Research Article Mechanical Performance Evaluation of ...downloads.hindawi.com/journals/amse/2013/572151.pdfe SERR refers to the energy dissipated during fracture per unit of newly

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Hindawi Publishing CorporationAdvances in Materials Science and EngineeringVolume 2013 Article ID 572151 9 pageshttpdxdoiorg1011552013572151

Research ArticleMechanical Performance Evaluation of Concrete BeamsStrengthened with Carbon Fiber Materials

Jun Ding12 Xia Huang1 Gang Zhu1 Song Chen1 and Guochao Wang1

1 College of Mechanical Engineering Chongqing University of Technology Chongqing 400054 China2 Key Laboratory of Manufacture and Test Techniques for Automobile Parts Ministry of Education Chongqing 400054 China

Correspondence should be addressed to Jun Ding dingjunawen126com

Received 16 July 2013 Accepted 12 September 2013

Academic Editor Xing Chen

Copyright copy 2013 Jun Ding 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

As a kind of novel material of high strength and light weight carbon fiber materials have been widely used in construction industryto repair the damaged bridges improving itsmechanical performance In this work the reinforced platesmade of carbon fibermate-rials (for short CFRP) are externally bonded to the bottom of concrete beams to enhance load capacity of beamsThe strain energyrelease rates are calculated at the interest crack in concrete beams based on virtual crack closure technology using FEM and arechosen as the criterion to determine whether themechanical properties of beams are strengthened by being externally bonded withCFRP The effects of main crack propagation on plain concrete beam on concrete beam strengthened with CFRP and on inclinedcrack are also discussedThe comparison between the beamswith andwithoutCFRP shows that theCFRP significantly increases theloading capacity and crack resistance It also shows that themain crack propagation can reduce loading capacity and crack resistanceregardless of strengtheningThe experiment observation also supports this It proves the validity of the method and it is concludedthat in order to increase the loading capacity and crack resistance effectively controlling over the crack propagation is necessary

1 Introduction

Due to the excellent mechanical properties such as lightweight high strength at tension and superior resistance toaggressive environment the reinforcement and strengthen-ing technique bonded with the reinforced plates made ofcarbon fiber materials (for short CFRP) to concrete beamis widely accepted and considered as an effective and con-venient method among various methods of strengtheninginfrastructure construction [1ndash4] According to structuralmechanics an important concern over the effectiveness andsafety for the method strengthening concrete beam withCFRP is the enormous potential for load capacity anddecrease in failure mode in service

So far many researches [5ndash11] have been conductedto investigate the mechanical properties of the retrofittedmembers externally bonded with CFRP It can be concludedthat the bonding of the CFRP plates has a significant effecton enhancing the overall performance even for plates witha relatively low modulus of elasticity The strengtheningtechnology externally bonded with CFRP can considerably

increase the strength at bending reduce the deflections aswell as crack width in area of tension and also change themechanical behavior and failure pattern during service Theresearches [5 8ndash10] demonstrated that the cracking momentin reinforced concrete beams with CFRP can significantlyincrease by 218 from 12 to 230 They also found thatin strengthened flexural members cracks are seen in biggernumbers and closer to each other however they are narrowerthan in beams without strengthened reinforcement

However the review for the previous works shows thatthe reinforced concrete beams used to be strengthened withCFRP are essentially intact that is without any greater cracksand obvious defects in advance As a matter of fact somebigger defects and macrocracks such as holes and cracksare already pre-existed in flexural members in constructionbefore it is strengthened Such imperfection can severelyreduce the structural strength and stiffness of members andit receives more and more attention for earlier design andlater maintenance Moreover the strengthened member failsoften in a brittle way due to the loss of connection between

2 Advances in Materials Science and Engineering

composite material and the concrete itself in the process ofstrengthening Consequently it is necessary to consider theeffect of pre existed macrocracks on the performance of theflexural members externally bonded with CFRP systemati-cally

For the convenience of investigating the mechanicalproperties theoretically the plain concrete beams not rein-forced with steel bars have been chosen for this workto examine the mechanical behaviors such as structuralstrength stiffness and strain energy release rates Consider-ing that the purpose of this work focuses on exploring theloading capacity and crack resistance for the flexural beamsexternally bonded with CFRP the choice of the purely plainbeam as being studied will not lose the generality of researchIn order to emphasize the influence of existence of crack onthe loading capacity of beams themacrocracks with differentlength were in advance made in the plain concrete beamsand subsequent researches serve for it According to fracturemechanics in order to predict crack propagation the strainenergy released rate (abbreviated as SERR for convenienceof later description) G must be evaluated and comparedwith the critical fracture energy of the constituent materialsG119868119862 determined from experimentsThe calculation for SERR

at the interest crack was based on virtual crack closuretechnology (VCCT) together with the finite element method(FEM) The values for SERR were then selected as a criterionto determine whether the mechanical properties of the beamare strengthened by being externally bonded with CFRPin comparison to those beams without bonded CFRP Wefocus upon the effect of crack propagation that occurredat the initial crack edge crack and the mutual interactionbetween them on the loading capacity and failuremechanismof strengthened with CFRP

2 Materials and Methods

21 The Description in the Model of the Concrete BeamIn order to experimentally investigate the improvement inmechanical properties of concrete beams due to the bond ofCFRP our previous work [11] made some experiments forbeams at the test setup as shown in Figure 1 Figure 2(a)shows the failure mode of the concrete beam without CFRPstrengthened and Figures 2(b)ndash2(d) show the failure modewith CFRP strengthened for the length of 01m 02m and035m respectively

According to the previous experimental measurementthe geometry model has been constructed shown as inFigure 3(a) It includes twomodels plain concrete beamwithand without CFRPThe geometry dimension for the concretebeam is about 01m times 01m times 04m each with an initialcrack length of 20mm marked as a

0in Figure 3 The plain

concrete beam externally bonded with CFRP is illustrated asin Figure 3(b) In experiment test the length of the CFRPvaries with the different case study to investigate the effect ofthe length of CFRP on the reinforcement The details aboutthe experiment can refer to the work [11]

Figure 1 The test setup for concrete beams

22 The Crack Criterion Based on SERR There are commonthree fracture criteriaK (stress intensity factor) J integral andStrain Energy Release Rate (SERR) in linearly elastic fracturemechanics which are used to determine whether and whencrack propagates [12]The critera for bothK and SERR are forlinearly elastic material while J is integral criterion for plasticmaterial Concrete behaves in brittle material behavior and itis considered as elastic material [13] Generally in numericalsimulation the calculation for the value of K requires strictconditions for the mesh to discrete the geometry modelespecially at the tip of the crack due to its numerical radiussingularity [12] Meanwhile the calculation for SERR usuallydoes not need a more special element type near the areaaround crack tip and it even employs fewer elements

The SERR refers to the energy dissipated during fractureper unit of newly created fracture surface area which is anessential quantity in fracture mechanics because the energythat must be supplied to a crack tip for it grows must bebalanced by the amount of energy dissipated due to theformation of new surfaces [12]

For calculation SERR is defined as

119866 = minus

120597 (119880 minus 119881)

120597119860

(1)

where U is the potential energy available for crack growthV is the work associated with any external forces and A isthe crack area (crack length for two-dimensional problems)The failure criterion of energy release rate states that the crackwill grow when the available energy release rate 119866 is greaterthan or equal to the critical value 119866

119862 The quantity 119866

119862is the

fracture energy and is considered to be a material propertywhich is independent of the applied loads and the geometryconfiguration of the body

For two-dimensional geometry configuration with iso-tropic elastic material property the value of K and SERR forthe case of problem of plane strain can follow (2) where119870

119868is

the stress intensity factor 119864 is Youngrsquos modulus and V is thePoisson ratio119866 is the fracture energy and is considered to be

Advances in Materials Science and Engineering 3

(a) (b)

(c) (d)

Figure 2 The photos of experimental measurement [11] (a) Failure mode of the beam without CFRP (b) Failure mode of the beam withCFRP of 01m (c) Failure mode of the beam with CFRP of 02m (d) Failure mode of the beam with CFRP of 035m

P

a0 w=01

t = 01L

(a)

P

a0 w=01

t = 01

L

L

0

(b)

Figure 3 The geometrical model of the concrete beam (a) The plain concrete beam (b) The beam strengthened with CFRP

a material property which is independent of the applied loadsand the geometry configuration of the body Consider

119866 =

1198702

119868

119864

(1 minus V2) (2)

For the three-point bending concrete beam illustratedas in Figure 3 the expression calculating the value for 119870

119868

follows (3) wherein119875 is the load applied at themidspan of theconcrete beam 119878 is the span distance between two supportsof the concrete beam 119905 is the thickness of the concrete beam119886 is the crack length and the expression for 119865(120572) can follow(4) Consider

119870119868=

3119878119875

21199051198822radic120587119886119865 (120572) (3)

119865 (120572) =

199 minus 120572 (1 minus 120572) (215 minus 393120572 + 271205722)

radic120587 (1 + 2120572) (1 minus 120572)32

where 120572 = 119886119908

(4)

In a word for the geometric configuration illustrated asin Figure 3 the value of K can be numerically calculated

by substituting the parameters such as the crack length thewidth of the cross section the span between supports andthe load Then taking on (2) can derive the value of SERR

23TheVirtual Crack Closure Technique (VCCT) Thevirtualcrack closure technique (VCCT) iswidely used for computingSERR based on results from finite element analysis to supplythe mode separation required when using the mixed-modefracture criterion [14]

This method is mainly based on Irwinrsquos crack closureintegral and on the assumption that the energy Δ119864 releasedwhen the crack is extended byΔ119886 from 119886 to 119886+Δ119886 is identicalto the energy required to close the crack between locations119897 and 119894 as illustrated in Figure 4 Additionally however it isassumed that a crack propagation of Δ119886 from Δ119886 to 119886 + Δ119886(node i) to 119886 + 2Δ119886 (node k) does not significantly alter thestate at the crack tipTherefore when the crack tip is located atnode k the displacements behind the crack tip at node i areapproximately equal to the displacements behind the cracktip at node l when the crack tip is located at node i Furtherthe energyΔ119864 releasedwhen the crack is extended byΔ119886 from


a Δa Δa

Δw1

Δul

Zi

Xi

i k

l

Crack closed

12

34

z w Z

x u X

Figure 4 The specimens were tested in the three-point bendingconfiguration and the load was applied with cylindrical rollers asline load across the top width of the beams To spread the load andavoid stress concentrations elastomeric pads were placed betweenthe roller supports and the lower surface of the specimens andbetween the loading device and the top surface of the specimensDuring experimental measurement the ultimate load which makescrack propagate or makes the reinforced concrete beam fail infiber in concrete or at the interface between them can be achievedaccording to instrument reading Virtual crack closure technique(VCCT)

119886 + Δ119886 to 119886 + 2Δ119886 is identical to the energy required to closethe crack between locations i and k

For a crack modeled with two dimensional four-nodeelements as shown in Figure 4 the workΔ119864 required to closethe crack along one element side therefore can be calculatedas follows

Δ119864 =

1

2

[119883119894Δ119906119897+ 119885119894Δ119908119897] (5)

where 119883119894and 119885

119894are the shear and opening forces at the

nodal point i and Δ119906119897and Δ119908

119897are the shear and opening

displacements at node l as shown in Figure 4Thus forces anddisplacements required to calculate the energyΔ119864 to close thecrackmay be obtained fromone single finite element analysis

However Figure 3(a) shows that the dominant separationmode at the central crack of the concrete beam shouldbe mode I crack The term of shearing force 119883

119894and of

shear displacement Δ119906119897can disappear in (5) only the terms

for opening force and displacement leave in the equationMeanwhile in Figure 3(b) both opening and shearing effectsmust be taken into account when calculating SERR for thecrack failure that occurred at the interface between concreteand CFRP and the failure at the bottom of CFRP

24 Calculation of SERR Based on VCCT In this section wediscuss the scheme employed to calculate SERR from finiteelement analysis based on VCCT The finite element modelhas firstly been created in ABAQUS software [15] Becausethe calculation for SERR based on VCCT only involves thenodal forces and nodal displacements at the element sidemore refinedmesh or special treatment element types are notnecessary for the discretization in FEM at the crack tip Thefewer elements with four-node plane strain elements are usedat the crack tip

Table 1 The value of SERR for various numbers of elements

Numbers of elements 300 600 1000 1400SERR from FEA 10404 9819 9232 9047Relative error 153 883 232 0265

The details of mesh configuration around the crack tipare also expressed as in Figure 4 The location of i is thecrack tip In ABAQUS software in order to obtain the valueof nodal force at node i it needs define elements 1 and 2as one element set using command lowastELSET and using lowastELPRINT command to output the nodal force Since the nodei is shared by elements 1 and 2 the total nodal force at nodei in Z direction is the sum of the contribution at node i byelement 1 and that at node i by element 2 The displacementsin Z direction at node l can also be known in finite elementanalysis Thus SERR at the crack tip based on VCCT usingfinite element analysis can be easily calculated

In this case the load applied on the concrete beam is5000N and the crack length is 20mm taken into (2)ndash(4)the value of SERR 119866 is 9024 Jm2 In order to check themesh dependency on computation four various numbersof elements 300 600 1000 and 1400 are used to discretethe plain concrete beam in FEA The calculated SERR fordifferent numbers of elements is summarized in Table 1Comparison between the value for SERR from FEA and the-oretical calculation indicates that 1400 elements are the bestchoice which results in the minor error with the theoreticalone

3 Results and Discussion

31 The Effect of Main Crack Propagation Firstly the effectsof crack propagation for the main crack seam (the locationillustrated as in Figure 3(a)) on the overall strength of theplain concrete beam (ie without externally bonded CFRPat the bottom of the beam) are studied as follows Thefinite element analysis for the concrete beam under threedifferent loading case is performed in advance and then thecalculation method the same as that previously described inSection 24 is employed to calculate the value of SERR Theinitial crack length is taken as 20mm We model the maincrack propagation by means of varying the lengths of cracks20 25 30 35 and 40mm to simulate crack growth in service

Figure 5 shows the plot of the calculated SERR for variousload cases against the crack propagation of the main crackSuppose that crack extends under the load from 20mm to40mm It can be seen from the figure that SERR for any caseof load can increase slightly with themain crack propagationThe bigger the magnitude of SERR is the closer it approaches119866119862of the concrete material It implies that the strength of the

beam decreases with the increase of SERR For a certain fixedcrack length the value for SERR also increases with the loadwhich indicates that the bigger load applied at the concretebeam can reduce the loading capacity and make the concretebeam closer to the critical load

According to Section 31 SERRwill increase with the loadapplied to the beam and also with the crack propagation


20 25 30 35 40

5

10

15

20

25

30Th

e cal

culat

ed st

rain

ener

gy re

leas

e rat

e (Jm

2)

The crack extension to different length a (mm)

3000 N4000N5000N

Figure 5 The plot of calculated SERR against the main crackpropagation

20 25 30 35 40358

101315182023252830

The crack length a (mm)

3000 N4000N

5000N

The s

trai

n en

ergy

rele

ase r

ate (

Jm2)

GIC

Figure 6 The comparison of SERR between the different load andthe critical case

Figure 6 shows a comparison made between the value ofSERR for various loads and the critical SERR For the cracklength of 20mm the crack propagation will not occur evenif the load exceeds 5000N However as the crack lengthincreases the magnitude of load at which SERR exceeds G

119868119862

decreases significantly indicating that the load capacity of thebeam weakens

32 The Prediction for the Critical Load According to Fig-ure 6 it can be seen that for a given length of crack theultimate load applied at the plain concrete beam can bepredicated in theory Figure 7 shows the calculated SERR forvarious loads at the crack length of 20mm The formula that

30 35 40 45 50 55 6023456789

1011121314

The loading magnitude applied at the beam for crack length of 20mm (KN)

The c

alcu

lated

stra

in en

ergy

rele

ase r

ate (

Jm2)

G = 03665P2 + 0008P minus 00018

Figure 7 The calculated SERR for different load at the crack lengthof 20mm

calculates the ultimate load could be derived based on curvefitting Consider

119866 = 036651198752+ 00008119875 minus 00018 (6)

The magnitude of ultimate load can be calculated whenthe value of 119866 ge 119866

119868119862 Then we can derive the magnitude of

load is 5580N which makes the initial crack propagates It isclose to the value of 5752N observed from the experimentalmeasurement [11 16] indicating the validity and accuracy ofthe method used to calculate the value for SERR

33TheDetermination for Location of Potential Crack In thissection we focus on the case in which the plain concretebeam was strengthened with CFRP externally bonded at thebottom surface to the beam Under the load there exist manycracks possibly initiated at the end of the CFRP or at theinterface between concrete and CFRP Such potential cracksin turn significantly influence the stability of the main crackConsequently the determination for the location of potentialcrack seems enormously important to investigate the increaseor decrease in strength of a concrete beam bonded externallywith CFRP

As illustrated in Figure 3(b) the length of CFRP bondedat the bottom of the concrete beam is taken as 119871

0= 02m

the initial crack length 1198860is 20mm and the length of the

total concrete beam is 119871 = 04m For the convenience ofcomparison with experimental measurements the loadapplied to the beam is P = 20KN Due to the symmetry ofthe model half of the concrete beam was considered andperformed in finite element analysis According to [11] themaximum tensile stress for concrete material is 515MPa Interms of strength theory for brittle material the concrete isconsidered to break or debond when the maximum tensilestress under the load in concrete beam exceeds the criticalstress 515MPa Figure 8 shows the distribution of maximumtensile stress in concrete beam while the dark black regionstands for the stress greater than critical stress value that ispotential crack initiation


Concrete beam

515Mpa515Mpa

515Mpa

CFRP

AB C

Figure 8Thedistribution of themaximum tensile stress in the plainconcrete beam strengthened with externally bonded CFRP

2120579

1

3

Figure 9Thepotential crack initiation in concrete beamwithCFRP

Seen from the figure the potential crack possibly occursat three sites near the main crack (denoted as A) at theinterface between concrete and CFRP (denoted as B) and atthe end of CFRP (denoted as C) respectively For CFRP it willnot break through since the maximum tensile stress in CFRPis much smaller than its critical tensile stress 4640MPaSubsequently suppose that the strength of bonding betweenconcrete material and epoxy resin and the bond forcebetween expoxy resin and CFRP are sufficient enough toresist the tensile stress The initiation of crack should occurat the side of the concrete beam Therefore the distributionof the potential crack in the concrete beam is sketched asin Figure 9 For the convenience of subsequent descriptionthe main crack in the middle of the concrete beam the crackprobably initiated at the end of the CFRP and the debondingcracks at the end of the CFRP are labeled as main crack 1inclined crack 2 and delamination crack respectively

Cracks 1 2 3 illustrated in Figure 9 are possible to coexistat the same time or one of them or both of them exist onetime Meanwhile they will probably influence another todecrease the strength and the stiffness of the concrete beam

34 The Effect of Main Crack Propagation on the StrengthenedBeam In order to investigate the effect of the main crackpropagation on the performance of the strengthened beamit is supposed in this section that the main crack only existsbut the other cracks such cracks 2 and 3 did not

Employing the similar method calculating SERR aspreviously described SERR for the strengthened concretebeam is calculated and the variation of SERR versus themain crack propagation in illustrated in as Figure 10 Thehorizontal line in figurewith the value of 1122 Jm2 represents

20 25 30 35 40 45456789

10111213141516

The main crack length a (mm)

15KN18KN20KN

25KNGIC

The s

trai

n en

ergy

rele

ase r

ate f

orm

ain

crac

k (J

m2)

Figure 10 The plot of SERR against the main crack propagation

the critical SERR for concrete material at which the concretecan probably break off It can be seen from Figure 10 thatwhen concrete beam is strengthened by CFRP the SERR cansignificantly increase with the applied load showing that theloading capacity of the beam is improved

As the load increase to the extent at which SERR is greaterthan critical value crack propagation will occur at the maincrack We can also find that the concrete beam strengthenedwith CFRP cannot sufficiently sustain the applied load at theinitial stage of loading because the main crack does not startto propagate and the subsequently increased load are mainlycarried by concrete beam itself

However as the main crack extends to some certainlength CFRP begins to carry the load and the maximum ten-sile stress in CFRP starts to increase As shown in Figure 10for each load SERR slightly increases as cracks extend butit decreases rapidly with crack propagation and the case forload of 20KN strongly supports this point

Figure 11 shows the comparison of SERR between beforeand after strengthening for various loads 3000 4000 and5000N respectively From the figure we can easily see thatthe strengthening beamwithCFRP significantly improves theperformance of the concrete beam supported by the SERR forstrengthened beam smaller than that without strengtheningThe value of SERR for the beam with CFRP is almost 110of that without CFRP It is CFRP that decreases the value ofSERR and increases the strength and stiffness of the concretebeam

35 The Effect of Main Crack Propagation on Inclined Crack

351 The Determination for the Inclined Angle Observedfrom Figure 9 as load applied at the midspan of the CFRPconcrete beam increases the deflection at the bottom of theconcrete beam continually increases and CFRP bonded at thebottom of the beam starts to sustain the tension force Since


20 25 30 35 400358

101315182023252830


3000N before strengthening4000N before strengthening5000N before strengthening3000N after strengthening4000N after strengthening5000N after strengthening

The s

trai

n en

ergy

rele

ase r

ate f

orm

ain

crac

k (J

m2)

Figure 11 The comparison of SERR before and after strengthening

the strength of CFRP is sufficient enough it will not break dueto a strength problemThe concrete material at the end of theCFRP can probably fail and the inclined crack can initiate atthis site According to experiment observation the directionof crack propagation makes an angle with the horizontal axisof a beam

Based on the predication for crack path in bimaterialcomposites plate under uniform mechanical load and tem-perature load suppose that an inclined crack can propagatealong the direction of angle with the horizontal axis by thelength of Δ119886 as illustrated in Figure 12

The SERR for inclined crack extended by Δ119886 can becalculated as follows

119866I (119886 997888rarr 119886 + Δ119886 120579) = 119860sin2120579 minus 119861 sin 120579 cos 120579 + 119862cos2120579

119866II (119886 997888rarr 119886 + Δ119886 120579) = 119860cos2120579 + 119861 sin 120579 cos 120579 + 119862sin2120579

119866 (119886 + Δ119886 120579) = 119866I (119886 997888rarr 119886 + Δ119886 120579)

+ 119866II (119886 997888rarr 119886 + Δ119886 120579) = 119860 + 119862

(7)

in which

119860 =

1

2119905Δ119886

[119865119891

119909(119906119890

119909minus 119906119891

119909) + 119865ℎ

119909(119906119892

119909minus 119906ℎ

119909)]

119861 =

1

2119905Δ119886

[119865119891

119909(119906119890

119910minus 119906119891

119910) + 119865ℎ

119909(119906119892

119910minus 119906ℎ

119910)

+119865119891

119910(119906119890

119909minus 119906119891

119909) + 119865ℎ

119910(119906119892

119909minus 119906ℎ

119909)]

119862 =

1

2119905Δ119886

[119865119891

119910(119906119890

119910minus 119906119891

119910) + 119865ℎ

119910(119906119892

119910minus 119906ℎ

119910)]

(8)

a c e

dbf

g i

h

j

X

Y

Δa

120579

Figure 12 The finite element mesh configuration at the tip of crackwith an angle against horizontal axis extended by the length of Δ119886

in which 119865119894119909 119865119894119910(119894 = 119891 ℎ) are the 119909 and 119910 components of

nodal force at the node 119894 before the crack extends by Δ119886and 119906119894

119909 119906119894119910(119894 = 119890 119891 119892 ℎ) are the 119909 and 119910 components of

displacement at the node 119894 once crack extends by Δ119886Combined with the results from finite element analysis

for concrete beam the SERR of 119866 119866I and 119866II can be easilycalculated from (7)-(8) In terms of brittle fracturemechanicsthe inclined crack always extends along the direction at which119866II = 0 when 119866I ge 119866119868119862 Since the angle is unknown now anarbitrary angle is assumed and the angle is desired if it justmakes 119866II = 0

First we forced 119866II(120579) to be zero to find a solution of theangle

119866II (120579) = 119860cos2120579 + 119861 sin 120579 cos 120579 + 119862sin2120579 = 0 (9)

In this case the length of CFRP is 02m the initial lengthof crack is 20mm and the load is 20KN Substituting suchknown parameters into (7)ndash(9) we can obtain the solution tothe angle 120579 = 45∘ along which the inclined crack extends

352 The Effect of Main Crack Propagation on Inclined CrackIn this section the effects of main crack propagation on theinclined crack for various lengths of inclined crack have beendiscussed Figures 13(a)ndash13(f) show the variation of SERRcalculated at the inclined crack 2 with the propagation ofmain crack (a) for the inclined crack length of 0mm (b) for1mm (c) for 2mm (d) for 3mm (e) for 4mm and (f) for5mm respectively The increase in the inclined crack lengthmeans the progressive propagation of inclined crack as theload The SERR calculated at the inclined crack is a goodchoice to reflect the influence of main crack propagation onthe inclined crack and on the overall stiffness and strength ofthe concrete beam

It can be seen from Figure 13 that for each given lengthof inclined crack the value for SERR increases as the maincrack grows approaching the critical SERR of the concretematerial and showing that the inclined crack with maincrack propagation tends to crack On the other hand asillustrated in Figure 13(a) the value for SERR firstly appearsto be a decrease as the main crack propagates and then itseems to be an increase with the propagation of main crackwhich implies that stress concentration exists in the concretematerial at the end of the bonded FRP As the main crack


0 5 10 15 2000

02

04

06

08

10

12


The S

ERR

for i

nclin

ed cr

ack

(Jm

2)

(a)

0 5 10 15 20

04

06

08

10

12

14

16


The S

ERR

for i

nclin

ed cr

ack

(Jm

2)

(b)

0 5 10 15 2000

05

10

15

20


The S

ERR

for i

nclin

ed cr

ack

(Jm

2)

(c)

0 5 10 15 2000

05

10

15

20

25


The S

ERR

for i

nclin

ed cr

ack

(Jm

2)

(d)

0 5 10 15 2000

05

10

15

20

25

3000N4000N

5000N6000N


The S

ERR

for i

nclin

ed cr

ack

(Jm

2)

(e)

0 5 10 15 2000

05

10

15

20

25

30

3000N4000N

5000N6000N


The S

ERR

for i

nclin

ed cr

ack

(Jm

2)

(f)

Figure 13The plot of SERR against main crack propagation for various length of inclined crack (a) 0mm of inclined crack length (b) 1mmof inclined crack length (c) 1mm of inclined crack length (d) 2mm of inclined crack length (e) 3mm of inclined crack length and (f) 0mmof inclined crack length


does not propagate the value for SERR at the end of the plateseems to be larger while it will decrease rapidly when maincrack starts to propagate Next the phenomenon of stressconcentration at the end of plate has been effectively releasedMeanwhile as the main crack progressively propagates to afixed crack length the magnitude of stress at the crack tip ofthe main crack can dramatically decrease due to the tensionof FRP externally bonded at the bottom of the concretebeam At this time CFRP begins to become tensile more thanbefore and it considerably releases the stress concentrationoccurring at the main crackThe higher tension in CFRP dueto its high stiffness and modulus can effectively prevent themain crack in concrete beam from propagating continuallyHowever SERR for inclined crack in turn increases slightlybecause the tensile stress in CFRP drastically increases theconcentration of stress at the end of CFRP building up therisk of crack propagationWith the inclined crack proceedingto propagate the value for SERR at the inclined crack tendsto increase slightly regardless of the applied load seen fromFigures 13(a)ndash13(f) showing that the propagation of inclinedcrack makes the SERR for itself much closer to the criticalvalue and increases the risk of crack formation

4 Conclusion

The concrete beams with and without CFRP strengthenedare considered to investigate the improvement in loadingcapacity and analyze the failure mechanism of the concretebeam The main conclusions can be drawn as follows

(1) Strain energy release rate (SERR) for a specifiedcrack has been chosen as an indicator to show thestrengthening effect of CFRP bonded at the bottomof concrete beams The strain energy release rate arecalculated at an interest crack based on virtual crackclosure technology (VCCT) using FEM to determinewhether the mechanical properties of the beam arestrengthened by being externally bonded with CFRP

(2) For plain concrete beam without being strengthenedby CFRP the calculated SERR for any case of loadcan increase slightly with themain crack propagationThe bigger the magnitude of SERR is the closer itapproaches119866

119862of the concretematerial It implies that

the strength of the beam decreases with the increaseof SERR

(3) For plain concrete beam externally bonded withCFRP the value for SERR can significantly increasewith the applied load showing that the loadingcapacity of the beam is improved However as themain crack extends to some certain length the valuefor SERR slightly increases as cracks extend but itdecreases rapidly with crack propagation

Acknowledgments

This work is financially supported by the Natural ScienceFoundation of China (11302272) by the Natural ScienceFoundation of China (11272368) by the Natural Science

Foundation Project of Chongqing Municipality (CSTC2010BB4085) and by the Science and Technology ProjectAffiliated to the EducationDepartment ofChongqingMunic-ipality ( KJ120801)

References

[1] A KM Anwarul Islam ldquoEffectivemethods of using CFRP barsin shear strengthening of concrete girdersrdquo Engineering Struc-tures vol 31 no 3 pp 709ndash714 2009

[2] M C Sundarraja and S Rajamohan ldquoStrengthening of RCbeams in shear using GFRP inclined stripsmdashan experimentalstudyrdquo Construction and Building Materials vol 23 no 2 pp856ndash864 2009

[3] H C Biscaia C Chastre and M A G Silva ldquoNolinear numer-ical analysis of the debonding failure process of FRP-to-concreteinterfacesrdquo Composites B vol 50 pp 210ndash223 2013

[4] J Michels R Christen and D Waldmann ldquoExperimental andnumerical investigation on postcracking behavior of steel fiberreinforced concreterdquo Engineering Fracture Mechanics vol 98pp 326ndash349 2013

[5] X-S Yang J M Lees and C T Morley ldquoModelling crackpropagation in structures comparison of numerical methodsrdquoCommunications in Numerical Methods in Engineering vol 24no 11 pp 1373ndash1392 2008

[6] J F Guan L B Qing and S B Zhao ldquoResearch on numericalsimulation on the whole cracking processes of three-pointbending notch concrete beamsrdquo Chinese Journal of Computa-tional Mechanics vol 30 pp 143ndash148 2013

[7] S S Pendhari T Kant and YM Desai ldquoApplication of polymercomposites in civil construction a general reviewrdquo CompositeStructures vol 84 no 2 pp 114ndash124 2008

[8] C Mazzotti M Savoia and B Ferracuti ldquoAn experimentalstudy on delamination of FRP plates bonded to concreterdquo Con-struction and Building Materials vol 22 no 7 pp 1409ndash14212008

[9] W Xue L Zeng and Y Tan ldquoExperimental studies on bondbehaviour of high strength CFRP platesrdquo Composites B vol 39no 4 pp 592ndash603 2008

[10] J Valivonis and T Skuturna ldquoCracking and strength of rein-forced concrete structures in flexure strengthened with carbonfibre laminatesrdquo Journal of Civil Engineering and Managementvol 13 no 4 pp 317ndash323 2007

[11] S S NiuThe experimental study on concrete beams strengthenedwith CFRP [MS thesis] Chongqing University ChongqingChina 2008

[12] T L Anderson FractureMechanics Fundamentals and Applica-tions CRC Press 2005

[13] A M Navier Material Properties of Concrete China BuildingIndustry Press 2011

[14] R Krueger ldquoVirtual crack closure technique history approachand applicationsrdquo Applied Mechanics Reviews vol 57 no 1ndash6pp 109ndash143 2004

[15] J Ding Crack failure study of CFRP reinforced concrete beams[MS thesis] Chongqing University Chongqing China 2008

[16] K P Herrmann andM Dong ldquoThermal cracking of two-phasecomposite structures under uniform and non-uniform tem-perature distributionsrdquo International Journal of Solids andStructures vol 29 no 14-15 pp 1789ndash1812 1992

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of



CeramicsJournal of


CompositesJournal of

NanoparticlesJournal of



International Journal of

Biomaterials


NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research



MetallurgyJournal of


BioMed Research International

MaterialsJournal of


Nano

materials


Journal ofNanomaterials


composite material and the concrete itself in the process ofstrengthening Consequently it is necessary to consider theeffect of pre existed macrocracks on the performance of theflexural members externally bonded with CFRP systemati-cally

For the convenience of investigating the mechanicalproperties theoretically the plain concrete beams not rein-forced with steel bars have been chosen for this workto examine the mechanical behaviors such as structuralstrength stiffness and strain energy release rates Consider-ing that the purpose of this work focuses on exploring theloading capacity and crack resistance for the flexural beamsexternally bonded with CFRP the choice of the purely plainbeam as being studied will not lose the generality of researchIn order to emphasize the influence of existence of crack onthe loading capacity of beams themacrocracks with differentlength were in advance made in the plain concrete beamsand subsequent researches serve for it According to fracturemechanics in order to predict crack propagation the strainenergy released rate (abbreviated as SERR for convenienceof later description) G must be evaluated and comparedwith the critical fracture energy of the constituent materialsG119868119862 determined from experimentsThe calculation for SERR

at the interest crack was based on virtual crack closuretechnology (VCCT) together with the finite element method(FEM) The values for SERR were then selected as a criterionto determine whether the mechanical properties of the beamare strengthened by being externally bonded with CFRPin comparison to those beams without bonded CFRP Wefocus upon the effect of crack propagation that occurredat the initial crack edge crack and the mutual interactionbetween them on the loading capacity and failuremechanismof strengthened with CFRP

2 Materials and Methods

21 The Description in the Model of the Concrete BeamIn order to experimentally investigate the improvement inmechanical properties of concrete beams due to the bond ofCFRP our previous work [11] made some experiments forbeams at the test setup as shown in Figure 1 Figure 2(a)shows the failure mode of the concrete beam without CFRPstrengthened and Figures 2(b)ndash2(d) show the failure modewith CFRP strengthened for the length of 01m 02m and035m respectively

According to the previous experimental measurementthe geometry model has been constructed shown as inFigure 3(a) It includes twomodels plain concrete beamwithand without CFRPThe geometry dimension for the concretebeam is about 01m times 01m times 04m each with an initialcrack length of 20mm marked as a

0in Figure 3 The plain

concrete beam externally bonded with CFRP is illustrated asin Figure 3(b) In experiment test the length of the CFRPvaries with the different case study to investigate the effect ofthe length of CFRP on the reinforcement The details aboutthe experiment can refer to the work [11]

Figure 1 The test setup for concrete beams

22 The Crack Criterion Based on SERR There are commonthree fracture criteriaK (stress intensity factor) J integral andStrain Energy Release Rate (SERR) in linearly elastic fracturemechanics which are used to determine whether and whencrack propagates [12]The critera for bothK and SERR are forlinearly elastic material while J is integral criterion for plasticmaterial Concrete behaves in brittle material behavior and itis considered as elastic material [13] Generally in numericalsimulation the calculation for the value of K requires strictconditions for the mesh to discrete the geometry modelespecially at the tip of the crack due to its numerical radiussingularity [12] Meanwhile the calculation for SERR usuallydoes not need a more special element type near the areaaround crack tip and it even employs fewer elements

The SERR refers to the energy dissipated during fractureper unit of newly created fracture surface area which is anessential quantity in fracture mechanics because the energythat must be supplied to a crack tip for it grows must bebalanced by the amount of energy dissipated due to theformation of new surfaces [12]

For calculation SERR is defined as

119866 = minus

120597 (119880 minus 119881)

120597119860

(1)

where U is the potential energy available for crack growthV is the work associated with any external forces and A isthe crack area (crack length for two-dimensional problems)The failure criterion of energy release rate states that the crackwill grow when the available energy release rate 119866 is greaterthan or equal to the critical value 119866

119862 The quantity 119866

119862is the

fracture energy and is considered to be a material propertywhich is independent of the applied loads and the geometryconfiguration of the body

For two-dimensional geometry configuration with iso-tropic elastic material property the value of K and SERR forthe case of problem of plane strain can follow (2) where119870

119868is

the stress intensity factor 119864 is Youngrsquos modulus and V is thePoisson ratio119866 is the fracture energy and is considered to be


(a) (b)

(c) (d)


P

a0 w=01

t = 01L

(a)

P

a0 w=01

t = 01

L

L

0

(b)



119866 =

1198702

119868

119864

(1 minus V2) (2)


119868


119870119868=

3119878119875

21199051198822radic120587119886119865 (120572) (3)

119865 (120572) =


radic120587 (1 + 2120572) (1 minus 120572)32

where 120572 = 119886119908

(4)






a Δa Δa

Δw1

Δul

Zi

Xi

i k

l

Crack closed

12

34

z w Z

x u X




Δ119864 =

1

2

[119883119894Δ119906119897+ 119885119894Δ119908119897] (5)

where 119883119894and 119885






119894and of














20 25 30 35 40

5

10

15

20

25

30Th

e cal

culat

ed st

rain

ener

gy re

leas

e rat

e (Jm

2)


3000 N4000N5000N


20 25 30 35 40358

101315182023252830


3000 N4000N

5000N

The s

trai

n en

ergy

rele

ase r

ate (

Jm2)

GIC



119868119862



30 35 40 45 50 55 6023456789

1011121314


The c

alcu

lated

stra

in en

ergy

rele

ase r

ate (

Jm2)

G = 03665P2 + 0008P minus 00018



119866 = 036651198752+ 00008119875 minus 00018 (6)






0= 02m




Concrete beam

515Mpa515Mpa

515Mpa

CFRP

AB C


2120579

1

3






20 25 30 35 40 45456789

10111213141516


15KN18KN20KN

25KNGIC

The s

trai

n en

ergy

rele

ase r

ate f

orm

ain

crac

k (J

m2)









20 25 30 35 400358

101315182023252830



The s

trai

n en

ergy

rele

ase r

ate f

orm

ain

crac

k (J

m2)







119866 (119886 + Δ119886 120579) = 119866I (119886 997888rarr 119886 + Δ119886 120579)

+ 119866II (119886 997888rarr 119886 + Δ119886 120579) = 119860 + 119862

(7)

in which

119860 =

1

2119905Δ119886

[119865119891

119909(119906119890

119909minus 119906119891

119909) + 119865ℎ

119909(119906119892

119909minus 119906ℎ

119909)]

119861 =

1

2119905Δ119886

[119865119891

119909(119906119890

119910minus 119906119891

119910) + 119865ℎ

119909(119906119892

119910minus 119906ℎ

119910)

+119865119891

119910(119906119890

119909minus 119906119891

119909) + 119865ℎ

119910(119906119892

119909minus 119906ℎ

119909)]

119862 =

1

2119905Δ119886

[119865119891

119910(119906119890

119910minus 119906119891

119910) + 119865ℎ

119910(119906119892

119910minus 119906ℎ

119910)]

(8)

a c e

dbf

g i

h

j

X

Y

Δa

120579








119866II (120579) = 119860cos2120579 + 119861 sin 120579 cos 120579 + 119862sin2120579 = 0 (9)





0 5 10 15 2000

02

04

06

08

10

12


The S

ERR

for i

nclin

ed cr

ack

(Jm

2)

(a)

0 5 10 15 20

04

06

08

10

12

14

16


The S

ERR

for i

nclin

ed cr

ack

(Jm

2)

(b)

0 5 10 15 2000

05

10

15

20


The S

ERR

for i

nclin

ed cr

ack

(Jm

2)

(c)

0 5 10 15 2000

05

10

15

20

25


The S

ERR

for i

nclin

ed cr

ack

(Jm

2)

(d)

0 5 10 15 2000

05

10

15

20

25

3000N4000N

5000N6000N


The S

ERR

for i

nclin

ed cr

ack

(Jm

2)

(e)

0 5 10 15 2000

05

10

15

20

25

30

3000N4000N

5000N6000N


The S

ERR

for i

nclin

ed cr

ack

(Jm

2)

(f)




4 Conclusion







Acknowledgments



References
























CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials




(a) (b)

(c) (d)


P

a0 w=01

t = 01L

(a)

P

a0 w=01

t = 01

L

L

0

(b)



119866 =

1198702

119868

119864

(1 minus V2) (2)


119868


119870119868=

3119878119875

21199051198822radic120587119886119865 (120572) (3)

119865 (120572) =


radic120587 (1 + 2120572) (1 minus 120572)32

where 120572 = 119886119908

(4)






a Δa Δa

Δw1

Δul

Zi

Xi

i k

l

Crack closed

12

34

z w Z

x u X




Δ119864 =

1

2

[119883119894Δ119906119897+ 119885119894Δ119908119897] (5)

where 119883119894and 119885






119894and of














20 25 30 35 40

5

10

15

20

25

30Th

e cal

culat

ed st

rain

ener

gy re

leas

e rat

e (Jm

2)


3000 N4000N5000N


20 25 30 35 40358

101315182023252830


3000 N4000N

5000N

The s

trai

n en

ergy

rele

ase r

ate (

Jm2)

GIC



119868119862



30 35 40 45 50 55 6023456789

1011121314


The c

alcu

lated

stra

in en

ergy

rele

ase r

ate (

Jm2)

G = 03665P2 + 0008P minus 00018



119866 = 036651198752+ 00008119875 minus 00018 (6)






0= 02m




Concrete beam

515Mpa515Mpa

515Mpa

CFRP

AB C


2120579

1

3






20 25 30 35 40 45456789

10111213141516


15KN18KN20KN

25KNGIC

The s

trai

n en

ergy

rele

ase r

ate f

orm

ain

crac

k (J

m2)









20 25 30 35 400358

101315182023252830



The s

trai

n en

ergy

rele

ase r

ate f

orm

ain

crac

k (J

m2)







119866 (119886 + Δ119886 120579) = 119866I (119886 997888rarr 119886 + Δ119886 120579)

+ 119866II (119886 997888rarr 119886 + Δ119886 120579) = 119860 + 119862

(7)

in which

119860 =

1

2119905Δ119886

[119865119891

119909(119906119890

119909minus 119906119891

119909) + 119865ℎ

119909(119906119892

119909minus 119906ℎ

119909)]

119861 =

1

2119905Δ119886

[119865119891

119909(119906119890

119910minus 119906119891

119910) + 119865ℎ

119909(119906119892

119910minus 119906ℎ

119910)

+119865119891

119910(119906119890

119909minus 119906119891

119909) + 119865ℎ

119910(119906119892

119909minus 119906ℎ

119909)]

119862 =

1

2119905Δ119886

[119865119891

119910(119906119890

119910minus 119906119891

119910) + 119865ℎ

119910(119906119892

119910minus 119906ℎ

119910)]

(8)

a c e

dbf

g i

h

j

X

Y

Δa

120579








119866II (120579) = 119860cos2120579 + 119861 sin 120579 cos 120579 + 119862sin2120579 = 0 (9)





0 5 10 15 2000

02

04

06

08

10

12


The S

ERR

for i

nclin

ed cr

ack

(Jm

2)

(a)

0 5 10 15 20

04

06

08

10

12

14

16


The S

ERR

for i

nclin

ed cr

ack

(Jm

2)

(b)

0 5 10 15 2000

05

10

15

20


The S

ERR

for i

nclin

ed cr

ack

(Jm

2)

(c)

0 5 10 15 2000

05

10

15

20

25


The S

ERR

for i

nclin

ed cr

ack

(Jm

2)

(d)

0 5 10 15 2000

05

10

15

20

25

3000N4000N

5000N6000N


The S

ERR

for i

nclin

ed cr

ack

(Jm

2)

(e)

0 5 10 15 2000

05

10

15

20

25

30

3000N4000N

5000N6000N


The S

ERR

for i

nclin

ed cr

ack

(Jm

2)

(f)




4 Conclusion







Acknowledgments



References
























CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials




a Δa Δa

Δw1

Δul

Zi

Xi

i k

l

Crack closed

12

34

z w Z

x u X




Δ119864 =

1

2

[119883119894Δ119906119897+ 119885119894Δ119908119897] (5)

where 119883119894and 119885






119894and of














20 25 30 35 40

5

10

15

20

25

30Th

e cal

culat

ed st

rain

ener

gy re

leas

e rat

e (Jm

2)


3000 N4000N5000N


20 25 30 35 40358

101315182023252830


3000 N4000N

5000N

The s

trai

n en

ergy

rele

ase r

ate (

Jm2)

GIC



119868119862



30 35 40 45 50 55 6023456789

1011121314


The c

alcu

lated

stra

in en

ergy

rele

ase r

ate (

Jm2)

G = 03665P2 + 0008P minus 00018



119866 = 036651198752+ 00008119875 minus 00018 (6)






0= 02m




Concrete beam

515Mpa515Mpa

515Mpa

CFRP

AB C


2120579

1

3






20 25 30 35 40 45456789

10111213141516


15KN18KN20KN

25KNGIC

The s

trai

n en

ergy

rele

ase r

ate f

orm

ain

crac

k (J

m2)









20 25 30 35 400358

101315182023252830



The s

trai

n en

ergy

rele

ase r

ate f

orm

ain

crac

k (J

m2)







119866 (119886 + Δ119886 120579) = 119866I (119886 997888rarr 119886 + Δ119886 120579)

+ 119866II (119886 997888rarr 119886 + Δ119886 120579) = 119860 + 119862

(7)

in which

119860 =

1

2119905Δ119886

[119865119891

119909(119906119890

119909minus 119906119891

119909) + 119865ℎ

119909(119906119892

119909minus 119906ℎ

119909)]

119861 =

1

2119905Δ119886

[119865119891

119909(119906119890

119910minus 119906119891

119910) + 119865ℎ

119909(119906119892

119910minus 119906ℎ

119910)

+119865119891

119910(119906119890

119909minus 119906119891

119909) + 119865ℎ

119910(119906119892

119909minus 119906ℎ

119909)]

119862 =

1

2119905Δ119886

[119865119891

119910(119906119890

119910minus 119906119891

119910) + 119865ℎ

119910(119906119892

119910minus 119906ℎ

119910)]

(8)

a c e

dbf

g i

h

j

X

Y

Δa

120579








119866II (120579) = 119860cos2120579 + 119861 sin 120579 cos 120579 + 119862sin2120579 = 0 (9)





0 5 10 15 2000

02

04

06

08

10

12


The S

ERR

for i

nclin

ed cr

ack

(Jm

2)

(a)

0 5 10 15 20

04

06

08

10

12

14

16


The S

ERR

for i

nclin

ed cr

ack

(Jm

2)

(b)

0 5 10 15 2000

05

10

15

20


The S

ERR

for i

nclin

ed cr

ack

(Jm

2)

(c)

0 5 10 15 2000

05

10

15

20

25


The S

ERR

for i

nclin

ed cr

ack

(Jm

2)

(d)

0 5 10 15 2000

05

10

15

20

25

3000N4000N

5000N6000N


The S

ERR

for i

nclin

ed cr

ack

(Jm

2)

(e)

0 5 10 15 2000

05

10

15

20

25

30

3000N4000N

5000N6000N


The S

ERR

for i

nclin

ed cr

ack

(Jm

2)

(f)




4 Conclusion







Acknowledgments



References
























CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials




20 25 30 35 40

5

10

15

20

25

30Th

e cal

culat

ed st

rain

ener

gy re

leas

e rat

e (Jm

2)


3000 N4000N5000N


20 25 30 35 40358

101315182023252830


3000 N4000N

5000N

The s

trai

n en

ergy

rele

ase r

ate (

Jm2)

GIC



119868119862



30 35 40 45 50 55 6023456789

1011121314


The c

alcu

lated

stra

in en

ergy

rele

ase r

ate (

Jm2)

G = 03665P2 + 0008P minus 00018



119866 = 036651198752+ 00008119875 minus 00018 (6)






0= 02m




Concrete beam

515Mpa515Mpa

515Mpa

CFRP

AB C


2120579

1

3






20 25 30 35 40 45456789

10111213141516


15KN18KN20KN

25KNGIC

The s

trai

n en

ergy

rele

ase r

ate f

orm

ain

crac

k (J

m2)









20 25 30 35 400358

101315182023252830



The s

trai

n en

ergy

rele

ase r

ate f

orm

ain

crac

k (J

m2)







119866 (119886 + Δ119886 120579) = 119866I (119886 997888rarr 119886 + Δ119886 120579)

+ 119866II (119886 997888rarr 119886 + Δ119886 120579) = 119860 + 119862

(7)

in which

119860 =

1

2119905Δ119886

[119865119891

119909(119906119890

119909minus 119906119891

119909) + 119865ℎ

119909(119906119892

119909minus 119906ℎ

119909)]

119861 =

1

2119905Δ119886

[119865119891

119909(119906119890

119910minus 119906119891

119910) + 119865ℎ

119909(119906119892

119910minus 119906ℎ

119910)

+119865119891

119910(119906119890

119909minus 119906119891

119909) + 119865ℎ

119910(119906119892

119909minus 119906ℎ

119909)]

119862 =

1

2119905Δ119886

[119865119891

119910(119906119890

119910minus 119906119891

119910) + 119865ℎ

119910(119906119892

119910minus 119906ℎ

119910)]

(8)

a c e

dbf

g i

h

j

X

Y

Δa

120579








119866II (120579) = 119860cos2120579 + 119861 sin 120579 cos 120579 + 119862sin2120579 = 0 (9)





0 5 10 15 2000

02

04

06

08

10

12


The S

ERR

for i

nclin

ed cr

ack

(Jm

2)

(a)

0 5 10 15 20

04

06

08

10

12

14

16


The S

ERR

for i

nclin

ed cr

ack

(Jm

2)

(b)

0 5 10 15 2000

05

10

15

20


The S

ERR

for i

nclin

ed cr

ack

(Jm

2)

(c)

0 5 10 15 2000

05

10

15

20

25


The S

ERR

for i

nclin

ed cr

ack

(Jm

2)

(d)

0 5 10 15 2000

05

10

15

20

25

3000N4000N

5000N6000N


The S

ERR

for i

nclin

ed cr

ack

(Jm

2)

(e)

0 5 10 15 2000

05

10

15

20

25

30

3000N4000N

5000N6000N


The S

ERR

for i

nclin

ed cr

ack

(Jm

2)

(f)




4 Conclusion







Acknowledgments



References
























CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials




Concrete beam

515Mpa515Mpa

515Mpa

CFRP

AB C


2120579

1

3






20 25 30 35 40 45456789

10111213141516


15KN18KN20KN

25KNGIC

The s

trai

n en

ergy

rele

ase r

ate f

orm

ain

crac

k (J

m2)









20 25 30 35 400358

101315182023252830



The s

trai

n en

ergy

rele

ase r

ate f

orm

ain

crac

k (J

m2)







119866 (119886 + Δ119886 120579) = 119866I (119886 997888rarr 119886 + Δ119886 120579)

+ 119866II (119886 997888rarr 119886 + Δ119886 120579) = 119860 + 119862

(7)

in which

119860 =

1

2119905Δ119886

[119865119891

119909(119906119890

119909minus 119906119891

119909) + 119865ℎ

119909(119906119892

119909minus 119906ℎ

119909)]

119861 =

1

2119905Δ119886

[119865119891

119909(119906119890

119910minus 119906119891

119910) + 119865ℎ

119909(119906119892

119910minus 119906ℎ

119910)

+119865119891

119910(119906119890

119909minus 119906119891

119909) + 119865ℎ

119910(119906119892

119909minus 119906ℎ

119909)]

119862 =

1

2119905Δ119886

[119865119891

119910(119906119890

119910minus 119906119891

119910) + 119865ℎ

119910(119906119892

119910minus 119906ℎ

119910)]

(8)

a c e

dbf

g i

h

j

X

Y

Δa

120579








119866II (120579) = 119860cos2120579 + 119861 sin 120579 cos 120579 + 119862sin2120579 = 0 (9)





0 5 10 15 2000

02

04

06

08

10

12


The S

ERR

for i

nclin

ed cr

ack

(Jm

2)

(a)

0 5 10 15 20

04

06

08

10

12

14

16


The S

ERR

for i

nclin

ed cr

ack

(Jm

2)

(b)

0 5 10 15 2000

05

10

15

20


The S

ERR

for i

nclin

ed cr

ack

(Jm

2)

(c)

0 5 10 15 2000

05

10

15

20

25


The S

ERR

for i

nclin

ed cr

ack

(Jm

2)

(d)

0 5 10 15 2000

05

10

15

20

25

3000N4000N

5000N6000N


The S

ERR

for i

nclin

ed cr

ack

(Jm

2)

(e)

0 5 10 15 2000

05

10

15

20

25

30

3000N4000N

5000N6000N


The S

ERR

for i

nclin

ed cr

ack

(Jm

2)

(f)




4 Conclusion







Acknowledgments



References
























CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials




20 25 30 35 400358

101315182023252830



The s

trai

n en

ergy

rele

ase r

ate f

orm

ain

crac

k (J

m2)







119866 (119886 + Δ119886 120579) = 119866I (119886 997888rarr 119886 + Δ119886 120579)

+ 119866II (119886 997888rarr 119886 + Δ119886 120579) = 119860 + 119862

(7)

in which

119860 =

1

2119905Δ119886

[119865119891

119909(119906119890

119909minus 119906119891

119909) + 119865ℎ

119909(119906119892

119909minus 119906ℎ

119909)]

119861 =

1

2119905Δ119886

[119865119891

119909(119906119890

119910minus 119906119891

119910) + 119865ℎ

119909(119906119892

119910minus 119906ℎ

119910)

+119865119891

119910(119906119890

119909minus 119906119891

119909) + 119865ℎ

119910(119906119892

119909minus 119906ℎ

119909)]

119862 =

1

2119905Δ119886

[119865119891

119910(119906119890

119910minus 119906119891

119910) + 119865ℎ

119910(119906119892

119910minus 119906ℎ

119910)]

(8)

a c e

dbf

g i

h

j

X

Y

Δa

120579








119866II (120579) = 119860cos2120579 + 119861 sin 120579 cos 120579 + 119862sin2120579 = 0 (9)





0 5 10 15 2000

02

04

06

08

10

12


The S

ERR

for i

nclin

ed cr

ack

(Jm

2)

(a)

0 5 10 15 20

04

06

08

10

12

14

16


The S

ERR

for i

nclin

ed cr

ack

(Jm

2)

(b)

0 5 10 15 2000

05

10

15

20


The S

ERR

for i

nclin

ed cr

ack

(Jm

2)

(c)

0 5 10 15 2000

05

10

15

20

25


The S

ERR

for i

nclin

ed cr

ack

(Jm

2)

(d)

0 5 10 15 2000

05

10

15

20

25

3000N4000N

5000N6000N


The S

ERR

for i

nclin

ed cr

ack

(Jm

2)

(e)

0 5 10 15 2000

05

10

15

20

25

30

3000N4000N

5000N6000N


The S

ERR

for i

nclin

ed cr

ack

(Jm

2)

(f)




4 Conclusion







Acknowledgments



References
























CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials




0 5 10 15 2000

02

04

06

08

10

12


The S

ERR

for i

nclin

ed cr

ack

(Jm

2)

(a)

0 5 10 15 20

04

06

08

10

12

14

16


The S

ERR

for i

nclin

ed cr

ack

(Jm

2)

(b)

0 5 10 15 2000

05

10

15

20


The S

ERR

for i

nclin

ed cr

ack

(Jm

2)

(c)

0 5 10 15 2000

05

10

15

20

25


The S

ERR

for i

nclin

ed cr

ack

(Jm

2)

(d)

0 5 10 15 2000

05

10

15

20

25

3000N4000N

5000N6000N


The S

ERR

for i

nclin

ed cr

ack

(Jm

2)

(e)

0 5 10 15 2000

05

10

15

20

25

30

3000N4000N

5000N6000N


The S

ERR

for i

nclin

ed cr

ack

(Jm

2)

(f)




4 Conclusion







Acknowledgments



References
























CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials





4 Conclusion







Acknowledgments



References
























CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials










CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials



Documents

Research Article Mechanical Performance Evaluation of ...downloads.hindawi.com/journals/amse/2013/572151.pdfe SERR refers to the energy dissipated during fracture per unit of newly