research article implementation of fictitious crack model

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 726317 8 pageshttpdxdoiorg1011552013726317

Research ArticleImplementation of Fictitious Crack Model UsingContact Finite Element Method for the Crack Propagation inConcrete under Cyclic Load

Lanhao Zhao12 Tianyou Yan3 Xin Bai2 Tongchun Li1 and Jing Cheng1

1 College of Water Conservancy and Hydropower Hohai University Xikang Road Nanjing 210098 China2 School of Engineering amp Materials Science Queen Mary University of London Mile End Road London E1 4NS UK3 Changjiang Survey Planning Design and Research Limited Liability Company Jiefang Dadao Road Wuhan 430010 China

Correspondence should be addressed to Tongchun Li zlhhhu163com

Received 19 July 2013 Accepted 24 August 2013

Academic Editor Song Cen

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

The mixed freedom finite element method proposed for contact problems was extended to simulate the fracture mechanics ofconcrete using the fictitious crack model Pairs of contact points were set along the potential developing path of the crack Thedisplacement of structure was chosen as the basic variable and the nodal contact force in contact region under local coordinatesystemwas selected as the iteration variable to confine the nonlinear iteration process in the potential contact surface which is morenumerically efficient The contact forces and the opening of the crack were obtained explicitly enabling the softening constitutiverelation for the concrete to be introduced conveniently by the fictitious crack model According to the states of the load and thecrack the constitutive relation of concrete under cyclic load is characterized by six contact states with each contact state denoting itsown displacement-stress relation In this paper the basic idea of themixed freedomfinite elementmethod as well as the constitutiverelation of concrete under cyclic load is presented A numerical method was proposed to simulate crack propagation process inconcrete The accuracy and capability of the proposed method were verified by a numerical example against experiment data

1 Introduction

Concrete is a highly heterogeneous material with its proper-ties varyong widely from point to point due to the presenceof high strength aggregates medium strength mortar andweakermortar-aggregate interfacesThe existence of geomet-ric voids which act as stress raisers can also contribute to thechanging property [1] Due to the low tensile strength of con-crete material cracking is the dominant failure mechanismin concrete structures and the mechanism of crack initiationand growth has received significant research effort

In recent years considerable amount of effort has beendevoted to develop numerical models to simulate the fracturebehavior of concrete used in civil engineering structures [2]A number of modeling approaches have been implementedsuch as plasticity models [3 4] microplanemodels [5 6] andfracture mechanics models Generally in the field of frac-ture mechanics the numerical methods based on the finite

element method (FEM) are classified into two groups [7]ldquosmeared crack approachrdquo and ldquodiscrete crack approachrdquo Inthe smeared crack approach the fracture is represented ina smeared manner in which an infinite number of parallelcracks with infinitely small opening are (theoretically) dis-tributed (smeared) over the finite element [8] Those cracksare usually modeled on a fixed finite element mesh and theirpropagation is simulated by the reduction of the stiffness andstrength of the material Most of the models in this approachtreat concrete as a quasi-brittle material that exhibits strain-softening behavior under tensile loading [9]The constitutivelaws defined by stress-strain relations are nonlinear andinvolve a strain softening behavior which can be challengingin the analysisThe systemof equationsmay become ill-posed[10] and can introduce localization instabilities and spuriousmesh sensitivity to the finite element calculations [8] In thefield of fracture mechanics there are a number of modelsthat can deal with the nonlinear zone ahead of a crack tip in

2 Mathematical Problems in Engineering

concrete including the fictitious crack model (FCM) [11] thecrack band model (CBM) [12] the two-parameter fracturemodel (TPFM) [13 14] the size effect model (SZM) [15]the effective crack model (ECM) [16 17] and the double-Kfracture model (DKFM) [18ndash20]

The discrete approach is preferred when there is onecrack or a finite number of cracks in the structure [2]Among them the cohesive zone model (CZM) whichassumes that stresses act across a narrowly open crack wasdeveloped by Hillerborg et al [11] under the name FCM Inthe literature it is generally accepted that the FCM is the bestsimple fracture mechanics model as it provides reasonableapproximation of the crack propagation in concrete usingonly simple calculations The model is mainly applicable tomode-I fracture but recently it has been extended to modelshear failure mode-II and mode-III cracks [21] The FCMhas been used to investigate the fracture process in concreteunder monotonic and cyclic loading Various constitutiverelationships for cyclic tension behaviour of concrete suchas the focal point model [22] and the continuous functionmodel [23] have been developed More recently linear andbilinear constitutive relationships describing the unloadingand reloading scenarios were developed [24 25] Althoughthese FE-based models claim to accurately simulate thefracture behaviour of concrete structures they are usu-ally computationally expensive This drawback makes themunsuitable for damage detection applicationswhere adaptableand computationally efficient models are needed [9]

This paper presents an efficient approach of modeling themonotonic and cyclic flexural cracking behaviour of concreteusing a FCM Under the cyclic load the constitutive relationof concrete fracture is very complex the same displacementmay correspond to three different kinds of stress states andnumerical conditions of the transformation between eachstress state are not easy to define In numerical analysisit will lead to instability problems during the overlappingprocess of positive and negative stiffness In the analysis ofconcrete crack propagation under cyclic load the states ofload (loading unloading and reloading) at joint interfaceshould be defined first because different states correspond todifferent displacement-stress curves And the softening loadthat fictitious crack plane transfer is a function of openingdisplacement on this plane based on fictitious crack modelbut the opening displacement is an unknown quantity still tobe determined whichmeans that the propagation calculationof concrete cracks under cyclic load is a nonlinear iterativeprocess This situation is very similar to contact problems inpractical engineering in which the state of load process con-tact forces and displacements on interfaces cannot be knownin advance The solution is to make the equation fulfill theequilibrium condition in general and satisfy the constitutiverelation between opening displacement and interface tensionon interface from the determined states of load and crack

The paper is structured as follows In the next sectionthe basic idea of the mixed freedom finite element methodfor contact problem is given Section 3 describes the mainfeatures of the proposed method including the improvedconstitutive relationships for the crack propagation simula-tion under cyclic load The numerical implementation and

X

Y

Z

F

w

f1

f2

Ω1

Ω2

1205851

2

120578

120577

Figure 1 Mechanical model

solution algorithms are given in Section 4 Section 5 reportsthe validation of themodel and discusses the findings Finallyconcluding remarks are summarized in Section 6

2 Basic Idea of the Mixed FreedomFinite Element Method

Pairs of contact points were set along the potential developingpath of the crack The mixed freedom FEM proposed forcontact problem was extended to simulate the fracturemechanics of concrete using the FCM The system of forcesacting on the structure was divided into two parts externalforces and contact forces The displacement of structure waschosen as the basic variable and the nodal contact force inpotential contact region under local coordinate system waschosen as the iteration variable so that the nonlinear iterationprocess was only limited in the contact surface and muchmore efficientThe contact forces and the opening of the crackcan be obtained explicitly enabling the softening constitutiverelation for the concrete to be introduced conveniently by thefictitious crack model

In Figure 1 the body is divided into two parts (Ω1 Ω2)

by the potential developing path of crack which was set inadvance Now we focus our attention on an arbitrary nodepair constituted by points 1 and 2119906119868 is the displacement of thenode pair and 119891119868 is the contact force Here the superscript119868 = 1 2 denotes the two body parts 120585120578120577 is the local coordinatesystem on the interface with 120585 being the normal directionand 120578 120577 the tangent directions Then the crack openingdisplacement (COD) in normal direction between point 1and 2 can be computed as 119908 = (119906

1 minus 1199062) sdot 120585Assuming that the analysis of 119899th step has been finished

the incremental static equilibrium equation for (119899 + 1)thstep is stated as follows by performing a finite elementdiscretization of each body Ω

1andΩ

2

119870Δ119906119899= (119865119899+1

+ 119891119899minus int119861

119879120590119899119889Ω) + Δ119891

119899 (1)

Mathematical Problems in Engineering 3

where 119870 is the global stiffness matrix Δ119906119899is the vector of

displacement increment at step 119899 119865119899+1

is the vector of totalexternal load at step 119899 + 1 119891

119899is the vector of total contact

force at step 119899 119861 is the strain-displacement matrix 120590119899is the

Cauchy stress tensor Δ119891119899is the vector of incremental contact

force Clearly the first term of the right-hand side (RHS) ofthe above equation that is the content in the bracket standsfor the external load increment of current load step since thecontribution of 119891

119899to the system is already included into 120590

119899

If a typical decomposition of global stiffnessmatrix119870wasperformed at the beginning of the analysis for one and onlyone time (1) could be rewritten as

Δ119906119899= 119870minus1(119865119899+1

+ 119891119899minus int119861

119879120590119899119889Ω) + 119870

minus1Δ119891119899 (2)

If Δ119906119899is defined as

Δ119906119899= 119870minus1(119865119899+1

+ 119891119899minus int119861

119879120590119899119889Ω) (3)

and the matrix 119862 is introduced into the above equation (3) isrewritten as

Δ119906119899= Δ119906119899+ 119862Δ119891

119899 (4)

where the matrix 119862 is the flexibility matrix which is definedon the possible contact boundary Γ119868

119888 An arbitrary component

of119862 119888119894119895 represents the flexibility coefficient corresponding to

the displacement at the freedom 119894 due to a unit force at thefreedom 119895 Here 119894 or 119895 is only limited in the freedom of theregion where contact is likely to take place

It is important to be pointed out that (3) and (4) and thefollowing equations in this section are only performed for thedegree of freedoms (DOFs) on the potential contact surfacenot for all the DOFs of the whole system

Applying (4) to points 1 and 2 of a given node pairrespectively gives

Δ1199061

119899= Δ1199061

119899+ 1198621Δ1198911

119899 (5)

Δ1199062

119899= Δ1199062

119899+ 1198622Δ1198912

119899 (6)

According to Newtonrsquos third law it is obvious that Δ1198911119899=

minusΔ1198912119899= Δ119891119899 and moreover incorporating flexibility matrix

1198621 and 1198622 into 119862 = 1198621 + 1198622 so that if we subtract (5) from(6) we obtain

119862Δ119891119899= (Δ119906

1

119899minus Δ1199062

119899) minus (Δ119906

1

119899minus Δ1199062

119899) (7)

Equation (7) is the finite element compliance equationof the mixed finite element method for the static contactproblemswith friction and initial gaps proposed in this paperIn this equation the second term in RHS (Δ1199061

119899minusΔ1199062

119899) stands

for the difference of incremental displacement only due tothe external load increment as can be seen easily from (3)therefore it has nothing to do with the current contact stateand can be obtained by back substitution directly Howeverfor the first term in RHS (Δ1199061

119899minusΔ1199062119899) it denotes the difference

of incremental displacement induced by both the external

Crack Fictitious crack

w

120590 = f(w) 120590 = f(120576)

120590 = ft

Figure 2 Fictitious crack model

load increment and the contact force increment From thispoint of view we can see that both the RHS and the left-hand side (LHS) of the above equation are associated withthe contact state An iterative method taking into accountdifferent contact states is necessary to solve (7) and this willbe given in detail in the next section

3 FCM for the Crack Propagation underCyclic Load

The strain softening behaviour can be applied to establish theso-called FCM according to Hillerborg [26] to simulate crackformation In Figure 2 a zone with more microcracks calledfracture process zone will appear before crack propagationbecause of the inhomogeneity of concrete The fictitiouscracks in this zone can transfer tensile stress and the stressvalue can be obtained from the 120590-119908 softening curve given itscrack widthThe nonlinear behavior of the propagating crackis described by a fictitious crack with cohesive forces actingbetween its interfaces [27] At the crack tip the maximumstress is equal to the tensile strength 119891

119905 and for all other

points along the crack path the transmitted stress 120590 dependson the crack opening119908 and is defined by the strain softeningdiagram The area under the 120590-119908 curve is equal to thespecific fracture energy 119866

119865 The fracture energy is the ratio

of the energy necessary for total fracture of a specimen tothe projection of the fracture surface area 119866

119865is related to

material characteristic constants such as concrete mixtureratio strength aggregate type particle diameters and cementgrade and will be determined by experiment According tothe widely adopted CEB-FIP Model Code 1990 119866

119865can be

determined by

119866119865= (

00204 + 00053119889095max8

)(119891119888

1198911198880

)

07

(8)

where1198911198880= 10MPa and 119889max119898119898 is themaximumaggregate

size The value of 1199080is of certain relation with the aggregate

size for example in CEB-FIP Model Code 1990 it is recom-mended that 119908

0= 120572119865119866119865119891119905 120572119865is the parameter relative to

the maximum aggregate sizeThere are different kinds of loadingunloading model

proposed by many researchers for crack propagation undercyclic load such as the simple elastic loading and unloadingmodel Hordijk-Reinhardt model [28 29] and the Toumimodel [24] see Figures 3(a)sim3(c)


120590

wD (w0)A

B

C

(a)

120590

w

U

R

L

A

B

C

ΔwUR

(b)

120590

ww0A

B

C

D

E

w(C)2

D998400

(c)Figure 3 Loadingunloadingmodel for crack propagation under cyclic load (a) Elastic loading and unloadingmodel (b) Hordijk-Reinhardtmodel and (c) Toumi model

120590

w

2

2

3

4

5 6

1

G (w0)A

B

C

D

E

F

Figure 4 The improved loadingunloading for crack propagationunder cyclic load

Based on the Toumimodel [24] an improved constitutivemodel shown in Figure 4 is proposed for the crack propa-gation under cyclic load The main improvement lies in thatthe loading and reloading curve equation of Toumi model isadjusted to ensure the continuity of displacement and stressduring the crack propagation which is convenient for FEMimplementation The equations used in this improved curveare shown as follows

(1) The softening curve of concrete is based on Cornelis-senrsquos (softening) function which can be expressed as [30]

120590 = 119891 (119908) = ([1 + (1198881119908

1199080

)

3

] 119890minus11988821199081199080

minus119908

1199080

(1 + 1198883

1) 119890minus1198882)119891119905

(9)

where coefficients 1198881 1198882takes the recommended values 10

and 564 respectively The ultimate crack developing width1199080was determined from the assumption that the area under

the softening curve is equal to the specific fracture energy119866119865

and can be approximated by 1199080= 5618119866

119891119891119905

(2)The equation of the unloading curve at point119862 shownin Figure 4 is

120590 = 119891119906(119908) =

119908

119908 (119888)[120590 (119888) minus 120572 (120590 (119888) minus 119891

119905)]

+ 120572 (120590 (119888) minus 119891119905)

(10)

where 120590(119888) 119908(119888) represent the stress at point 119862 and dis-placement of interface opening during unloading and 120572 is amaterial characteristic parameter describing the relationshipbetween the compressive stress required to reclose the crackwith unloading stress and the tensile strength in fractureprocess zone According to the experimental results ofReinhardt et al [31] a recommended value of120572 = 035 is usedin this paper In cracked areas when unloading if 119908 gt 119908

0

then 119891119906(119908) = 0 Otherwise if 0 lt 119908 lt 1199080 we have 120590(119888) = 0

and the stress 120590 can be calculated using (10)(3) The equation of reloading curve at point 119863 in the

proposed model is

120590 = 119891119897(119908) = (

119908 minus 119908 (119863)

119908 (119864) minus 119908 (119863))

120574

(120590 (119864) minus 120590 (119863)) + 120590 (119863)

(11)

where 120574 is the shape parameter of reloading curve which canbe approximated by 120574 = 1(1 + 12119908(119864)119908

0) according to the

experimental data of [31]The stress value at point 119864 is governed by the reduction

factor together with the stress value at point 119862 and can bedetermined using the following equation

120590 (119864) = (1 minus 120573) 120590 (119862) (12)

The value of 120573 takes as a constant of 005 according toHorrirsquos suggestion [29]

4 Numerical Implementation

In the propagation of the crack in concrete there arethree loading process states that is loading unloading and


reloading For model I crack there are three interface gapstates that is closure opening and virtual opening In orderto describe the whole loadingunloading process of model Icrack in a physical model six contact states are employedIn Figure 4 the corresponding positions of the six contactstates in the constitutive relation of concrete are shownrespectively A the initial closed state in noncrack zone Bfictitious crack extension state in fracture process zone whileloadingC fictitious crack unloading state in fracture processzone and macro crack zone while unloadingD closed crackstate in fracture process zone and macrocrack zone whilebeing fully reclosed E fictitious crack reloading state infracture process zone while reloading F the opening statein macro crack zone For the crack of models II and III wecan add an extra state which is the sliding state of the fractureprocess zone

When using mixed freedom FEM to simulate fractureprocess the initial condition was taken from the convergedresult (contact state and contact force between crack surfaces)of former FEM step Using the overall displacement fieldobtained from (1) the joint opening displacement can bedetermined and employed to solve the new contact forceon joint interface iteratively Assuming that the 119894th contactiteration has been finished the displacement and stressexpression for (119894 + 1)th iteration is stated as follows

119908119894+1

119899+1= 119908119894

119899+1+ (Δ119906

1119894

119899minus Δ1199062119894

119899) sdot 120585

(120590119894+1

119899+1)120585= 120590119894+1

119899+1sdot 120585

(13)

Here the superscript denotes the contact iteration stepwhile the subscript stands for the loading increment stepThe displacement and stress from the above equation areprediction results and must be checked against the contactstate after (119894 + 1)th iteration Based on the calculated resultfrom the former loading step and 119894th contact iteration aswell as the joint opening displacement and contact stressafter (119894 + 1)th iteration the contact states were processed andupdated which will be explained in detail as follows

A If a contact state is at the initial closed state where theinitial condition is 119908 = 0 radic1198912

120578+ 1198912120577lt minus120583119891

120585+ 119860119888 119891

1= minus1198912

119891119903119905=119891119905 After (119894+1)th iteration if the normal stress of the crack

surfaces satisfied (120590119894+1119899+1)120585gt 119891119905 the contact state is updated

to fictitious crack extension state and 119891119894+1119899+1

= 119891119905119860 is set for

the next iteration otherwise if radic1198912120578+ 1198912120577ge minus120583119891

120585+ 119860119888 was

satisfied the current state is changed to sliding state and anew condition of radic1198912

120578+ 1198912120577= minus120583119891

120585+ 119860119888 is set accordingly

119891120585is the normal contact force component 119891

120578 119891120577are two

tangential components of the contact forces 120583 is the frictioncoefficient119860 is the controlled area of pair nodes 119888 is cohesivestrength 119891

119905is the tensile strength of the crack surface 119891119903

119905is

the residual tensile strength of the crack surfaceB If the current state is at the fictitious crack extension

state where the condition is 0 lt 119908 lt 1199080 the normal stress

must satisfy (9) If the opening width reaches the conditionof 119908119894+1119899+1

le 119908119899 the current state is adjusted to fictitious crack

unloading state and contact force is set as 119891119894+1119899+1

= (120590119899)120585119860

stress at the unloading point is set as 120590(119862) = (120590119899)120585 and

the residual tensile strength of the contact pair is changed to119891119903119905= (1 minus 120573)120590(119862) If (119908119894+1

119899+1ge 1199080) is met the current state

is changed to opening state and new values are set to make119891119894+1119899+1

= 0 119891119903119905= 0 (120590

119899)120585is the normal stress of crack at 119899th

loading stepC If the current state is at the fictitious crack unloading

state where the condition is 0 lt 119908 le 119908(119862) the normal stressmust satisfy (10) If the opening width of crack after iterationmeet 119908119894+1

119899+1ge 119908119899 the current state is changed to fictitious

crack reloading state and contact force is set as119891119894+1119899+1

= (120590119899)120585119860

and stress at the reloading point is set as 120590(119863) = (120590119899)120585

120590(119864) = 119891119903119905 If (119908119894+1

119899+1lt 0) is satisfied the current state is

changed to crack closed state and the width is set to 119908 = 0

to avoid crack surface embeddingD If current state is at the crack closed state where the

condition is 119908 = 0 119891119903119905lt 119891119905 and 120590

119894+1

119899+1lt 119891119906(0) If the

normal stress after iteration satisfies 120590119894+1119899+1

gt 119891119906(0) the contactstate is changed to fictitious crack reloading state and is set as119891119894+1119899+1

= 119891119906(0)119860 where 119891119906(0) is the stress when the crack isfully closed

E If the current state is at the fictitious crack reloadingstate where the condition is 0 lt 119908 le 119908(119864) the normal stressis satisfied in (11)120590 = 119891119897(119908) If the normal stress after iterationsatisfies 120590119894+1

119899+1= 119891119903119905 and the opening width of crack surface

after iteration meets the condition 119908119894+1119899+1

gt 119908(119864) then thecurrent state is changed to the fictitious crack extension state

F If the current state is at the opening state where thecondition is 119908 ge 119908

0 1198911= 1198912= 0 If the opening degree

of joints after iteration meets 119908 lt 1199080 the current state is

changed to the fictitious crack unloading state and the stressat loading point is set 120590(119862) = 0

G If the current state is at the sliding state where thecondition isradic1198912

120578+ 1198912120577= minus120583119891

120585+119860119888 Ifradic1198912

120578+ 1198912120577lt minus120583119891

120585+119860119888

is met after iteration the current state is changed to the initialclosed state

Having updated the new contact state of one loadingincrement step the RHS of (7) is obtained and can be usedto get the overall displacement field from (1)Then the widthof the crack surface can be updated and the new contact forcevector can be determined from the stress-displacement curveThe updated contact force and state are taken into the nextiteration and the above process is repeated until both the stateand force have converged which will lead to the start of thenext loading step

5 Numerical Experiments

The simulation of crack propagation of Toumirsquos [24] beamproblem under cyclic load was carried out to verify theproposed scheme The numerical setup is present in Figure 5along with themesh configurations Similarly an initial crackis set in BC section and AB is the pair of contact points Themesh size near the crack is 1mm and other parameters are

length = 320mmdepth = 80mm


A

B

C

Figure 5 Finite element model for Toumirsquos beam

1050

900

750

600

450

300

150

002502015010050

Load

(N)

Deflection (mm)

Numerical results

Upper envelop of experimental resultsLower envelop of experimental results

Figure 6 Experimental and FEM results for monotonous load

thickness = 50mminitial crack length = 40mmtensile strength = 52MPafracture energy = 342Nmelastic modulus = 316GPapoissonrsquos ratio = 02

The load-displacement curve under monotonous loadwas shown in Figure 6 and compared against the exper-imental data [24] The simulated result lies between thetwo experimental curves and followed the trend closelyIn Figure 7 the load-displacement curve under cyclic loadconditionwas plotted andmatcheswell the experimental datafor each loading cycle

Figure 8 shows the distributions of normal stress alongthe height of crack under three different loading conditionswhen the vertical displacement is 004mm and 007mmrespectively No abrupt stress change has been observed asthe stress curve is in a continuous and smooth way andthe distribution of normal tensile stress of crack fits nicelywith the concrete softening curve adopted When unloadingthe crack surface below the tip of fracture process zonetransited gradually from tensile state to compressive state andthe compressive stress increases gradually towards the endHowever if the crack zone is long (119863119910 = 007mm) andthe opening width of the bottom crack surface is too large

900

750

600

450

300

150

0

02502015010050

Deflection (mm)

Numerical results

minus150

Load

(N)

Experimental results

Figure 7 Experimental and FEM results for cyclic load

the compressive stress tends to decreaseWhen reloading thecrack surface below the tip is mainly in the tensile state Bothfindings match well general understanding and demonstrategreat accuracy and capability of the current scheme

For simulating the crack propagation the size ofmesh is afactor of great importance To fully understand the influenceof themesh size on calculated results and investigate themeshdependence of the suggested method the load-displacementcurves near the crack were plotted using different meshingsizes varying from 1mm to 8mm (about 140sim15 of the crackheight) in Figure 9 It can be seen that for a meshing size lessthan 110 of the crack height the result of different meshesshows little change while if themeshing size and crack heightratio is bigger than 15 the load-displacement curve is notsmooth anymore In this case the trend of the curve ingeneral agreed well with the result of fine mesh despite thebig error in the far end

6 Conclusion

(1) The mixed freedom finite element method proposedfor contact problems is extended to investigate thecrack propagation under cyclic load using the con-stitutive relation of model I crack According to thestates of the load and the crack the constitutiverelation is characterized by six contact states andsolved iteratively The current scheme is verifiedagainst a number of experimental cases and provedto be accurate and reliable

(2) Using the mixed freedom finite element method tosimulate the crack propagation the softening con-stitutive relation of concrete is implemented directlyinto the contact iteration It will not lead to numericalinstability during the overlapping process of positiveand negative stiffness which is convenient to adoptthe softening constitutive relation of random shapeMoreover the dependence of the result on the size ofmesh is not obvious


A

B

(a)

A

B

(b)

A

B

(c)

A

B

(d)

A

B

(e)

A

B

(f)Figure 8 Normal stress distribution along the crack (a) Loading 119863119910 = 004mm (b) unloading 119863119910 = 004mm (c) reloading 119863119910 =

004mm (d) loading119863119910 = 007mm (e) unloading119863119910 = 007mm and (f) reloading119863119910 = 007mm

900

750

600

450

300

150

002502015010050

Deflection (mm)

Load

(N)

1mm2mm

4mm8mm

(a)

900

750

600

450

300

150

0

02502015010050

Deflection (mm)

minus150

Load

(N)

1mm2mm

4mm8mm

(b)

Figure 9 Load displacement curve for different mesh sizes (a) Monotonous load and (b) cyclic load

(3) As there are limited resources available about thesoftening relation ofmixed type crack this paper onlyshowed simulations on the propagation of model Itype crack However the authors believe the proposedmethod is applicable for other type crack as well Dueto the fact that mixed freedom finite element methodneeds to preset pairs of contact points in the potentialcontact area the application of the current methodis confined to problems which the paths of the crackpropagation are already known Future work involvescontinuing the current work on different types ofcracks and a more detailed analysis of the calculationparameters of the constitutive relations is presented inthis paper

Conflict of Interests

The authors do not have any conflict of interests with thecontent of the paper

Acknowledgments

The present work is supported by the National NaturalScience Foundation of China (no 51279050) the NationalHigh-tech RampD Program of China (863 Program) (no2012BAK10B04) the National Key Technology RampD Pro-gram in 12th Five-Year Plan (no SS2012AA112507) andthe Non-Profit Industry Financial Program of MWR (Min-istry of Water Resources of China) (no 201301058)


References

[1] B K Raghu Prasad and M V Renuka Devi ldquoExtension ofFCM to plain concrete beams with vertical tortuous cracksrdquoEngineering Fracture Mechanics vol 74 no 17 pp 2758ndash27692007

[2] J C Galvez J Cervenka D A Cendon and V Saouma ldquoA dis-crete crack approach to normalshear cracking of concreterdquoCement and Concrete Research vol 32 no 10 pp 1567ndash15852002

[3] T M Abu-Lebdeh and G Z Voyiadjis ldquoPlasticity-damagemodel for concrete under cyclic multiaxial loadingrdquo Journal ofEngineering Mechanics vol 119 no 7 pp 1465ndash1485 1993

[4] J-Y Kim H-G Park and S-T Yi ldquoPlasticity model fordirectional nonlocality by tension cracks in concrete planarmembersrdquo Engineering Structures vol 33 no 3 pp 1001ndash10122011

[5] J Ozbolt and Z P Bazant ldquoMicroplane model for cyclic triaxialbehavior of concreterdquo Journal of EngineeringMechanics vol 118no 7 pp 1365ndash1386 1992

[6] Z P Bazant and G Di Luzio ldquoNonlocal microplane model withstrain-softening yield limitsrdquo International Journal of Solids andStructures vol 41 no 24-25 pp 7209ndash7240 2004

[7] M Elices and J Planas ldquoMaterialmodelsrdquo inFractureMechanicsof Concrete Structures L Elfgren Ed pp 16ndash66 Chapman andHall London UK 1989

[8] Z P Bazant and J Planas Fracture and Size Effect in Concreteand Other Quasibrittle Materials CRC Press New York NYUSA 1998

[9] W I Hamad J S Owen and M F M Hussein ldquoAn efficientapproach of modelling the flexural cracking behaviour of un-notched plain concrete prisms subject to monotonic and cyclicloadingrdquo Engineering Structures vol 51 pp 36ndash50 2013

[10] I Carol P C Prat and C M Lopez ldquoNormalshear crackingmodel application to discrete crack analysisrdquo Journal of Engi-neering Mechanics vol 123 no 8 pp 765ndash773 1997

[11] A Hillerborg M Modeer and P-E Petersson ldquoAnalysis ofcrack formation and crack growth in concrete by means offracture mechanics and finite elementsrdquo Cement and ConcreteResearch vol 6 no 6 pp 773ndash781 1976

[12] Z P Bazant and B H Oh ldquoCrack band theory for fracture ofconcreterdquoMaterials and Structures vol 16 pp 155ndash177 1983

[13] Y S Jenq and S P Shah ldquoA Fracture toughness criterion forconcreterdquo Engineering Fracture Mechanics vol 21 no 5 pp1055ndash1069 1985

[14] Y S Jenq and S P Shah ldquoTwo parameter fracture model forconcreterdquo Journal of Engineering Mechanics vol 111 no 10 pp1227ndash1241 1985

[15] Z P Bazant ldquoSize effect in blunt fracture concrete rock metalrdquoJournal of Engineering Mechanics vol 110 no 4 pp 518ndash5351984

[16] P Nallathambi and B L Karihaloo ldquoDetermination of speci-men-size independent fracture toughness of plain concreterdquoMagazine of Concrete Research vol 38 no 135 pp 67ndash76 1986

[17] S E Swartz and T M E Refai ldquoInfluence of size on openingmode fracture parameters for precracked concrete beams inbendingrdquo in Proceedings of the SEM-RILEM International Con-ference on Fracture of Concrete and Rock pp 242ndash254 HoustonTex USA June 1987

[18] S Xu and H W Reinhardt ldquoDetermination of double-Kcriterion for crack propagation in quasi-brittle fracture Part I

experimental investigation of crack propagationrdquo InternationalJournal of Fracture vol 98 no 2 pp 111ndash149 1999

[19] S Xu and H W Reinhardt ldquoDetermination of double-Kcriterion for crack propagation in quasi-brittle fracture Part Iexperimental investigation of crack propagationrdquo InternationalJournal of Fracture vol 98 no 2 pp 151ndash177 1999

[20] X Zhang and S Xu ldquoA comparative study on five approachesto evaluate double-K fracture toughness parameters of concreteand size effect analysisrdquo Engineering FractureMechanics vol 78no 10 pp 2115ndash2138 2011

[21] Z Shi ldquoNumerical analysis of mixed-mode fracture in concreteusing extended fictitious crack modelrdquo Journal of StructuralEngineering vol 130 no 11 pp 1738ndash1747 2004

[22] D S Yankelevsky and H W Reinhardt ldquoUniaxial behavior ofconcrete in cyclic tensionrdquo Journal of Structural Engineering vol115 no 1 pp 166ndash182 1989

[23] D A Hordijk ldquoTensile and tensile fatigue behaviour of con-crete experiments modelling and analysesrdquo Heron vol 37 no1 pp 3ndash77 1992

[24] A Toumi and A Bascoul ldquoMode I crack propagation in con-crete under fatigue microscopic observations and modellingrdquoInternational Journal for Numerical and Analytical Methods inGeomechanics vol 26 no 13 pp 1299ndash1312 2002

[25] J F Sima P Roca and C Molins ldquoCyclic constitutive modelfor concreterdquo Engineering Structures vol 30 no 3 pp 695ndash7062008

[26] A Hillerborg ldquoAnalysis of one single crackrdquo in Fracture Mecha-nics of Concrete F H Wittmann Ed pp 223ndash249 ElsevierAmsterdam The Netherlands 1983

[27] K Andreev and H Harmuth ldquoFEM simulation of the thermo-mechanical behaviour and failure of refractories a case studyrdquoJournal of Materials Processing Technology vol 143-144 no 1pp 72ndash77 2003

[28] D A Hordijk and H W Reinhardt ldquoGrowth of discrete cracksin concrete under fatigue Loadingrdquo in Toughening Mechanismin Qusi-Brittle Materials S P Shah Ed pp 541ndash554 KluwerAcademic Publishers Dordrecht The Netherlands 1991

[29] H Horii H C Shin and T M Pallewatta ldquoAn analytical modelof Fatigue crack growth in concreterdquo Japan Concrete Institutevol 12 pp 835ndash840 1990

[30] H A W Cornelissen D A Hordijk and H W ReinhardtldquoExperimental determination of crack softening characteristicsof normal weight and lightweight concreterdquo Heron vol 31 no2 pp 45ndash56 1985

[31] H W Reinhardt H A W Cornelissen and D A HordijkldquoTensile tests and failure analysis of concreterdquo Journal ofStructural Engineering vol 112 no 11 pp 2462ndash2477 1986

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


concrete including the fictitious crack model (FCM) [11] thecrack band model (CBM) [12] the two-parameter fracturemodel (TPFM) [13 14] the size effect model (SZM) [15]the effective crack model (ECM) [16 17] and the double-Kfracture model (DKFM) [18ndash20]

The discrete approach is preferred when there is onecrack or a finite number of cracks in the structure [2]Among them the cohesive zone model (CZM) whichassumes that stresses act across a narrowly open crack wasdeveloped by Hillerborg et al [11] under the name FCM Inthe literature it is generally accepted that the FCM is the bestsimple fracture mechanics model as it provides reasonableapproximation of the crack propagation in concrete usingonly simple calculations The model is mainly applicable tomode-I fracture but recently it has been extended to modelshear failure mode-II and mode-III cracks [21] The FCMhas been used to investigate the fracture process in concreteunder monotonic and cyclic loading Various constitutiverelationships for cyclic tension behaviour of concrete suchas the focal point model [22] and the continuous functionmodel [23] have been developed More recently linear andbilinear constitutive relationships describing the unloadingand reloading scenarios were developed [24 25] Althoughthese FE-based models claim to accurately simulate thefracture behaviour of concrete structures they are usu-ally computationally expensive This drawback makes themunsuitable for damage detection applicationswhere adaptableand computationally efficient models are needed [9]

This paper presents an efficient approach of modeling themonotonic and cyclic flexural cracking behaviour of concreteusing a FCM Under the cyclic load the constitutive relationof concrete fracture is very complex the same displacementmay correspond to three different kinds of stress states andnumerical conditions of the transformation between eachstress state are not easy to define In numerical analysisit will lead to instability problems during the overlappingprocess of positive and negative stiffness In the analysis ofconcrete crack propagation under cyclic load the states ofload (loading unloading and reloading) at joint interfaceshould be defined first because different states correspond todifferent displacement-stress curves And the softening loadthat fictitious crack plane transfer is a function of openingdisplacement on this plane based on fictitious crack modelbut the opening displacement is an unknown quantity still tobe determined whichmeans that the propagation calculationof concrete cracks under cyclic load is a nonlinear iterativeprocess This situation is very similar to contact problems inpractical engineering in which the state of load process con-tact forces and displacements on interfaces cannot be knownin advance The solution is to make the equation fulfill theequilibrium condition in general and satisfy the constitutiverelation between opening displacement and interface tensionon interface from the determined states of load and crack

The paper is structured as follows In the next sectionthe basic idea of the mixed freedom finite element methodfor contact problem is given Section 3 describes the mainfeatures of the proposed method including the improvedconstitutive relationships for the crack propagation simula-tion under cyclic load The numerical implementation and

X

Y

Z

F

w

f1

f2

Ω1

Ω2

1205851

2

120578

120577

Figure 1 Mechanical model

solution algorithms are given in Section 4 Section 5 reportsthe validation of themodel and discusses the findings Finallyconcluding remarks are summarized in Section 6

2 Basic Idea of the Mixed FreedomFinite Element Method

Pairs of contact points were set along the potential developingpath of the crack The mixed freedom FEM proposed forcontact problem was extended to simulate the fracturemechanics of concrete using the FCM The system of forcesacting on the structure was divided into two parts externalforces and contact forces The displacement of structure waschosen as the basic variable and the nodal contact force inpotential contact region under local coordinate system waschosen as the iteration variable so that the nonlinear iterationprocess was only limited in the contact surface and muchmore efficientThe contact forces and the opening of the crackcan be obtained explicitly enabling the softening constitutiverelation for the concrete to be introduced conveniently by thefictitious crack model

In Figure 1 the body is divided into two parts (Ω1 Ω2)

by the potential developing path of crack which was set inadvance Now we focus our attention on an arbitrary nodepair constituted by points 1 and 2119906119868 is the displacement of thenode pair and 119891119868 is the contact force Here the superscript119868 = 1 2 denotes the two body parts 120585120578120577 is the local coordinatesystem on the interface with 120585 being the normal directionand 120578 120577 the tangent directions Then the crack openingdisplacement (COD) in normal direction between point 1and 2 can be computed as 119908 = (119906

1 minus 1199062) sdot 120585Assuming that the analysis of 119899th step has been finished

the incremental static equilibrium equation for (119899 + 1)thstep is stated as follows by performing a finite elementdiscretization of each body Ω

1andΩ

2

119870Δ119906119899= (119865119899+1

+ 119891119899minus int119861

119879120590119899119889Ω) + Δ119891

119899 (1)










119899


Δ119906119899= 119870minus1(119865119899+1

+ 119891119899minus int119861

119879120590119899119889Ω) + 119870

minus1Δ119891119899 (2)


Δ119906119899= 119870minus1(119865119899+1

+ 119891119899minus int119861

119879120590119899119889Ω) (3)


Δ119906119899= Δ119906119899+ 119862Δ119891

119899 (4)







Δ1199061

119899= Δ1199061

119899+ 1198621Δ1198911

119899 (5)

Δ1199062

119899= Δ1199062

119899+ 1198622Δ1198912

119899 (6)




119862Δ119891119899= (Δ119906

1

119899minus Δ1199062

119899) minus (Δ119906

1

119899minus Δ1199062

119899) (7)


119899minusΔ1199062

119899) stands





w

120590 = f(w) 120590 = f(120576)

120590 = ft









119865is related to


119865can be

determined by

119866119865= (

00204 + 00053119889095max8

)(119891119888

1198911198880

)

07

(8)








120590

wD (w0)A

B

C

(a)

120590

w

U

R

L

A

B

C

ΔwUR

(b)

120590

ww0A

B

C

D

E

w(C)2

D998400


120590

w

2

2

3

4

5 6

1

G (w0)A

B

C

D

E

F




120590 = 119891 (119908) = ([1 + (1198881119908

1199080

)

3

] 119890minus11988821199081199080

minus119908

1199080

(1 + 1198883

1) 119890minus1198882)119891119905

(9)





119891119891119905


120590 = 119891119906(119908) =

119908

119908 (119888)[120590 (119888) minus 120572 (120590 (119888) minus 119891

119905)]

+ 120572 (120590 (119888) minus 119891119905)

(10)


0



proposed model is

120590 = 119891119897(119908) = (

119908 minus 119908 (119863)

119908 (119864) minus 119908 (119863))

120574

(120590 (119864) minus 120590 (119863)) + 120590 (119863)

(11)


0) according to the



120590 (119864) = (1 minus 120573) 120590 (119862) (12)







119908119894+1

119899+1= 119908119894

119899+1+ (Δ119906

1119894

119899minus Δ1199062119894

119899) sdot 120585

(120590119894+1

119899+1)120585= 120590119894+1

119899+1sdot 120585

(13)



120578+ 1198912120577lt minus120583119891

120585+ 119860119888 119891

1= minus1198912




= 119891119905119860 is set for


120585+ 119860119888 was


120578+ 1198912120577= minus120583119891



120578 119891120577are two



119905is






= (120590119899)120585119860





= 0 119891119903119905= 0 (120590






= (120590119899)120585119860


120590(119864) = 119891119903119905 If (119908119894+1





119894+1

119899+1lt 119891119906(0) If the













120578+ 1198912120577= minus120583119891

120585+119860119888 Ifradic1198912

120578+ 1198912120577lt minus120583119891

120585+119860119888







A

B

C


1050

900

750

600

450

300

150

002502015010050

Load

(N)

Deflection (mm)

Numerical results






900

750

600

450

300

150

0

02502015010050

Deflection (mm)

Numerical results

minus150

Load

(N)





6 Conclusion




A

B

(a)

A

B

(b)

A

B

(c)

A

B

(d)

A

B

(e)

A

B



900

750

600

450

300

150

002502015010050

Deflection (mm)

Load

(N)

1mm2mm

4mm8mm

(a)

900

750

600

450

300

150

0

02502015010050

Deflection (mm)

minus150

Load

(N)

1mm2mm

4mm8mm

(b)





Acknowledgments



References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra


















119899


Δ119906119899= 119870minus1(119865119899+1

+ 119891119899minus int119861

119879120590119899119889Ω) + 119870

minus1Δ119891119899 (2)


Δ119906119899= 119870minus1(119865119899+1

+ 119891119899minus int119861

119879120590119899119889Ω) (3)


Δ119906119899= Δ119906119899+ 119862Δ119891

119899 (4)







Δ1199061

119899= Δ1199061

119899+ 1198621Δ1198911

119899 (5)

Δ1199062

119899= Δ1199062

119899+ 1198622Δ1198912

119899 (6)




119862Δ119891119899= (Δ119906

1

119899minus Δ1199062

119899) minus (Δ119906

1

119899minus Δ1199062

119899) (7)


119899minusΔ1199062

119899) stands





w

120590 = f(w) 120590 = f(120576)

120590 = ft









119865is related to


119865can be

determined by

119866119865= (

00204 + 00053119889095max8

)(119891119888

1198911198880

)

07

(8)








120590

wD (w0)A

B

C

(a)

120590

w

U

R

L

A

B

C

ΔwUR

(b)

120590

ww0A

B

C

D

E

w(C)2

D998400


120590

w

2

2

3

4

5 6

1

G (w0)A

B

C

D

E

F




120590 = 119891 (119908) = ([1 + (1198881119908

1199080

)

3

] 119890minus11988821199081199080

minus119908

1199080

(1 + 1198883

1) 119890minus1198882)119891119905

(9)





119891119891119905


120590 = 119891119906(119908) =

119908

119908 (119888)[120590 (119888) minus 120572 (120590 (119888) minus 119891

119905)]

+ 120572 (120590 (119888) minus 119891119905)

(10)


0



proposed model is

120590 = 119891119897(119908) = (

119908 minus 119908 (119863)

119908 (119864) minus 119908 (119863))

120574

(120590 (119864) minus 120590 (119863)) + 120590 (119863)

(11)


0) according to the



120590 (119864) = (1 minus 120573) 120590 (119862) (12)







119908119894+1

119899+1= 119908119894

119899+1+ (Δ119906

1119894

119899minus Δ1199062119894

119899) sdot 120585

(120590119894+1

119899+1)120585= 120590119894+1

119899+1sdot 120585

(13)



120578+ 1198912120577lt minus120583119891

120585+ 119860119888 119891

1= minus1198912




= 119891119905119860 is set for


120585+ 119860119888 was


120578+ 1198912120577= minus120583119891



120578 119891120577are two



119905is






= (120590119899)120585119860





= 0 119891119903119905= 0 (120590






= (120590119899)120585119860


120590(119864) = 119891119903119905 If (119908119894+1





119894+1

119899+1lt 119891119906(0) If the













120578+ 1198912120577= minus120583119891

120585+119860119888 Ifradic1198912

120578+ 1198912120577lt minus120583119891

120585+119860119888







A

B

C


1050

900

750

600

450

300

150

002502015010050

Load

(N)

Deflection (mm)

Numerical results






900

750

600

450

300

150

0

02502015010050

Deflection (mm)

Numerical results

minus150

Load

(N)





6 Conclusion




A

B

(a)

A

B

(b)

A

B

(c)

A

B

(d)

A

B

(e)

A

B



900

750

600

450

300

150

002502015010050

Deflection (mm)

Load

(N)

1mm2mm

4mm8mm

(a)

900

750

600

450

300

150

0

02502015010050

Deflection (mm)

minus150

Load

(N)

1mm2mm

4mm8mm

(b)





Acknowledgments



References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










120590

wD (w0)A

B

C

(a)

120590

w

U

R

L

A

B

C

ΔwUR

(b)

120590

ww0A

B

C

D

E

w(C)2

D998400


120590

w

2

2

3

4

5 6

1

G (w0)A

B

C

D

E

F




120590 = 119891 (119908) = ([1 + (1198881119908

1199080

)

3

] 119890minus11988821199081199080

minus119908

1199080

(1 + 1198883

1) 119890minus1198882)119891119905

(9)





119891119891119905


120590 = 119891119906(119908) =

119908

119908 (119888)[120590 (119888) minus 120572 (120590 (119888) minus 119891

119905)]

+ 120572 (120590 (119888) minus 119891119905)

(10)


0



proposed model is

120590 = 119891119897(119908) = (

119908 minus 119908 (119863)

119908 (119864) minus 119908 (119863))

120574

(120590 (119864) minus 120590 (119863)) + 120590 (119863)

(11)


0) according to the



120590 (119864) = (1 minus 120573) 120590 (119862) (12)







119908119894+1

119899+1= 119908119894

119899+1+ (Δ119906

1119894

119899minus Δ1199062119894

119899) sdot 120585

(120590119894+1

119899+1)120585= 120590119894+1

119899+1sdot 120585

(13)



120578+ 1198912120577lt minus120583119891

120585+ 119860119888 119891

1= minus1198912




= 119891119905119860 is set for


120585+ 119860119888 was


120578+ 1198912120577= minus120583119891



120578 119891120577are two



119905is






= (120590119899)120585119860





= 0 119891119903119905= 0 (120590






= (120590119899)120585119860


120590(119864) = 119891119903119905 If (119908119894+1





119894+1

119899+1lt 119891119906(0) If the













120578+ 1198912120577= minus120583119891

120585+119860119888 Ifradic1198912

120578+ 1198912120577lt minus120583119891

120585+119860119888







A

B

C


1050

900

750

600

450

300

150

002502015010050

Load

(N)

Deflection (mm)

Numerical results






900

750

600

450

300

150

0

02502015010050

Deflection (mm)

Numerical results

minus150

Load

(N)





6 Conclusion




A

B

(a)

A

B

(b)

A

B

(c)

A

B

(d)

A

B

(e)

A

B



900

750

600

450

300

150

002502015010050

Deflection (mm)

Load

(N)

1mm2mm

4mm8mm

(a)

900

750

600

450

300

150

0

02502015010050

Deflection (mm)

minus150

Load

(N)

1mm2mm

4mm8mm

(b)





Acknowledgments



References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119908119894+1

119899+1= 119908119894

119899+1+ (Δ119906

1119894

119899minus Δ1199062119894

119899) sdot 120585

(120590119894+1

119899+1)120585= 120590119894+1

119899+1sdot 120585

(13)



120578+ 1198912120577lt minus120583119891

120585+ 119860119888 119891

1= minus1198912




= 119891119905119860 is set for


120585+ 119860119888 was


120578+ 1198912120577= minus120583119891



120578 119891120577are two



119905is






= (120590119899)120585119860





= 0 119891119903119905= 0 (120590






= (120590119899)120585119860


120590(119864) = 119891119903119905 If (119908119894+1





119894+1

119899+1lt 119891119906(0) If the













120578+ 1198912120577= minus120583119891

120585+119860119888 Ifradic1198912

120578+ 1198912120577lt minus120583119891

120585+119860119888







A

B

C


1050

900

750

600

450

300

150

002502015010050

Load

(N)

Deflection (mm)

Numerical results






900

750

600

450

300

150

0

02502015010050

Deflection (mm)

Numerical results

minus150

Load

(N)





6 Conclusion




A

B

(a)

A

B

(b)

A

B

(c)

A

B

(d)

A

B

(e)

A

B



900

750

600

450

300

150

002502015010050

Deflection (mm)

Load

(N)

1mm2mm

4mm8mm

(a)

900

750

600

450

300

150

0

02502015010050

Deflection (mm)

minus150

Load

(N)

1mm2mm

4mm8mm

(b)





Acknowledgments



References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










A

B

C


1050

900

750

600

450

300

150

002502015010050

Load

(N)

Deflection (mm)

Numerical results






900

750

600

450

300

150

0

02502015010050

Deflection (mm)

Numerical results

minus150

Load

(N)





6 Conclusion




A

B

(a)

A

B

(b)

A

B

(c)

A

B

(d)

A

B

(e)

A

B



900

750

600

450

300

150

002502015010050

Deflection (mm)

Load

(N)

1mm2mm

4mm8mm

(a)

900

750

600

450

300

150

0

02502015010050

Deflection (mm)

minus150

Load

(N)

1mm2mm

4mm8mm

(b)





Acknowledgments



References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










A

B

(a)

A

B

(b)

A

B

(c)

A

B

(d)

A

B

(e)

A

B



900

750

600

450

300

150

002502015010050

Deflection (mm)

Load

(N)

1mm2mm

4mm8mm

(a)

900

750

600

450

300

150

0

02502015010050

Deflection (mm)

minus150

Load

(N)

1mm2mm

4mm8mm

(b)





Acknowledgments



References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article implementation of fictitious crack model

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