895119899(119896)

119896119899(119896)

119897

(4)

where 119875+

119894119895119896119897and 119875

minus

119894119895119896119897= 119868119894119895119896119897

minus 119875+

119894119895119896119897are the projection tensors

that describe the orientation of the tensile and compressiveprincipal stresses respectively where 119868

119894119895119896119897is the identity

fourth-order tensor119872119894119895119896119897

is the damage-effect tensor 120593+ and120593minus are the damage variables describing the evolution of cracks

due to tensile principal stresses and cracks due to compressiveprincipal stresses respectively 119867(

120590

(119896)

) is the Heaviside stepfunction

120590

(119896)

are the principal values of 120590119894119895 and 119899

(119896)

119894are

the corresponding directions The model has been validatedusing several experimental data for concrete see [4]

32 Plasticity Yield Surface Rock materials show plasticdeformation before failure especially under high confine-ment pressures In plastic deformation there is a need todefine three elements (a) yield criterion to describe theonset of inelastic deformation (b) flow rule and plasticdeformation function for calculating the magnitude anddirection of plastic strain rate and (c) hardeningsofteninglaw for evolution of the yield stressThe yield criterion chosenis that developed by Lubliner et al [37] due to its capability toaccount for different tensile and compressive behaviors andconfinement pressure effects The nonassociative plasticityflow rule is used to ensure realistic modelling of volumetricexpansion of rock under compression The yield criterion isexpressed in the effective configuration (see Figure 1) This isbecause plasticity evolution is actually driven by the stressesand strains in the intact material It is expressed as

119891 = radic31198692+ 1205721198681+ 120573 (120576

+

eq 120576minus

eq)119867 (120590max)

120590max

minus (1 minus 120572) 119888minus(120576minus

eq) le 0

(5)

where 1198692

= 1199041198941198951199041198941198952 is the second-invariant of the effective

deviatoric stress tensor 119904119894119895

= 120590119894119895

minus 1205901198961198961205751198941198953 where 120575

119894119895is

the Kronecker delta 1198681

= 120590119896119896

is the first invariant of theeffective Cauchy stress tensor 120590

119894119895 which incorporates the

hydrostatic pressure effect 120590max is the maximum principal


effective stress 119867(120590max) is the Heaviside step function and

120572 and 120573 are dimensionless constants given by

120572 =

(1198911198870119891minus

0) minus 1

2 (1198911198870119891minus

0) minus 1

120573 = (1 minus 120572)

119888minus(120576minus

eq)

119888+(120576+

eq)minus (1 + 120572)

(6)

where 1198911198870

is the initial equibiaxial yield strength 119891minus0is the

uniaxial compressive yield strength 119888+ and 119888

minus are tensileand compressive isotropic hardening functions and theequivalent plastic strains are 120576

+

eq and 120576minus

eq in tension andcompression respectively The isotropic hardening functionsare given by

119888minus= 119891minus

0+ 119876minus[1 minus exp (minus119887

minus120576eq)]

119888+= 119891+

0+ ℎ+120576+

eq(7)

where 119891minus

0and 119891

+

0are the compressive and tensile initial yield

stresses respectively 119876minus 119887minus and ℎ+ are material constants

obtained from the effective configuration of the uniaxialstress-strain diagramThe equivalent plastic strains and theirrates are expressed as

120576+

eq = int

119905

0

120576+

eq119889119905 120576+

eq = 119903 (120590119894119895) 120576

pmax

120576minus

eq = int

119905

0

120576minus

eq119889119905 120576minus

eq = minus (1 minus 119903 (120590119894119895)) 120576

pmin

(8)

where

119903 (120590119894119895) =

sum3

119896=1⟨120590

(119896)

⟩

sum3

119896=1

1003816100381610038161003816100381610038161003816

120590

(119896)1003816100381610038161003816100381610038161003816

(9)

The superimposed dot indicates derivative with respect totime The maximum and minimum principal values of theplastic strain rate 120576

p119894119895such that 120576p

1gt 120576

p2gt 120576

p3are represented

respectively by 120576

pmax = 120576

p1and 120576

pmin = 120576

p3 (sdot) indicate a

principal value and as stated earlier (sdot)+ and (sdot)minus indicate

tensile and compressive variables respectively 119903(120590119894119895) is a

weight factor for tension and compression and ⟨sdot⟩ representsthe Macaulay bracket taken as ⟨119909⟩ = 12(|119909| + 119909)

The nonassociative plasticity flow rule is described by

120576p119894119895=

120582

p 120597119865p

120597120590119894119895

119865p= radic3119869

2+ 120572

p1198681 (10)

where 120582

p is the plastic Lagrange multiplier 119865p is the plasticpotential and 120572

p is the dilation material constantThe plasticmultiplier

120582

p is obtained using the consistency condition

120582

p119891 = 0 119891 le 0

120582

pge 0 (11)

33 Tensile and Compression Damage Surfaces The damagegrowth function adopted in this model incorporates bothtensile and compressive damage It is as follows

119892plusmn= radic

1

2

119884plusmn

119894119895119884plusmn

119894119895minus 119870plusmn(120593plusmn

eq) le 0 (12)

where 119870plusmn is the tensile or compressive damage isotropic

hardening functionWhen there is no damage119870plusmn equals thedamage threshold119870

0

plusmn 119884plusmn119894119895is the damage driving force or the

energy release rate expressed as [4 38]

119884plusmn

119903119904= minus

1

2

119864

minus1

119894119895119886119887120590119886119887

120597119872119894119895119901119902

120597120593plusmn

119903119904

120590119901119902 (13)

The evolution equation for plusmn119894119895is as follows

plusmn

119894119895=

120582

plusmn

119889

120597119892plusmn

120597119884plusmn

119894119895

(14)

where 120582

plusmn

119889= plusmn

eq is the damage multiplier given by

plusmn

eq = radicplusmn

119894119895plusmn

119894119895

120593plusmn

eq = int

119905

0

plusmn

eq119889119905

(15)

34 Tensile and Compressive Damage Evolution Laws Bothexponential and power damage laws could be used forevolution of the damage variables While the exponentiallaw has less number of material constants the power law isproven to give more accurate results [4] Hence in this studythe power law is adopted for evolution of damage both intension and in compression It is expressed for tension andcompression respectively as

120593+

eq = 119861+(

119870+

0

119870+)(

119870+

119870+

0

minus 1)

119902+

120593minus

eq = 119861minus(

119870minus

119870minus

0

minus 1)

119902minus

(16)

where 119861plusmn and 119902

plusmn are material constants Under uniaxialloading the tensile damage isotropic hardening function119870+ and tensile damage threshold 119870

0

+ are respectivelyexpressed as

119870+= radic119864120590

+

119894119895120576119890+

119894119895= radic119864120590

+(

120590+

119864

) = 120590+

119870+

0= 119891+

0cong 119891+

u

(17)

where 119891+

0is the tensile yield strength which is almost

equal to the ultimate tensile strength 119891+

u for rocks at whichtensile damage initiates The compressive damage isotropic


hardening function119870minus and compressive damage threshold119870minus

0 are respectively

119870minus= radic3119869

minus

2+ 120572119868minus

1= [1 + (

120572

3

)] 120590minus (18)

119870minus

0= [1 + (

120572

3

)]1198910

minus (19)

where 1198910

minus is the uniaxial compressive stress at which damagestarts

4 Model Calibration

Abu Al-Rub and Kim [4] proposed a method of obtainingunique material parameters based on data from cyclic testsThe proposed method was used to obtain the starting valuesin this work the parameters had to be adjusted further toobtain a close fit with experimental data Carrara marble isused as the rock material

41 Carrara Marble as a Representative Rock Material Var-ious experimental data exist but calibration of the modelrequires uniaxial cyclic compression and tension data forunique calibration of material constants [4] Data for cyclictensile test of rocks is not readily available because it poses achallenge to experimentalists It is difficult to perform tensiletests on rocks without introducing spurious stresses Henceindirect tests such as the Brazilian disc test are used [39]Chen et al [40] explained the suitability of the Brazilian disctest in determining the tensile strength of both isotropic andanisotropic rocks However they argued that elastic isotropicrelations cannot and should not be used for analysis oftests on anisotropic rocks They used analytical methods inaddition to experiments to determine the elastic constantsand indirect tensile strength of transversely isotropic rocksTo strike a balance between accuracy and available datait would be strategic to choose data for an approximatelyisotropic rock for the Brazilian disc test Even though somestudies reveal some level of anisotropy [41] Carrara marblecould be reasonably assumed to be isotropic [42]

Wong et al [43] recently studied the tensile behaviorof Carrara marble using the Brazilian disc test MoreoverCarrara marble has been and is being widely studied byvarious researchers (eg [2 16 17 28 43ndash48]) Walton etal [46] studied the strength deformability and dilatancy ofcarbonate rocks including Carrara marble Triaxial test dataat different confinements were presented In this work thetensilematerial parameters of theAKModelwill be calibratedusing data fromBrazilian disc test byWong et al [43] For thecompressive material parameters triaxial test data presentedby Walton et al [46] will be used In addition experimentaldata for precracked Carrara marble specimens with variousflaw (artificially made preexisting crack) geometries exist forvalidation [2]

42 Calibration Based on Data from Uniaxial Compressionand Uniaxial Tension Tests Data for monotonic uniaxialtests are used in this study The sources of utilized data for

Carrara marble have been presented in the previous sectionBased on compressive yield strength the compressive damagethreshold was calculated following (19) The tensile stress-strain curve with a tensile strength 69MPa as in Wonget al [43] is adopted here For compression the stress-strain curve presented by Walton et al [46] is used Theobtained material properties are unconfined compressivestrength = 943MPa Youngrsquos modulus = 453GPa Poissonrsquosratio = 019

To ensure that the selected values are representativeof Carrara marble properties a brief review of propertiesreported by other researchers was carried out According toEvans et al [49] the compressive yield strength of Carraramarble is approximately 76MPa with no confinement Thetensile strength of Carrara marble was obtained as 75MPa[45] and with varying Brazilian disc diameter a range of 6ndash8MPa was reported [50] A compressive strength of 92MPawas also obtained by other experimenters [51] Accordingto Siegesmund et al [52] the properties of Carrara marbleare as follows unconfined compressive strength = 846MPaYoungrsquos Modulus = 49GPa Poissonrsquos ratio = 019 tensilestrength = 69MPa These values confirm that the selectedexperimental data fall well within range for Carrara marbleproperties

421 Simulation Setup For both uniaxial tension and com-pression a single 1mm by 1mm plane stress element is usedfor calibration Each element is supported by rollers on the leftand bottom edges and a top displacement is imposed eitherupward (tension) or downward (compression) as shown inFigure 2

422 Material Parameters To obtain the initial materialparameters the calibration procedure outlined by Abu Al-Rub and Kim [4] was adopted using the cyclic compressiondata in [46] However because the confining pressure forthis data set was unclear and most likely nonzero trial anderror was further used to make the parameters fit the data forunconfined uniaxial compression For tension trial and errorwas used based on the data by Wong et al [43] The materialparameters that provide a good fit for the data are detailed inTable 1

A comparison between the model prediction using thesematerial constants and experimental data is presented inFigure 3(a) for compression and Figure 3(b) for tension Themodel gives a good prediction of the compressive behaviorof Carrara marble as seen in Figure 3(a) based on data foruniaxial compression In the case of tension (Figure 3(b))the data used was extracted from a Brazilian disc test whichis indirect tension Experiments reveal the progressive devel-opment of white patches and an initial nonlinear behaviorThe current model assumes a linear elastic behavior untilyield Thus specimen behavior on the onset of linear elasticbehavior is used to aid calibration by a shift in the strainvalues A more detailed approach would have been thesimulation of a complete Brazilian test for a more accuratecalibration however the current fitting yielded satisfactorilyresults


Table 1 Material constants for Carrara marble

Material constant Value119891minus

0(compressive yield strength) 76MPa

119870minus

0(compressive damage threshold) 79MPa

119876minus (compressive hardening modulus) 16462MPa

119887minus (compressive hardening rate constant) 9609

119861minus (compressive strength where nonlinearity starts) 044

119902minus (compressive power damage law constant) 1148

119891+

0(tensile yield strength) 69MPa

119870+

0(tensile damage threshold) 69MPa

ℎ+ (tensile hardening modulus) 15000MPa

119861+ (tensile strength where nonlinearity starts) 125

119902+ (tensile power damage law constant) 032

(a) Uniaxial compression (b) Uniaxial tension

Figure 2 One element geometry (1mm by 1mm) for calibration showing boundary conditions

0102030405060708090

100

Stre

ss (M

Pa)

Experiment (Walton et al 2015)Model prediction

0001 0002 0003 0004 0005 0006 0007 00080Strain (mmmm)

0009

(a)

Experiment (Wong et al 2014)Model prediction (shifted)

00001 0000500003 000060 00002 00004Strain (mmmm)

0

1

2

3

4

5

6

7

8

Stre

ss (M

Pa)

(b)

Figure 3 Prediction of stress-strain data for Carrara marble (a) tension and (b) compression

5 Model Validation

51 Simulation Setup To validate the model tests on single-flawed Carrara marble were used Prismatic specimens witha dimension of 152 times 76 times 32mm were tested by Wong and

Einstein [16] The nomenclature associated with the flaw andspecimen geometry is presented in Figure 4

A plane stress representation of prismatic Carrara marblespecimens with a single flaw is used for validation of themodel As shown in Figure 5 the bottom of the specimen is


152mm

13mm

125mm

32mm

76mm

(a)

L

120573

120572

(b)

Figure 4 Dimensions of prismatic specimens tested by Wong and Einstein [16] (a) Single-flawed specimen (b) double-flawed specimenshowing the ligament length (L) bridging angle (120572) and flaw inclination angle (120573)

Figure 5 Plane stress representation of a specimen with 60∘ flaw

Figure 6 Finite element mesh for specimen with 60∘ flaw

pinned and displacement is imposed on the top An exampleof a finite element mesh is presented in Figure 6 and consistsof linear (CPE3) and bilinear (CPE4R) triangular elements(or triangular elements) Due to stress concentration at the

flaw tips the mesh is made finer at those regions Cases withflaw angles (120572) of 0∘ 30∘ 45∘ 60∘ and 75∘ are considered

52 Predicted Fracture Patterns Figure 7 shows the varioustypes of cracks observed in all experiments based on themode of fracture involved [16] There are tensile cracks (dueto tensile forces) shear cracks (due to shear forces) andmixed tensile shear cracks Also types 1 2 and 3 havebeen identified for both tensile and shear cracks to help indistinguishing various crack geometries

Newly evolved crack patterns from simulations are com-pared qualitativelywith experiments fromWong [2] as shownin Figure 8 Although there are some cracks that are notcaptured by themodel overall there is a very good qualitativematch between the simulations and experiments As shownCWPmeans curvilinear white patch and TWCmeans tensilewing crack [2] Clearly for 120572 = 0

∘ the model is even able tocapture the evolution of CWP Also TWC tensile (T) andshear cracks (S) are all captured by the model for different 120572One can notice that TWC dominates the crack pattern as 120572

increases which is in agreement with the model predictionsTherefore themodel considers bothwhite patches and cracksin the form of damage The force-displacement curves fordifferent single flaw geometries are presented in Figure 9Thisfigure shows the different softening mechanisms for differentflaw geometries

Double flaws are also considered as shown in Figure 10for 120572 = 30

∘ and size of the ligament lengths (L) of 2aand 4a (see Figure 10) The predictions are not as accurateas in the case of the single-flawed geometry SpecificallyFigure 10(a) shows that the simulations do not capture crackcoalescence between the two preexisting cracks as comparedto experimental observations This is obvious because thestress distribution in a double-flawed specimen is morecomplex However better agreement is seen in Figure 10(b)The force-displacement curves for the two different double


T

T

(a) Type1 tensilecrack(tensilewingcrack)

T

T

(b) Type2 tensilecrack

T

T

(c) Type 3 tensile crack

SS

T

T

(d) Mixed tensileshear crack

S

S

(e) Type 1 shearcrack

S

(f) Type 2 shearcrack

S

S(g) Type 3 shearcrack

Figure 7 Various crack types initiated from the preexisting flaw of Carrara marble specimen as observed in experiments by Wong andEinstein [28] T is tensile crack opening and S is shear displacement

flaw geometries are presented in Figure 9 which shows thatas L increases stiffer response is obtained

It is worthy to mention that experimental results varyfrom one test to another for the same flaw geometry Thismakes it quite difficult to get a definite fracture patternmost recurrent patterns are taken as representative resultsHence a qualitative comparison would be sufficient Moreaccurate results could be obtained if data from the samesample is used for both tension and compression While thevalidation experiments use a certain Carrara marble sample

the tensile data was based on a different sample and the datafor compression was based on yet another sample Althoughall are samples of Carrara marble there could be slightvariations in material behavior

6 Incorporating the Effects ofMaterial Heterogeneity

Rocks as observed using microscopy have a heterogeneousmicrostructure their heterogeneous nature is also visible


Flaw

angl

eFl

aw an

gle

Flaw

angl

eFl

aw an

gle

Flaw

angl

e

Experimental observation Model prediction

(a)

(b)

(c)

(d)

(e)

0∘

30∘

45∘

60∘

75∘

Figure 8 Qualitative comparison of crack patterns between simulations and experiments as observed by Wong [2] Note that T indicatestensile opening S = shearing TWC = tensile wing crack CWP = curvilinear white patch A B C and D are all white patches indicatingdamage localization


0

500

1000

1500

2000

2500

3000

3500

4000

Forc

e (N

)

002 004 006 008 01 012 014 0160

Displacement (mm)

Flaw angle = 0 (single)Flaw angle = 30 (single)Flaw angle = 45 (single)Flaw angle = 60 (single)

Flaw angle = 75 (single)Ligament = 2a (double)Ligament = 4a (double)

Figure 9 Force-displacement diagram from simulations of single-flawed (single) and double-flawed (double) Carrara marble specimensSingle-flawed specimens have varying flaw angles (in degrees) while double-flawed specimens are coplanar with a flaw angle of 30∘ anddifferent ligament lengths

Experiments Model prediction

(a) Coplanar flaws with an inclination angle of 30∘and 119871 = 2119886


(b) Coplanar flaws with an inclination angle of 30∘ and 119871 = 4119886

Figure 10 Prediction of crack propagation in marble specimen with double flaws Experimental results are fromWong [2]


from outcrops Stress-induced and preexisting defects in theform of voids microcracks and weak interfaces contributeto rock heterogeneity and randomness in microstructurethereby influencing rock behavior and crack propagationStudies have shown that the compressive strength of car-bonates is a function of the grain size porosity and elasticproperties (see [11 53]) In marbles and limestones theuniaxial compressive strength increases linearly with theinverse square root of mean grain size [54 55] These studiesindicate that heterogeneity in its various manifestationsis very important in the study and modelling of fracturepropagation in rocks [56] The importance of capturingheterogeneity in hydraulic fracture modelling has also beendemonstrated in the work of Sirat et al [57]

DeMarsily et al [58] presented a history of heterogeneityand how it applies to hydrogeology They generalized themethods of treating heterogeneity into (1) averaging thatis defining equivalent homogeneous properties and (2)describing spatial variability of rock properties from geologicobservations or local measurements These descriptions areimplemented through either continuous geostatistical mod-els or discontinuous facies (distinctive rock unit) models

Liu et al [56] pointed out various methods used toincorporate heterogeneity in numerical modelling includ-ing (1) randomly assigning different properties through aprobabilistic distribution or otherwise (2) use of a meshwith random geometry but equal element properties (3)projecting a generated microstructure on a regular elementnetwork and then assigning element properties dependingon the position of each element (4) combining a randomgeometry with a generated microstructure (5) use of thehomogenization theory to obtain effective properties througha representative volume element (RVE)

Randomly assigning properties involves statistical mod-elling which requires a probability distribution function Inthis regard the Weibull distribution which was originallydeveloped for modelling the breaking strength of materialsis widely used [56] On a different note studies usingthe random finite element method (RFEM) developed byFenton and Griffiths (eg [59]) for random behavior ofsoils have also considered the spatial variation of propertiesusing a lognormal distribution [59] Results from statisticalmodelling of rock heterogeneity and the homogenizationtheory as implemented in the interaction code R-T2D showeda close match with experiments [56] This builds some levelof confidence in adopting the two methods and also verifiesthe R-T2D code R-T2D or ldquorock and tool interaction coderdquois a numerical approach that was developed based on RockFracture Process Analysis (RFPA) and the Finite ElementMethod (FEM)

In this work theAKModel is extended to study the effectsof heterogeneity on rock fracture propagation while adoptingthe Weibull distribution function for the elastic propertiesspecifically Youngrsquos modulus

61 Weibull Distribution Function as a Representative Func-tion for Heterogeneity The effect of heterogeneity is incor-porated in the model using the Weibull distribution This

000E + 00

500E minus 06

100E minus 05

150E minus 05

200E minus 05

250E minus 05

300E minus 05

350E minus 05

400E minus 05

Prob

abili

ty d

ensit

y fu

nctio

n

20E

+04

40E+04

60E+04

80E

+04

10E

+05

12E

+05

14E+05

16E+05

18E

+05

20E

+05

00E

+00

Youngrsquos modulus x (MPa)

m = 10

m = 15

m = 20

m = 25

m = 40

Figure 11 Variation of the probability density function of Youngrsquosmodulus with the shape parameter119898 according to theWeibull dis-tribution As119898 increases the material becomes more homogeneousand the distribution of Youngrsquos modulus approaches the base valueof 45300MPa

is partly because it was developed to model the strength ofmaterials [60] which is similar to our application and partlybecause of its flexibility it was also used successfully for rockfracture propagation in the RFPA code [19] and is generallywidely used [19 56] TheWeibull distribution is given by [19]

119882(119909) =

119898

119909

(

119909

1199090

)

119898minus1

exp [minus(

119909

1199090

)

119898

] (20)

where 119909 represents a mechanical property such as Youngrsquosmodulus 119909

0is the scale parameter and the shape of the

distribution function is described by the parameter 119898 Tang[12] andWang et al [19] considered119898 to be the homogeneityindex that increases with an increase in material homogene-ity According to [19]where theWeibull distribution is used toassign element properties 119898 signifies material heterogeneityaccounting for cracks and pores in the microscale [19] Wanget al [19] considered homogeneity index of 06 and 11 asvalues for a heterogeneous rock 119898 values of 15 and 2were considered to be relatively homogeneous Coarse andmedium-grained marble were assigned values of 15 and 2respectively [19]

Studies have shown that Youngrsquos modulus (119864) varies spa-tially in rocks [61 62] Hence it is taken as the property to bevaried in this study that is 119909 = 119864 The variation of 119864with theshape parameter (119898) is illustrated in Figure 11 As the shapeparameter119898 increases Youngrsquos modulus approaches the basevalue 119864

0= 45300MPa that is homogeneity increases To

implement the Weibull distribution a built-in function inMATLAB was used


(a) m = 2 (b) m = 4

(c) m = 7 (d) m = 15

(e) No heterogeneity effects (f) Experiment

Figure 12 Crack patterns with variation of homogeneous index (119898) High values of m which represent homogeneous Carrara marble givesatisfactory results in comparison with experiments even though all cases show a failure region around the flaw as expected Experimentalresult is fromWong [2]

62 Effect of Varying the Homogeneity Index (Shape Factor)The Weibull distribution is used to spatially vary Youngrsquosmodulus To do this a random distribution of Youngrsquosmodulus is generated using theWeibull distribution and eachelement in FEM is assigned a value from the distributionResults are presented in Figure 12 for the 60∘ flawed specimensetup as used for validation in the previous section For low119898 values (le7) Youngrsquos modulus varies significantly therebygiving interesting failure patterns With an increase in 119898

(ge15) however favorable comparison with experimental datais obtained Only high values of119898 (very homogeneous) showacceptable results in simulationsThis confirms earlier studiesasserting that Carrara marble could be reasonably assumedto be a homogeneous material [63 64] Only high values of119898 show acceptable results in simulations

The force-displacement diagrams (Figure 13) show anincrease in the average elastic modulus (more values are closeto the base value) with an increase in homogeneity index119898


002 004 006 008 01 012 014 0160Displacement (mm)

m = 2

m = 4

m = 7

No m effectm = 15

0

500

1000

1500

2000

2500

3000

3500

Forc

e (N

)

Figure 13 Force-displacement diagrams from simulations of singlyflawed specimenswith a flaw angle of 60∘ and a varying homogeneityindex (119898) With a high value of 119898 the curve matches the case withno homogeneity effect

Also Figure 13 shows thatmaterialswith higher heterogeneityare weaker due to increase potential of cracking in differentregions

7 Conclusion

A coupled elastoplasticity-damage model developed forquasi-brittle materials has been calibrated and validated forrocks using Carrara marble data The modelrsquos prediction offracture patterns matches closely with experimental resultsthereby verifying the model as a capable tool for fracture pre-diction in rocks Thus the model AK Model could be usedto aid better reservoir management whether hydrocarbongeothermal or underground water reservoirs Although themodel is capable of describing the conferment effect this wasnot taken into account in this study and will be the focusof future work It is important to investigate such effects asconfinement significantly affects rock behavior The Weibulldistribution function has also been used to incorporateheterogeneity effects in the coupled elastoplasticity-damagemodel by varying Youngrsquos modulus Results hint that Carraramarble could be reasonably assumed to be a homogeneousmaterial

Competing Interests

The authors declare that they have no competing interests

References

[1] P Jager S M Schmalholz D W Schmid and E KuhlldquoBrittle fracture during folding of rocks a finite element studyrdquoPhilosophical Magazine vol 88 no 28-29 pp 3245ndash3263 2008

[2] N Y Wong Crack Coalescence in Molded Gypsum and CarraraMarble Massachusetts Institute of Technology 2008

[3] U Cicekli G Z Voyiadjis and R K Abu Al-Rub ldquoA plasticityand anisotropic damagemodel for plain concreterdquo InternationalJournal of Plasticity vol 23 no 10-11 pp 1874ndash1900 2007

[4] R K Abu Al-Rub and S-M Kim ldquoComputational applicationsof a coupled plasticity-damage constitutive model for simulat-ing plain concrete fracturerdquo Engineering Fracture Mechanicsvol 77 no 10 pp 1577ndash1603 2010

[5] H Xu and C Arson ldquoAnisotropic damage models for geo-materials theoretical and numerical challengesrdquo InternationalJournal of Computational Methods vol 11 no 2 Article ID1342007 23 pages 2014

[6] L Chen J F Shao and H W Huang ldquoCoupled elastoplas-tic damage modeling of anisotropic rocksrdquo Computers andGeotechnics vol 37 no 1-2 pp 187ndash194 2010

[7] E Kuhl E Ramm and R de Borst ldquoAn anisotropic gradientdamage model for quasi-brittle materialsrdquo Computer Methodsin Applied Mechanics and Engineering vol 183 no 1-2 pp 87ndash103 2000

[8] L Chen J F Shao Q Z Zhu and G Duveau ldquoInducedanisotropic damage and plasticity in initially anisotropic sed-imentary rocksrdquo International Journal of Rock Mechanics andMining Sciences vol 51 pp 13ndash23 2012

[9] J W Focke and D Munn ldquoCementation exponents in middleeastern carbonate reservoirsrdquo SPE Formation Evaluation vol 2no 2 pp 155ndash167 1987

[10] S C M Krevor R Pini B Li and S M Benson ldquoCapillaryheterogeneity trapping of CO

2in a sandstone rock at reservoir

conditionsrdquo Geophysical Research Letters vol 38 no 15 2011[11] V Palchik ldquoInfluence of porosity and elastic modulus on

uniaxial compressive strength in soft brittle porous sandstonesrdquoRock Mechanics and Rock Engineering vol 32 no 4 pp 303ndash309 1999

[12] C Tang ldquoNumerical simulation of progressive rock failure andassociated seismicityrdquo International Journal of Rock Mechanicsand Mining Sciences amp Geomechanics Abstracts vol 34 no 2pp 249ndash261 1997

[13] Y Fujii and Y Ishijima ldquoConsideration of fracture growth froman inclined slit and inclined initial fracture at the surface ofrock and mortar in compressionrdquo International Journal of RockMechanics and Mining Sciences vol 41 no 6 pp 1035ndash10412004

[14] S Q Yang Y Z Jiang W Y Xu and X Q Chen ldquoExperimentalinvestigation on strength and failure behavior of pre-crackedmarble under conventional triaxial compressionrdquo InternationalJournal of Solids and Structures vol 45 no 17 pp 4796ndash48192008

[15] Y-P Li L-Z Chen andY-HWang ldquoExperimental research onpre-cracked marble under compressionrdquo International Journalof Solids and Structures vol 42 no 9-10 pp 2505ndash2516 2005

[16] L N YWong andHH Einstein ldquoCrack coalescence inmoldedgypsum and carrara marble part 1 macroscopic observationsand interpretationrdquo Rock Mechanics and Rock Engineering vol42 no 3 pp 475ndash511 2009

[17] L N YWong andHH Einstein ldquoCrack coalescence inmoldedgypsum and carrara marble part 2mdashmicroscopic observationsand interpretationrdquo Rock Mechanics and Rock Engineering vol42 no 3 pp 513ndash545 2009

[18] Z Z Liang H Xing S Y Wang D J Williams and CA Tang ldquoA three-dimensional numerical investigation of the


fracture of rock specimens containing a pre-existing surfaceflawrdquo Computers and Geotechnics vol 45 pp 19ndash33 2012

[19] S YWang SW Sloan D C Sheng S Q Yang and C A TangldquoNumerical study of failure behaviour of pre-cracked rock spec-imens under conventional triaxial compressionrdquo InternationalJournal of Solids and Structures vol 51 no 5 pp 1132ndash1148 2014

[20] Z Z Liang C A Tang H X Li T Xu and Y B ZhangldquoNumerical simulation of 3-d failure process in heterogeneousrocksrdquo International Journal of Rock Mechanics and MiningSciences vol 41 supplement 1 pp 323ndash328 2004

[21] A Bobet and H H Einstein ldquoNumerical modeling of fracturecoalescence in a model rock materialrdquo International Journal ofFracture vol 92 no 3 pp 221ndash252 1998

[22] A Bobet and H H Einstein ldquoFracture coalescence in rock-type materials under uniaxial and biaxial compressionrdquo Inter-national Journal of Rock Mechanics andMining Sciences vol 35no 7 pp 863ndash888 1998

[23] B Goncalves Da Silva and H H Einstein ldquoModeling of crackinitiation propagation and coalescence in rocksrdquo InternationalJournal of Fracture vol 182 no 2 pp 167ndash186 2013

[24] H Li and L N Y Wong ldquoInfluence of flaw inclination angleand loading condition on crack initiation and propagationrdquoInternational Journal of Solids and Structures vol 49 no 18 pp2482ndash2499 2012

[25] X-P Zhang and L N Y Wong ldquoLoading rate effects oncracking behavior of flaw-contained specimens under uniaxialcompressionrdquo International Journal of Fracture vol 180 no 1pp 93ndash110 2013

[26] HAKhairDCooke andMHand ldquoNatural fracture networksenhancing unconventional reservoirsrsquo producibilitymappingamppredictingrdquo American Geophysical Union Fall Meeting 2012

[27] L R Alejano and A Bobet ldquoDruckerndashPrager criterionrdquo RockMechanics and Rock Engineering vol 45 no 6 pp 995ndash9992012

[28] L N Y Wong and H H Einstein ldquoSystematic evaluation ofcracking behavior in specimens containing single flaws underuniaxial compressionrdquo International Journal of Rock Mechanicsand Mining Sciences vol 46 no 2 pp 239ndash249 2009

[29] A M Al-Mukhtar and B Merkel ldquoSimulation of the crackpropagation in rocks using fracture mechanics approachrdquoJournal of Failure Analysis and Prevention vol 15 no 1 pp 90ndash100 2015

[30] A R Ingraffea ldquoTheory of crack initiation and propagation inrockrdquo in Fracture Mechanics of Rock B K Atkinson Ed pp71ndash110 Academic Press New York NY USA 1987

[31] T Rabczuk ldquoComputational methods for fracture in brittle andquasi-brittle solids state-of-the-art review and future perspec-tivesrdquo ISRN Applied Mathematics vol 2013 Article ID 84923138 pages 2013

[32] B G da Silva and H H Einstein ldquoFinite element study offracture initiation in flaws subject to internal fluid pressure andvertical stressrdquo International Journal of Solids and Structuresvol 51 no 23-24 pp 4122ndash4136 2014

[33] M Elices G V Guinea J Gomez and J Planas ldquoThe cohesivezone model advantages limitations and challengesrdquo Engineer-ing Fracture Mechanics vol 69 no 2 pp 137ndash163 2001

[34] S Busetti K Mish and Z Reches ldquoDamage and plasticdeformation of reservoir rocks part 1 Damage fracturingrdquoAAPG Bulletin vol 96 no 9 pp 1687ndash1709 2012

[35] D Krajcinovic Damage Mechanics vol 41 Elsevier 1996

[36] J Cordebois and F Sidoroff ldquoAnisotropic damage in elasticityand plasticityrdquo Journal deMecaniqueTheorique et Appliquee pp45ndash60 1982

[37] J Lubliner J Oliver S Oller and E Onate ldquoA plastic-damagemodel for concreterdquo International Journal of Solids and Struc-tures vol 25 no 3 pp 299ndash326 1989

[38] R K Abu Al-Rub and G Z Voyiadjis ldquoOn the coupling ofanisotropic damage and plasticity models for ductile materialsrdquoInternational Journal of Solids and Structures vol 40 no 11 pp2611ndash2643 2003

[39] MA Etheridge ldquoDifferential stressmagnitudes during regionaldeformation and metamorphism upper bound imposed bytensile fracturingrdquo Geology vol 11 no 4 pp 231ndash234 1983

[40] C-S Chen E Pan and B Amadei ldquoDetermination of deforma-bility and tensile strength of anisotropic rock using Braziliantestsrdquo International Journal of Rock Mechanics and MiningSciences vol 35 no 1 pp 43ndash61 1998

[41] B Leiss and T Weiss ldquoFabric anisotropy and its influenceon physical weathering of different types of Carrara marblesrdquoJournal of Structural Geology vol 22 no 11-12 pp 1737ndash17452000

[42] M R H Ramez and S A F Murrell ldquoA petrofabric analysis ofCarrara marblerdquo International Journal of Rock Mechanics andMining Sciences amp Geomechanics Abstracts vol 1 no 2 pp 217ndash229 1964

[43] L N Y Wong C Zou and Y Cheng ldquoFracturing and failurebehavior of carrara marble in quasistatic and dynamic braziliandisc testsrdquo Rock Mechanics and Rock Engineering vol 47 no 4pp 1117ndash1133 2014

[44] M Migliazza A M Ferrero and A Spagnoli ldquoExperimentalinvestigation on crack propagation in Carrara marble subjectedto cyclic loadsrdquo International Journal of Rock Mechanics andMining Sciences vol 48 no 6 pp 1038ndash1044 2011

[45] J M Ramsey and F M Chester ldquoHybrid fracture and thetransition from extension fracture to shear fracturerdquo Naturevol 428 no 6978 pp 63ndash66 2004

[46] G Walton J Arzua L R Alejano and M S Diederichs ldquoAlaboratory-testing-based study on the strength deformabilityand dilatancy of carbonate rocks at low confinementrdquo RockMechanics and Rock Engineering vol 48 no 3 pp 941ndash9582015

[47] G Cardani and A Meda ldquoMarble behaviour under monotonicand cyclic loading in tensionrdquo Construction and Building Mate-rials vol 18 no 6 pp 419ndash424 2004

[48] E Cadoni A Bragov M Dotta D Forni A Konstantinovand A Lomunov ldquoMechanical characterization of rocks at highstrain raterdquo EPJ Web of Conferences vol 26 Article ID 01021 6pages 2012

[49] B Evans J T Fredrich andT FWong ldquoThebrittle-ductile tran-sition in rocks recent experimental and theoretical progressrdquoin The Brittle-Ductile Transition in Rocks pp 1ndash20 AmericanGeophysical Union 1990

[50] M Molenda F Stockhert S Brenne and M Alber ldquoCom-parison of hydraulic and conventional tensile strength testsrdquoin Effective and Sustainable Hydraulic Fracturing chapter 50InTech 2013

[51] D M Cruden ldquoSingle-increment creep experiments on rockunder uniaxial compressionrdquo International Journal of RockMechanics and Mining Sciences amp Geomechanics Abstracts vol8 no 2 pp 127ndash142 1971


[52] S Siegesmund K Ullemeyer T Weiss and E K TscheggldquoPhysical weathering of marbles caused by anisotropic thermalexpansionrdquo International Journal of Earth Sciences vol 89 no1 pp 170ndash182 2000

[53] Y H Hatzor and V Palchik ldquoA microstructure-based failurecriterion forAminadav dolomitesrdquo International Journal of RockMechanics andMining Sciences vol 35 no 6 pp 797ndash805 1998

[54] J T Fredrich B Evans and T-F Wong ldquoEffect of grain size onbrittle and semibrittle strength implications for micromechan-ical modelling of failure in compressionrdquo Journal of GeophysicalResearch Solid Earth vol 95 no 7 pp 10907ndash10920 1990

[55] W A Olsson ldquoGrain size dependence of yield stress in marblerdquoJournal of Geophysical Research vol 79 no 32 pp 4859ndash48621974

[56] H Y Liu M Roquete S Q Kou and P-A Lindqvist ldquoChar-acterization of rock heterogeneity and numerical verificationrdquoEngineering Geology vol 72 no 1-2 pp 89ndash119 2004

[57] M Sirat M Ahmed and X Zhang ldquoPredicting hydraulicfracturing in a carbonate gas reservoir in Abu Dhabi using1D mechanical earth model uncertainty and constraintsrdquo inProceedings of the SPE Middle East Unconventional ResourcesConference and Exhibition Society of Petroleum EngineersMuscat Oman January 2015

[58] G De Marsily F Delay J Goncalves P Renard V Teles andS Violette ldquoDealing with spatial heterogeneityrdquo HydrogeologyJournal vol 13 no 1 pp 161ndash183 2005

[59] D V Griffiths and G A Fenton ldquoProbabilistic settlementanalysis by stochastic and random finite-element methodsrdquoJournal of Geotechnical and Geoenvironmental Engineering vol135 no 11 pp 1629ndash1637 2009

[60] W Weibull ldquoA statistical distribution function of wide appli-cabilityrdquo Journal of Applied Mechanics no 51-A-6 pp 293ndash2971951

[61] A Gudmundsson ldquoEffects of Youngrsquos modulus on fault dis-placementrdquo Comptes RendusmdashGeoscience vol 336 no 1 pp85ndash92 2004

[62] N S Rao B Al-Qadeeri and V K Kidambi ldquoBuilding aseismic-driven 3D geomechanical model in a deep carbonatereservoirrdquo in Proceedings of the SEG 2011 Annual Meeting SanAntonio Tex USA September 2011

[63] N Herz and N E Dean ldquoStable isotopes and archaeologicalgeology the Carrara marble northern Italyrdquo Applied Geochem-istry vol 1 no 1 pp 139ndash151 1986

[64] A Barnhoorn M Bystricky L Burlini and K Kunze ldquoTherole of recrystallisation on the deformation behaviour of calciterocks large strain torsion experiments on Carrara marblerdquoJournal of Structural Geology vol 26 no 5 pp 885ndash903 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


Subsequently (Section 3) the constitutive model is conciselydescribed without detailing its numerical implementationUsing Carrara marble as a representative rock material themodel is then calibrated (Section 4) and with experimentson single-flawed rock specimens the capability of the modelin predicting fracture patterns is demonstrated in Section 5(model validation) In Section 6 some methods of incorpo-rating the effects of heterogeneity in modelling geologicalmedia are highlighted before the spatial variation of elasticproperties is used to study heterogeneity effects through theWeibull distribution function

2 Fracture Prediction in Rocks

Crack initiation and propagation in rocks have been thesubject of various past and current studies The approachis to study propagation of cracks from precracked (flawed)specimens using experiments (eg [13ndash17]) and numeri-cal simulations (see [18ndash21]) Bobet and Einstein [21 22]used both experiments and numerical simulations to studycrack initiation propagation and coalescence in rock Theyobtained good simulation results with the upgraded codeFROCK which is based on the Displacement DiscontinuityMethod (DDM) A later work by Goncalves Da Silva andEinstein [23] improved the capabilities of the code by intro-ducing a new strain-based criterion as well as a normal stress-dependent criterion for crack development

AUTODYN is a nonlinear hydrodynamics code compat-ible with ANSYS Li and Wong [24] used it to study theinfluence of flaw inclination angle and loading condition oncrack propagation Tang [12] developed the 2D finite elementcode RFPA 2D The code incorporates the effect of bothdamage and heterogeneity and its extended 3D version hasbeen used successfully in predicting fracture of specimensin triaxial compression [19] In another study Zhang andWong [25] used the Bonded Particle Method (BPM) toinvestigate the effect of loading rate on crack behavior offlawed specimens Crack coalescence mode was observed tochange from that dominated by the tensile segment to thatdominated by the shear-band

FROCK is based on DDM which is a boundary elementmethod (BEM) and according to Khair et al [26] the FiniteElement Method (FEM) is superior to BEM in predictingsubsurface fractures While the AUTODYN-based modeladopted by [24] captures damage like the RFPA it relieson Drucker-Prager criterion which has been proven tooverestimate intact rock strength [27] Additionally RFPAdoes not consider anisotropy and plastic deformation inrocks This calls for an elastoplastic-damage model based onFEM The model by Abu Al-Rub and Kim [4] AK Modelfor plain concrete considers damage effects anisotropy andplastic deformation It also adopts the Lubliner yield criterionwhich is an improvement on the Drucker-Prager criterionThis model is implemented as a user material subroutine inAbaqus a commercial finite element code

It is noteworthy that Linear Elastic Fracture Mechanics(LEFM) has also been used to simulate crack propagation inrocks (eg see [29] which uses FRANC2D crack propagationsimulator) However LEFM is not able to predict fracture

initiation and when a crack is assumed the stress intensityand fracture toughness on which LEFM is based could bemeaningless for the assumed flaw size [30] Also the materialbehavior near the crack tip region (fracture process zone)could be inelastic and nonlinear [30] making LEFM onlyapplicable when the size of process zone (L) is significantlysmall with respect to the smallest critical dimension of thestructure (D) ndash DL gt 100 [31] This influenced the decisionof [32] in their elastic Abaqus analysis to take stress measure-ments at a distance from the flaw tip that is to avoid stressesin the process zone which make no physical sense Whilecohesive zone modelling has been introduced to addressthese issues in materials like concrete [33] increased numberor complexity of fractures might not be easily handled bycohesive zone modelling The continuum damage mechanics(CDM) approach adopted by the AK Model is generallymeritorious because (1) it accounts for localized damage(which actually happens in rock deformation) and micro-scopic initiation of cracks unlike Fracture Mechanics whichconsiders a clearly defined discrete macroscopic cracks (2)it is capable of analyzing complex fractures and networkswhich is not possible in elastic analysis (such as LinearElastic Fracture Mechanics (LEFM)) (3) there is no need forany special initial assumptions such as initial perturbationsand (4) it has no computational limitations on number offractures like the Extended Finite Element Method (XFEM)[34]

3 Elastoplastic-Damage Constitutive Model

A coupled elastoplasticity-damage model (AK Model) isadopted here It was developed by Abu Al-Rub and Kim[4] based on an earlier work by Cicekli et al [3] hencethe name AK For a full description of the model pleasesee [3 4] The model stands out because it presents acoupled anisotropic damage and plasticity constitutivemodelthat predicts rockrsquos distinct behavior in compression andtension with the following (1) a modified continuum damagemechanics framework to include quadratic isotropic andanisotropic variation of the effective (undamaged) stress interms of the nominal (damaged) stress The nominal andeffective configurations are explained in Section 31 and (2)two novel and different damage (power) evolution laws forboth tension and compression for a more accurate predictionof rock behavior after damage initiation

An overview of the modelrsquos main constitutive relationsis presented here to give a general idea without detailing itsnumerical implementation

31 Anisotropic Damage Model In damage mechanics thedamaged (nominal) configuration of a material is the normalstate of thematerial with imperfections like voids (pores) andcracks To analyze this an imaginary state of the materialcalled the effective (undamaged) configuration is assumed(see Figure 1)

The effective (undamaged) configuration of the materialconsiders is to be intact without any voids or cracks Strainis assumed to be constant in both configurations (ie strainequivalence hypothesis) and a damage internal state variable


AA



120593 =A minus A

A=

AD

A

0 le 120593 le 1




120590119894119895= (1 minus 120593)

2120590119894119895 (1)

where 120590119894119895and 120590



119864119894119895119896119897

= (1 minus 120593)2119864119894119895119896119897

(2)


120590119894119895= 120590+

119894119895+ 120590minus

119894119895

120590119894119895= 120590+

119894119895+ 120590minus

119894119895

(3)




120590119894119895= 119872119894119895119896119897

120590119896119897

119872119894119895119896119897

= (1 minus 120593+)2

119875+

119894119895119896119897+ (1 minus 120593

minus)2

119875minus

119894119895119896119897

119875+

119894119895119896119897=

3

sum

119896=1

119867(120590

(119896)

) 119899(119896)

119894119899(119896)

119895119899(119896)

119896119899(119896)

119897

(4)

where 119875+

119894119895119896119897and 119875

minus

119894119895119896119897= 119868119894119895119896119897

minus 119875+



119894119895119896119897is the identity




120590

(119896)


120590

(119896)


(119896)

119894are



119891 = radic31198692+ 1205721198681+ 120573 (120576

+

eq 120576minus

eq)119867 (120590max)

120590max


eq) le 0

(5)

where 1198692



= 120590119894119895

minus 1205901198961198961205751198941198953 where 120575

119894119895is


= 120590119896119896







120572 =

(1198911198870119891minus

0) minus 1

2 (1198911198870119891minus

0) minus 1

120573 = (1 minus 120572)


eq)

119888+(120576+

eq)minus (1 + 120572)

(6)

where 1198911198870




+

eq and 120576minus




minus120576eq)]

119888+= 119891+

0+ ℎ+120576+

eq(7)

where 119891minus

0and 119891

+




120576+

eq = int

119905

0

120576+

eq119889119905 120576+

eq = 119903 (120590119894119895) 120576

pmax

120576minus

eq = int

119905

0

120576minus

eq119889119905 120576minus

eq = minus (1 minus 119903 (120590119894119895)) 120576

pmin

(8)

where

119903 (120590119894119895) =

sum3

119896=1⟨120590

(119896)

⟩

sum3

119896=1

1003816100381610038161003816100381610038161003816

120590

(119896)1003816100381610038161003816100381610038161003816

(9)


p119894119895such that 120576p

1gt 120576

p2gt 120576

p3are represented


pmax = 120576

p1and 120576

pmin = 120576






120576p119894119895=

120582

p 120597119865p

120597120590119894119895

119865p= radic3119869

2+ 120572

p1198681 (10)

where 120582



120582


120582

p119891 = 0 119891 le 0

120582

pge 0 (11)


119892plusmn= radic

1

2

119884plusmn

119894119895119884plusmn


eq) le 0 (12)



0



119884plusmn

119903119904= minus

1

2

119864

minus1

119894119895119886119887120590119886119887

120597119872119894119895119901119902

120597120593plusmn

119903119904

120590119901119902 (13)


plusmn

119894119895=

120582

plusmn

119889

120597119892plusmn

120597119884plusmn

119894119895

(14)

where 120582

plusmn

119889= plusmn


plusmn

eq = radicplusmn

119894119895plusmn

119894119895

120593plusmn

eq = int

119905

0

plusmn

eq119889119905

(15)


120593+

eq = 119861+(

119870+

0

119870+)(

119870+

119870+

0

minus 1)

119902+

120593minus

eq = 119861minus(

119870minus

119870minus

0

minus 1)

119902minus

(16)



0


119870+= radic119864120590

+

119894119895120576119890+

119894119895= radic119864120590

+(

120590+

119864

) = 120590+

119870+

0= 119891+

0cong 119891+

u

(17)

where 119891+






0 are respectively


minus

2+ 120572119868minus

1= [1 + (

120572

3

)] 120590minus (18)

119870minus

0= [1 + (

120572

3

)]1198910

minus (19)

where 1198910


4 Model Calibration














119870minus






119891+


119870+







0102030405060708090

100

Stre

ss (M

Pa)


0001 0002 0003 0004 0005 0006 0007 00080Strain (mmmm)

0009

(a)


00001 0000500003 000060 00002 00004Strain (mmmm)

0

1

2

3

4

5

6

7

8

Stre

ss (M

Pa)

(b)


5 Model Validation





152mm

13mm

125mm

32mm

76mm

(a)

L

120573

120572

(b)













T

T


T

T


T

T


SS

T

T


S

S


S


S









Flaw

angl

eFl

aw an

gle

Flaw

angl

eFl

aw an

gle

Flaw

angl

e


(a)

(b)

(c)

(d)

(e)

0∘

30∘

45∘

60∘

75∘



0

500

1000

1500

2000

2500

3000

3500

4000

Forc

e (N

)

002 004 006 008 01 012 014 0160

Displacement (mm)
















000E + 00

500E minus 06

100E minus 05

150E minus 05

200E minus 05

250E minus 05

300E minus 05

350E minus 05

400E minus 05

Prob

abili

ty d

ensit

y fu

nctio

n

20E

+04

40E+04

60E+04

80E

+04

10E

+05

12E

+05

14E+05

16E+05

18E

+05

20E

+05

00E

+00


m = 10

m = 15

m = 20

m = 25

m = 40



119882(119909) =

119898

119909

(

119909

1199090

)

119898minus1

exp [minus(

119909

1199090

)

119898

] (20)








(a) m = 2 (b) m = 4

(c) m = 7 (d) m = 15







002 004 006 008 01 012 014 0160Displacement (mm)

m = 2

m = 4

m = 7

No m effectm = 15

0

500

1000

1500

2000

2500

3000

3500

Forc

e (N

)



7 Conclusion


Competing Interests


References













































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










AA



120593 =A minus A

A=

AD

A

0 le 120593 le 1




120590119894119895= (1 minus 120593)

2120590119894119895 (1)

where 120590119894119895and 120590



119864119894119895119896119897

= (1 minus 120593)2119864119894119895119896119897

(2)


120590119894119895= 120590+

119894119895+ 120590minus

119894119895

120590119894119895= 120590+

119894119895+ 120590minus

119894119895

(3)




120590119894119895= 119872119894119895119896119897

120590119896119897

119872119894119895119896119897

= (1 minus 120593+)2

119875+

119894119895119896119897+ (1 minus 120593

minus)2

119875minus

119894119895119896119897

119875+

119894119895119896119897=

3

sum

119896=1

119867(120590

(119896)

) 119899(119896)

119894119899(119896)

119895119899(119896)

119896119899(119896)

119897

(4)

where 119875+

119894119895119896119897and 119875

minus

119894119895119896119897= 119868119894119895119896119897

minus 119875+



119894119895119896119897is the identity




120590

(119896)


120590

(119896)


(119896)

119894are



119891 = radic31198692+ 1205721198681+ 120573 (120576

+

eq 120576minus

eq)119867 (120590max)

120590max


eq) le 0

(5)

where 1198692



= 120590119894119895

minus 1205901198961198961205751198941198953 where 120575

119894119895is


= 120590119896119896







120572 =

(1198911198870119891minus

0) minus 1

2 (1198911198870119891minus

0) minus 1

120573 = (1 minus 120572)


eq)

119888+(120576+

eq)minus (1 + 120572)

(6)

where 1198911198870




+

eq and 120576minus




minus120576eq)]

119888+= 119891+

0+ ℎ+120576+

eq(7)

where 119891minus

0and 119891

+




120576+

eq = int

119905

0

120576+

eq119889119905 120576+

eq = 119903 (120590119894119895) 120576

pmax

120576minus

eq = int

119905

0

120576minus

eq119889119905 120576minus

eq = minus (1 minus 119903 (120590119894119895)) 120576

pmin

(8)

where

119903 (120590119894119895) =

sum3

119896=1⟨120590

(119896)

⟩

sum3

119896=1

1003816100381610038161003816100381610038161003816

120590

(119896)1003816100381610038161003816100381610038161003816

(9)


p119894119895such that 120576p

1gt 120576

p2gt 120576

p3are represented


pmax = 120576

p1and 120576

pmin = 120576






120576p119894119895=

120582

p 120597119865p

120597120590119894119895

119865p= radic3119869

2+ 120572

p1198681 (10)

where 120582



120582


120582

p119891 = 0 119891 le 0

120582

pge 0 (11)


119892plusmn= radic

1

2

119884plusmn

119894119895119884plusmn


eq) le 0 (12)



0



119884plusmn

119903119904= minus

1

2

119864

minus1

119894119895119886119887120590119886119887

120597119872119894119895119901119902

120597120593plusmn

119903119904

120590119901119902 (13)


plusmn

119894119895=

120582

plusmn

119889

120597119892plusmn

120597119884plusmn

119894119895

(14)

where 120582

plusmn

119889= plusmn


plusmn

eq = radicplusmn

119894119895plusmn

119894119895

120593plusmn

eq = int

119905

0

plusmn

eq119889119905

(15)


120593+

eq = 119861+(

119870+

0

119870+)(

119870+

119870+

0

minus 1)

119902+

120593minus

eq = 119861minus(

119870minus

119870minus

0

minus 1)

119902minus

(16)



0


119870+= radic119864120590

+

119894119895120576119890+

119894119895= radic119864120590

+(

120590+

119864

) = 120590+

119870+

0= 119891+

0cong 119891+

u

(17)

where 119891+






0 are respectively


minus

2+ 120572119868minus

1= [1 + (

120572

3

)] 120590minus (18)

119870minus

0= [1 + (

120572

3

)]1198910

minus (19)

where 1198910


4 Model Calibration














119870minus






119891+


119870+







0102030405060708090

100

Stre

ss (M

Pa)


0001 0002 0003 0004 0005 0006 0007 00080Strain (mmmm)

0009

(a)


00001 0000500003 000060 00002 00004Strain (mmmm)

0

1

2

3

4

5

6

7

8

Stre

ss (M

Pa)

(b)


5 Model Validation





152mm

13mm

125mm

32mm

76mm

(a)

L

120573

120572

(b)













T

T


T

T


T

T


SS

T

T


S

S


S


S









Flaw

angl

eFl

aw an

gle

Flaw

angl

eFl

aw an

gle

Flaw

angl

e


(a)

(b)

(c)

(d)

(e)

0∘

30∘

45∘

60∘

75∘



0

500

1000

1500

2000

2500

3000

3500

4000

Forc

e (N

)

002 004 006 008 01 012 014 0160

Displacement (mm)
















000E + 00

500E minus 06

100E minus 05

150E minus 05

200E minus 05

250E minus 05

300E minus 05

350E minus 05

400E minus 05

Prob

abili

ty d

ensit

y fu

nctio

n

20E

+04

40E+04

60E+04

80E

+04

10E

+05

12E

+05

14E+05

16E+05

18E

+05

20E

+05

00E

+00


m = 10

m = 15

m = 20

m = 25

m = 40



119882(119909) =

119898

119909

(

119909

1199090

)

119898minus1

exp [minus(

119909

1199090

)

119898

] (20)








(a) m = 2 (b) m = 4

(c) m = 7 (d) m = 15







002 004 006 008 01 012 014 0160Displacement (mm)

m = 2

m = 4

m = 7

No m effectm = 15

0

500

1000

1500

2000

2500

3000

3500

Forc

e (N

)



7 Conclusion


Competing Interests


References













































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












120572 =

(1198911198870119891minus

0) minus 1

2 (1198911198870119891minus

0) minus 1

120573 = (1 minus 120572)


eq)

119888+(120576+

eq)minus (1 + 120572)

(6)

where 1198911198870




+

eq and 120576minus




minus120576eq)]

119888+= 119891+

0+ ℎ+120576+

eq(7)

where 119891minus

0and 119891

+




120576+

eq = int

119905

0

120576+

eq119889119905 120576+

eq = 119903 (120590119894119895) 120576

pmax

120576minus

eq = int

119905

0

120576minus

eq119889119905 120576minus

eq = minus (1 minus 119903 (120590119894119895)) 120576

pmin

(8)

where

119903 (120590119894119895) =

sum3

119896=1⟨120590

(119896)

⟩

sum3

119896=1

1003816100381610038161003816100381610038161003816

120590

(119896)1003816100381610038161003816100381610038161003816

(9)


p119894119895such that 120576p

1gt 120576

p2gt 120576

p3are represented


pmax = 120576

p1and 120576

pmin = 120576






120576p119894119895=

120582

p 120597119865p

120597120590119894119895

119865p= radic3119869

2+ 120572

p1198681 (10)

where 120582



120582


120582

p119891 = 0 119891 le 0

120582

pge 0 (11)


119892plusmn= radic

1

2

119884plusmn

119894119895119884plusmn


eq) le 0 (12)



0



119884plusmn

119903119904= minus

1

2

119864

minus1

119894119895119886119887120590119886119887

120597119872119894119895119901119902

120597120593plusmn

119903119904

120590119901119902 (13)


plusmn

119894119895=

120582

plusmn

119889

120597119892plusmn

120597119884plusmn

119894119895

(14)

where 120582

plusmn

119889= plusmn


plusmn

eq = radicplusmn

119894119895plusmn

119894119895

120593plusmn

eq = int

119905

0

plusmn

eq119889119905

(15)


120593+

eq = 119861+(

119870+

0

119870+)(

119870+

119870+

0

minus 1)

119902+

120593minus

eq = 119861minus(

119870minus

119870minus

0

minus 1)

119902minus

(16)



0


119870+= radic119864120590

+

119894119895120576119890+

119894119895= radic119864120590

+(

120590+

119864

) = 120590+

119870+

0= 119891+

0cong 119891+

u

(17)

where 119891+






0 are respectively


minus

2+ 120572119868minus

1= [1 + (

120572

3

)] 120590minus (18)

119870minus

0= [1 + (

120572

3

)]1198910

minus (19)

where 1198910


4 Model Calibration














119870minus






119891+


119870+







0102030405060708090

100

Stre

ss (M

Pa)


0001 0002 0003 0004 0005 0006 0007 00080Strain (mmmm)

0009

(a)


00001 0000500003 000060 00002 00004Strain (mmmm)

0

1

2

3

4

5

6

7

8

Stre

ss (M

Pa)

(b)


5 Model Validation





152mm

13mm

125mm

32mm

76mm

(a)

L

120573

120572

(b)













T

T


T

T


T

T


SS

T

T


S

S


S


S









Flaw

angl

eFl

aw an

gle

Flaw

angl

eFl

aw an

gle

Flaw

angl

e


(a)

(b)

(c)

(d)

(e)

0∘

30∘

45∘

60∘

75∘



0

500

1000

1500

2000

2500

3000

3500

4000

Forc

e (N

)

002 004 006 008 01 012 014 0160

Displacement (mm)
















000E + 00

500E minus 06

100E minus 05

150E minus 05

200E minus 05

250E minus 05

300E minus 05

350E minus 05

400E minus 05

Prob

abili

ty d

ensit

y fu

nctio

n

20E

+04

40E+04

60E+04

80E

+04

10E

+05

12E

+05

14E+05

16E+05

18E

+05

20E

+05

00E

+00


m = 10

m = 15

m = 20

m = 25

m = 40



119882(119909) =

119898

119909

(

119909

1199090

)

119898minus1

exp [minus(

119909

1199090

)

119898

] (20)








(a) m = 2 (b) m = 4

(c) m = 7 (d) m = 15







002 004 006 008 01 012 014 0160Displacement (mm)

m = 2

m = 4

m = 7

No m effectm = 15

0

500

1000

1500

2000

2500

3000

3500

Forc

e (N

)



7 Conclusion


Competing Interests


References













































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











0 are respectively


minus

2+ 120572119868minus

1= [1 + (

120572

3

)] 120590minus (18)

119870minus

0= [1 + (

120572

3

)]1198910

minus (19)

where 1198910


4 Model Calibration














119870minus






119891+


119870+







0102030405060708090

100

Stre

ss (M

Pa)


0001 0002 0003 0004 0005 0006 0007 00080Strain (mmmm)

0009

(a)


00001 0000500003 000060 00002 00004Strain (mmmm)

0

1

2

3

4

5

6

7

8

Stre

ss (M

Pa)

(b)


5 Model Validation





152mm

13mm

125mm

32mm

76mm

(a)

L

120573

120572

(b)













T

T


T

T


T

T


SS

T

T


S

S


S


S









Flaw

angl

eFl

aw an

gle

Flaw

angl

eFl

aw an

gle

Flaw

angl

e


(a)

(b)

(c)

(d)

(e)

0∘

30∘

45∘

60∘

75∘



0

500

1000

1500

2000

2500

3000

3500

4000

Forc

e (N

)

002 004 006 008 01 012 014 0160

Displacement (mm)
















000E + 00

500E minus 06

100E minus 05

150E minus 05

200E minus 05

250E minus 05

300E minus 05

350E minus 05

400E minus 05

Prob

abili

ty d

ensit

y fu

nctio

n

20E

+04

40E+04

60E+04

80E

+04

10E

+05

12E

+05

14E+05

16E+05

18E

+05

20E

+05

00E

+00


m = 10

m = 15

m = 20

m = 25

m = 40



119882(119909) =

119898

119909

(

119909

1199090

)

119898minus1

exp [minus(

119909

1199090

)

119898

] (20)








(a) m = 2 (b) m = 4

(c) m = 7 (d) m = 15







002 004 006 008 01 012 014 0160Displacement (mm)

m = 2

m = 4

m = 7

No m effectm = 15

0

500

1000

1500

2000

2500

3000

3500

Forc

e (N

)



7 Conclusion


Competing Interests


References













































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













119870minus






119891+


119870+







0102030405060708090

100

Stre

ss (M

Pa)


0001 0002 0003 0004 0005 0006 0007 00080Strain (mmmm)

0009

(a)


00001 0000500003 000060 00002 00004Strain (mmmm)

0

1

2

3

4

5

6

7

8

Stre

ss (M

Pa)

(b)


5 Model Validation





152mm

13mm

125mm

32mm

76mm

(a)

L

120573

120572

(b)













T

T


T

T


T

T


SS

T

T


S

S


S


S









Flaw

angl

eFl

aw an

gle

Flaw

angl

eFl

aw an

gle

Flaw

angl

e


(a)

(b)

(c)

(d)

(e)

0∘

30∘

45∘

60∘

75∘



0

500

1000

1500

2000

2500

3000

3500

4000

Forc

e (N

)

002 004 006 008 01 012 014 0160

Displacement (mm)
















000E + 00

500E minus 06

100E minus 05

150E minus 05

200E minus 05

250E minus 05

300E minus 05

350E minus 05

400E minus 05

Prob

abili

ty d

ensit

y fu

nctio

n

20E

+04

40E+04

60E+04

80E

+04

10E

+05

12E

+05

14E+05

16E+05

18E

+05

20E

+05

00E

+00


m = 10

m = 15

m = 20

m = 25

m = 40



119882(119909) =

119898

119909

(

119909

1199090

)

119898minus1

exp [minus(

119909

1199090

)

119898

] (20)








(a) m = 2 (b) m = 4

(c) m = 7 (d) m = 15







002 004 006 008 01 012 014 0160Displacement (mm)

m = 2

m = 4

m = 7

No m effectm = 15

0

500

1000

1500

2000

2500

3000

3500

Forc

e (N

)



7 Conclusion


Competing Interests


References













































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










152mm

13mm

125mm

32mm

76mm

(a)

L

120573

120572

(b)













T

T


T

T


T

T


SS

T

T


S

S


S


S









Flaw

angl

eFl

aw an

gle

Flaw

angl

eFl

aw an

gle

Flaw

angl

e


(a)

(b)

(c)

(d)

(e)

0∘

30∘

45∘

60∘

75∘



0

500

1000

1500

2000

2500

3000

3500

4000

Forc

e (N

)

002 004 006 008 01 012 014 0160

Displacement (mm)
















000E + 00

500E minus 06

100E minus 05

150E minus 05

200E minus 05

250E minus 05

300E minus 05

350E minus 05

400E minus 05

Prob

abili

ty d

ensit

y fu

nctio

n

20E

+04

40E+04

60E+04

80E

+04

10E

+05

12E

+05

14E+05

16E+05

18E

+05

20E

+05

00E

+00


m = 10

m = 15

m = 20

m = 25

m = 40



119882(119909) =

119898

119909

(

119909

1199090

)

119898minus1

exp [minus(

119909

1199090

)

119898

] (20)








(a) m = 2 (b) m = 4

(c) m = 7 (d) m = 15







002 004 006 008 01 012 014 0160Displacement (mm)

m = 2

m = 4

m = 7

No m effectm = 15

0

500

1000

1500

2000

2500

3000

3500

Forc

e (N

)



7 Conclusion


Competing Interests


References













































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










T

T


T

T


T

T


SS

T

T


S

S


S


S









Flaw

angl

eFl

aw an

gle

Flaw

angl

eFl

aw an

gle

Flaw

angl

e


(a)

(b)

(c)

(d)

(e)

0∘

30∘

45∘

60∘

75∘



0

500

1000

1500

2000

2500

3000

3500

4000

Forc

e (N

)

002 004 006 008 01 012 014 0160

Displacement (mm)
















000E + 00

500E minus 06

100E minus 05

150E minus 05

200E minus 05

250E minus 05

300E minus 05

350E minus 05

400E minus 05

Prob

abili

ty d

ensit

y fu

nctio

n

20E

+04

40E+04

60E+04

80E

+04

10E

+05

12E

+05

14E+05

16E+05

18E

+05

20E

+05

00E

+00


m = 10

m = 15

m = 20

m = 25

m = 40



119882(119909) =

119898

119909

(

119909

1199090

)

119898minus1

exp [minus(

119909

1199090

)

119898

] (20)








(a) m = 2 (b) m = 4

(c) m = 7 (d) m = 15







002 004 006 008 01 012 014 0160Displacement (mm)

m = 2

m = 4

m = 7

No m effectm = 15

0

500

1000

1500

2000

2500

3000

3500

Forc

e (N

)



7 Conclusion


Competing Interests


References













































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Flaw

angl

eFl

aw an

gle

Flaw

angl

eFl

aw an

gle

Flaw

angl

e


(a)

(b)

(c)

(d)

(e)

0∘

30∘

45∘

60∘

75∘



0

500

1000

1500

2000

2500

3000

3500

4000

Forc

e (N

)

002 004 006 008 01 012 014 0160

Displacement (mm)
















000E + 00

500E minus 06

100E minus 05

150E minus 05

200E minus 05

250E minus 05

300E minus 05

350E minus 05

400E minus 05

Prob

abili

ty d

ensit

y fu

nctio

n

20E

+04

40E+04

60E+04

80E

+04

10E

+05

12E

+05

14E+05

16E+05

18E

+05

20E

+05

00E

+00


m = 10

m = 15

m = 20

m = 25

m = 40



119882(119909) =

119898

119909

(

119909

1199090

)

119898minus1

exp [minus(

119909

1199090

)

119898

] (20)








(a) m = 2 (b) m = 4

(c) m = 7 (d) m = 15







002 004 006 008 01 012 014 0160Displacement (mm)

m = 2

m = 4

m = 7

No m effectm = 15

0

500

1000

1500

2000

2500

3000

3500

Forc

e (N

)



7 Conclusion


Competing Interests


References













































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0

500

1000

1500

2000

2500

3000

3500

4000

Forc

e (N

)

002 004 006 008 01 012 014 0160

Displacement (mm)
















000E + 00

500E minus 06

100E minus 05

150E minus 05

200E minus 05

250E minus 05

300E minus 05

350E minus 05

400E minus 05

Prob

abili

ty d

ensit

y fu

nctio

n

20E

+04

40E+04

60E+04

80E

+04

10E

+05

12E

+05

14E+05

16E+05

18E

+05

20E

+05

00E

+00


m = 10

m = 15

m = 20

m = 25

m = 40



119882(119909) =

119898

119909

(

119909

1199090

)

119898minus1

exp [minus(

119909

1199090

)

119898

] (20)








(a) m = 2 (b) m = 4

(c) m = 7 (d) m = 15







002 004 006 008 01 012 014 0160Displacement (mm)

m = 2

m = 4

m = 7

No m effectm = 15

0

500

1000

1500

2000

2500

3000

3500

Forc

e (N

)



7 Conclusion


Competing Interests


References













































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















000E + 00

500E minus 06

100E minus 05

150E minus 05

200E minus 05

250E minus 05

300E minus 05

350E minus 05

400E minus 05

Prob

abili

ty d

ensit

y fu

nctio

n

20E

+04

40E+04

60E+04

80E

+04

10E

+05

12E

+05

14E+05

16E+05

18E

+05

20E

+05

00E

+00


m = 10

m = 15

m = 20

m = 25

m = 40



119882(119909) =

119898

119909

(

119909

1199090

)

119898minus1

exp [minus(

119909

1199090

)

119898

] (20)








(a) m = 2 (b) m = 4

(c) m = 7 (d) m = 15







002 004 006 008 01 012 014 0160Displacement (mm)

m = 2

m = 4

m = 7

No m effectm = 15

0

500

1000

1500

2000

2500

3000

3500

Forc

e (N

)



7 Conclusion


Competing Interests


References













































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(a) m = 2 (b) m = 4

(c) m = 7 (d) m = 15







002 004 006 008 01 012 014 0160Displacement (mm)

m = 2

m = 4

m = 7

No m effectm = 15

0

500

1000

1500

2000

2500

3000

3500

Forc

e (N

)



7 Conclusion


Competing Interests


References













































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










002 004 006 008 01 012 014 0160Displacement (mm)

m = 2

m = 4

m = 7

No m effectm = 15

0

500

1000

1500

2000

2500

3000

3500

Forc

e (N

)



7 Conclusion


Competing Interests


References













































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra

































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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