Abstract

maximum. Smith et al. [2] developed a generalized maximum tensile stress criterion by taking T-stress intoconsideration. Alternatively, the minimum strain energy density (MSED) criterion introduced by Sih [3,4]

* Corresponding author. Tel./fax: +86 21 54747252.E-mail address: jqxu@sjtu.edu.cn (J.-q. Xu).

Engineering Fracture Mechanics 73 (2006) 12491263

www.elsevier.com/locate/engfracmech0013-7944/$ - see front matter 2006 Elsevier Ltd. All rights reserved.1. Introduction

The fracture criterion is the primary basis of the safe evaluation in cracked materials. A theoretically inte-grated fracture criterion shall be capable of predicting both fracture direction and fracture loadings of crackedstructures under various combined loading conditions. For the pure-mode cases, it has been commonlyaccepted that fracture will occur when the corresponding SIF reaches its critical value, but the fracture direc-tion is not indicated, though we know empirically that fracture occurs along the extension of the crack undermode I or III, while branch fracture occurs under mode II. For a crack under the mixed-mode I/II loadingconditions, a number of fracture criteria have been developed through a concerted eort by many researchersin the past decades. Erdogan and Sih [1] proposed the maximum circumferential stress (MCS) criterion, whichassumed that fracture occurs in the direction where the circumferential stress surrounding the crack tip is theIn order to predict the fracture direction and fracture loadings of cracked materials under the general mixed-mode state,this paper presents a new general mixed-mode brittle fracture criterion based on the concept of maximum potential energyrelease rate (MPERR). This criterion can be easily degraded to the pure-mode fracture criterion, and can also be reducedto the commonly accepted fracture criteria for the mixed-mode I/II state. In order to validate the proposed criterion, wehave carried out the experiments with aluminium alloy specimens under various mixed-mode loading conditions. Theexperimental results agree well with the predictions of the proposed criterion. 2006 Elsevier Ltd. All rights reserved.

Keywords: Crack; Stress intensity factor (SIF); Fracture criterion; Mixed-mode; Potential energy release rateA general mixed-mode brittle fracture criterionfor cracked materials

Jun Chang a, Jin-quan Xu a,*, Yoshiharu Mutoh b

a School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiaotong University, Shanghai 200240, Chinab Department of Mechanical Engineering, Nagaoka University of Technology, Niigata 1603-1, Japan

Received 17 October 2005; received in revised form 8 December 2005; accepted 22 December 2005Available online 9 February 2006doi:10.1016/j.engfracmech.2005.12.011

1250 J. Chang et al. / Engineering Fracture Mechanics 73 (2006) 12491263Nomenclature

a crack lengthE Youngs modulusl shear modulusm Poissons ratioj = 3 4m for plane strain, = (3 m)/(1 + m) for plane stressE 0 =E/(1 m2) for plane strain, =E for plane stressG energy release rateGC critical value of energy release rateGh energy release rate in h directionKi SIF of mode i(i = I, II, III)KiC fracture toughness (i = I,II, III)Ke eective SIFS strain energy density factorSC critical value of strain energy density factorW strain energy densityc mode ratio angle, tan1(KII/KI)

w mode ratio angle, tan1 KIII=K2I K2II

q dk opening displacement of the kinked crack face in k direction (k = r,h,z)Da crack extensionhf fracture angleassumed that fracture occurs in the direction where the strain energy density is the minimum. Hussain et al. [5],Palaniswamy and Knauss [6], Nuismer [7] and Wu [8] proposed the maximum energy release rate (MERR)criterion based on Griths theory [9]. The aforementioned criteria are reviewed and discussed in Gdoutossbook [10] reecting on their limitations and applications. Recently, Sutton et al. [11] developed the crack tipopening displacement (CTOD) criterion based on a detailed analysis of crack kinking in arbitrary directions,and assumed that the crack growth occurs when the current CTOD reaches a critical value. Besides, to assessthe crack propagation in elasticplastic materials under mixed-mode loading, Li et al. [12] developed the JMp

based criterion using J-integral and the plastic parameter Mp as basic fracture parameters. The simplied J-estimation equations, proposed by Kim et al. [13] based on the engineering treatment model [14], developedthe J-integral fracture criterion for homogeneous and mismatched structures. On the other hand, for a crackunder the mixed-mode I/III, or under three-mode mixed loading conditions, which are common in engineeringstructures, only few fracture criteria have been proposed so far. Tian [15] has proposed a fracture criterionbased on the principal stress factor for mixed-mode I/III crack problems. Deng et al. [16] have tried to extendtheir two-dimensional CTOD-Based fracture criterion [11,17] and simulation tools into the three-dimensionalrealm. Sih [18] has extended the MSED criterion to the three-dimensional problems. Schollmann [19,20]recently developed the maximum principle stress r01 criterion for the three-dimensional crack growth, andthe criterion has been examined by computational simulation and experiments [21]. In order to carry outmixed I/II fracture tests with a wide range of mixtures, Richard [22] developed a loading xture, and Schwalbeand Cornec [23] proposed a method using specimens with an inclined crack. However, the fracture theory andexperimental method for mixed three-mode conditions have not been yet well-established.

The aim of this paper is to establish a brittle fracture theory applicable for the general mixed-mode loadingconditions. Such a criterion, of course, shall be continuous to the well-known pure or mixed-mode fracturecriterion. Therefore, we investigate the advantages and limitations of the existing mixed-mode I/II fracturecriteria rstly in Section 2, to check if they are appropriate to be extended to the three-mode mixed loadingconditions. Based on the investigations, it is found that the MERR theory has clear physical meanings, though

H function determining the fracture direction

the mathematical expression of the energy release rate Gh in an arbitrary direction is dicult to be obtained.By introducing the approximate form of the energy release rate Gh, the general mixed-mode fracture criterionis presented in terms of the SIFs KI, KII and KIII in Section 3. Finally, the proposed criterion is examined byexperiments with the cracked aluminium alloy specimens under various mixed-mode loading conditions. It isfound that the theoretical predictions and experimental data agree well with each other.

2. Investigations on the existing mixed-mode criteria

rrh sin sin cos 3 cos

rion cthe fr

Hererial c(2), ththe ploadi

J. Chang et al. / Engineering Fracture Mechanics 73 (2006) 12491263 12512.2. Maximum circumferential stress criterion

The fracture criterion assumed that fracture occurs in the direction where the circumferential stress is themaximum surrounding the crack tip. Buczek and Herakovich [24], Saouma et al. [25], Beuth and Herakovich[26] and Carloni and Nobile [27], etc., have extended this criterion to cracked orthotropic materials for planeproblems. The criterion can be expressed as

O x

y

z

ran be certainly described by them. Under the three-mode mixed loading conditions, the general form ofacture criterion can be expressed as

hf HKI;KII;KIII; f KI;KII;KIII C 2KI, KII and KIII are the SIFs of the pure-mode I, II and III, respectively, C is a critical value of the mate-onstant, hf denotes the fracture initiation direction, and H, f are the functions to be determined. In Eq.e rst part determines the fracture direction, and the second part determines the fracture loadings. Bothure-mode and the existing mixed-mode fracture criteria are just the special cases of Eq. (2) for dierentng conditions.rhz KIII2pr

p cos h2; rrz KIII

2prp sin h

2

Since the stress and displacement elds near the crack tip can be expressed by SIFs, the brittle fracture crite-2pr 4 2 2 4 2 22.1. General remarks

The crack-tip stress and displacement elds are the bases to establish fracture criteria for cracked struc-tures. We consider the general mixed-mode loading conditions with three modes I, II and III. In terms ofthe polar coordinates r and h at the crack tip located at point O (as shown in Fig. 1), the stress componentsof the crack-tip stress eld can be written as

rrr 12pr

p KI4

5 cosh2 cos 3h

2

KII

45 sin h

2 3 sin 3h

2

rhh 12pr

p KI4

3 cosh2 cos 3h

2

KII

43 sin h

2 3 sin 3h

2

1p KI h 3h KII h 3h 1Fig. 1. The coordinate at the crack tip.

orhhoh

hhf

0 o2rhhoh2

hhf

< 0

!; rhhmax rhhjhhf

Khmax2pr

p ; Khmax P KIC 3a

By substituting Eq. (1) into Eq. (3a), one can get the fracture criterion for cracked homogenous materials as

H KI sin hf KII3 cos hf 1 0

f 14

KI 3 coshf2 cos 3

2hf

3KII sin hf

2 sin 3

2hf

KIC

3b

Here KIC is the fracture toughness of the material. This criterion can well-predict the fracture direction andloadings for the mixed-mode I/II conditions. However, if three modes coexist, it is clear that KIII cannot beconsidered since the mode III has no eect on the circumferential stress. Therefore, the MCS criterion is justa special theory applicable only for the mixed-mode I/II conditions, and not suitable to be extended into three-mode mixed cases.

According to reference [3,4], the MSED criterion can be expressed as

wheralong

=45 =60

1252 J. Chang et al. / Engineering Fracture Mechanics 73 (2006) 12491263-150 -100 -50 50 100 15000 (Degree) =02 =30becomes extreme is not unique. The relative smaller angle (absolute value) corresponds to larger extreme,and is near to the MCS direction. Thus, by taking this angle as the fracture direction, one can obtain the cri-terion almost similar to the MCS criterion, except for the pure-mode II case. However, the angle where S be-comes the minimum is quite dierent from the fracture direction. For example, when KI = KII, the angle whereS becomes the minimum is h = 118, which is dierent from the fracture direction h = 62.

4

6 16S/( K2I+K2

II)

=90

=0.3Plane strainoSoh

hhf

0 o2S

oh2

hhf

> 0

!; Smin Sjhhf SC 5

e SC is the critical value of strain energy density factor. The distribution of strain energy density factorsthe h direction is shown in Fig. 2, where m is Poissons ratio. It can be found that the angle where SII

where S is the strain energy density factor, and eij(i, j = r,h,z) are strain components.S 116pl

1 cos hj cos hK2I 2 sin h2 cos h j 1KIKII

j 11 cos h 1 cos h3 cos h 1K2 42.3. Minimum strain energy density criterion

The MSED criterion states that fracture occurs in the direction where the strain energy density becomes theminimum. For the mixed-modes I and II, the strain energy density near the crack tip can be written as

W 12rijeij Sr ;Fig. 2. The distribution of strain energy density factors in the h direction.

On the other hand, Sih [18] has extended the MSED criterion into three-dimensional realm. Wang andHadeld [28,29] have adopted the extended criterion to predict the crack growth direction of ring crack in sil-icon nitride. Obviously, such an extension is valid only by taking the relative smaller angle where S becomesextreme as the fracture direction.

The strain energy density factor under mixed three modes can be expressed as [18]

S 116pl

1 cos hj cos hK2I 2 sin h2 cos h j 1KIKII

j 11 cos h 1 cos h3 cos h 1K2 4K2 6

lead

Howe

where

J. Chang et al. / Engineering Fracture Mechanics 73 (2006) 12491263 1253

ax

h

Crack length ayused directly since the practical fracture direction under mixed-mode conditions is generally not the extensiondirection. Hence, many researchers [58] have proposed the MERR fracture criterion by taking the fracturedirection into account. The MERR criterion states that fracture will occur along the direction where the en-ergy release rate is the maximum, and when the energy release rate reaches a critical value. Here, the energyrelease rate is dened as the released energy when crack propagates a kinked unit length. The criterion can bedescribed as

hf hjGhGhmax ; Ghmax P GC 9However, it is very dicult to calculate the energy release rate Gh. Based on the physical meanings, if the crackkinks into the h direction, G can be expressed as (referred to Fig. 3):G E0KI KII 2lKIII; E E=1 m2 Plane strain 8

E, m are Youngs modulus and Poissons ratio, respectively. However, this energy release rate cannot bedict the fracture direction. Furthermore, the relationship of three fracture toughness values is not claried inthe criterion.

2.5. Maximum energy release rate criterion

Assuming a crack propagates a unit length along its extension, the energy release rate can be given as

1 2 2 1 2 0 E Plane stress

KIC

KIIC

KIIIC 1 7

ver, the elliptical criterion has some limitations. It is not theoretically integrated because it cannot pre-KII). Therefore, the MSED criterion is also not appropriate to be extended to three-mode mixed conditions.

2.4. The empirical elliptical criterion

Besides the fracture criteria mentioned above, the elliptical criterion is widely applied in engineering for itssimple form. This criterion is summarized from the experimental results, and can be described as

KI 2 KII 2 KIII 2III

to an obvious contradiction for KIII dominating fracture (e.g., the case of large KIII with very smallII III

From the above expression, if one extends the minimum S theory to three-mode mixed criterion, one can eas-ily nd that K only has eect on fracture loadings, but has no eect on fracture direction. This result willFig. 3. Sketch for calculating Gh.

of GhUsuaDa! 0.

existed criteria, it is more suitable to be extended to the case of three-mode mixity, if one can give the mathe-maticeral b

3. A

3.1. T

K K h 3 K h 3

1254 J. Chang et al. / Engineering Fracture Mechanics 73 (2006) 12491263oGhoh

0 o2Ghoh2 < 0

!; Ghf GC 14Eq. (11) implies that the stresses in the h direction are, respectively, equivalent to the crack-tip stresses withSIFs KIe, KIIe, KIIIe. Hence, the energy release rate Gh in h direction for a crack under mixed KI, KII, KIIIstates can be approximately assumed to be the same as the energy release rate of a crack under KIe, KIIe,KIIIe with propagation in its extension direction. With this postulate, we can easily obtain G(h) from Eq. (8)

Gh j 18l

K2Ieff K2IIeff 1

2lK2IIIeff

j 18l

KI4

3 cosh2 cos 3

2h

KII

43 sin h

2 3 sin 3

2h

2

j 18l

KI4

sinh2 sin 3

2h

KII

4cos

h2 3 cos 3

2h

2 K

2III

2lcos2

h2

12

The above equation can be simplied as

Gh 12l

cos2h2

j 18

K2I 1 cos h 4KIKII sin h K2II5 3 cos h K2III

13

with the approximate expression of G(h), we can establish the general fracture criterion based on the MERRcriterion asrhh Ieff2pr

p ; KIeff I4

3 cos2 cos

2h II

43 sin

2 3 sin

2h

rrh KIIeff2pr

p ; KIIeff KI4

sinh2 sin 3

2h

KII

4cos

h2 3 cos 3

2h

rhz KIIIeff2pr

p ; KIIIeff KIII cos h2

11To establish the fracture criterion based on MPERR concept, we have to give the expression of Gh. Here,we focus on an approximate form of Gh.

Introducing the eective SIFs, KIe, KIIe and KIIIe, the stress components in the h direction can be rewrit-ten as al expression of Gh. Therefore, we select the MERR criterion as the theoretical basis to establish the gen-rittle fracture criterion, by introducing an approximate expression of Gh.

general fracture criterion

he approximate expression of energy release rate and fracture criterionFor all that, the MERR criterion has a rm theoretical basis of energy balance. Compared with otheris nearly impossible to be obtained, and may even lose its physical meanings when considering Da! 0.lly, Gh can be only calculated by huge numerical analyses with taking the approximate limitation ofwhere rij(a) is the crack-tip stress eld before the crack kinks, dk(k = r,h,z) is the opening displacement of thekinked crack face in k direction. As Gdoutos [10] has pointed out, however, the strict mathematical expressionGh limDa!0

1

2Da

Z Da0

1

2rhhadha Da rrhadra Da rhzadza Dadn 10hhf hhf

By su

2 2

f KIKIII0 II IC II IIC IC

j 1 2That is, the general fracture criterion is continuous to commonly accepted pure-mode fracture criteria. More-

in EqFo

h

I f II f

Fromwhen

Eqs.

MCS and MERR (considering branched extension) theories with experiment results. Their results show thatthe dibothSinceotherfor th

Fo

J. Chang et al. / Engineering Fracture Mechanics 73 (2006) 12491263 1255hf 0; Ghf j 1K2I K2III GC j 1K2IC 21erence between MCS and MERR fracture curves is not so large even for the KII dominant cases, thoughthese two theories are somewhat dierent from the experiment results for the cases near to pure-mode II.the present criterion is identical to MCS theory for the pure-mode, such a dierence still remains. Inwords, MERR theory, with or without considering the branched extension, may need to be improvede KII dominant cases if high evaluation accuracy is required.r the mixed-mode I/III loading conditions, Eq. (15) gives outdegraded form of the criterion presented here.It is worth to point out that the approximate form of energy release rate shown in Eq. (20) is strictly dif-

ferent from that obtained by considering the branched extension Da! 0 [30]. However, the dierence appearsmeaningfully only for the KII dominant cases. Hussiain et al. [5] has compared the fracture curves based onthe above equation, we can work out the fracture angle hf (where srh = 0 in fact). The fracture occurs

f Ghf j 18l

KI4

3 cosh2 cos 3

2h

KII

43 sin h

2 3 sin 3

2h

2 GC j 1

8lK2IC 20

(19b) and (20) are identical with the MCS criterion [1]. Therefore, the MCS criterion can be regarded as aKI sin hf KII3 cos hf 1KI1 cos hf KII sin hf cos f2 0 19a

in which the maximum value condition of Gh is

H K sin h K 3 cos h 1 0 19bs. (16)(18).r the mixed-mode I/II case, the extreme value condition of Gh can be deduced from Eq. (15)over, the toughness KIC, KIIC and KIIIC are not independent, they have relationships with each other as shown3 2

hf jKIKII0 0;4

K2III K2IC or KIII KIIIC j 1p

KIC 18hf jKIIKIII0 0; 8l KI GC 8l KIC or KI KIC 16

h j cos11=3; 4K2 K2 or K K 3

pK 17H j 18

K2I sinhf2 sin 3

2hf 4KIKII cos 3

2hf K2II 3 sin

3

2hf 5 sin hf

2 K2III sin

hf2 0

f 12l

cos2hf2

j 18

K2I 1 cos hf 4KIKII sin hf K2II5 3 cos hf K2III

GC j 1

8lK2IC

15

3.2. Comparisons with the existing criteria

Considering the pure-mode I, II and III loading conditions, respectively, we can obtain the fracture direc-tion and fracture loadings from Eq. (15)

j 1 j 1bstituting Eq. (13) into (14), we can get the general fracture criterion as8l 2l 8l

The f

The e

Suchcalcuness.fractucreas

Inthree

havethe eand s

1256 J. Chang et al. / Engineering Fracture Mechanics 73 (2006) 12491263lysis. For the convenience to measure the crack shapes and initial failure points, we marked the specimens on

theirbeen carried out to make fatigue propagation so that the inuence of fused eect can be eliminated. Tond, the relatively regular initial annular cracks can be introduced as shown in Fig. 9. The practical sizehape of the pre-crack are determined from the cross-sections after failure experiments for numerical ana-the test machine MTS809a/809t, by which the tensile and torsion loadings can be applied simultaneously.Cylindrical specimens are made of aluminium alloy with the material constants E = 70 GPa and m = 0.35.To introduce a regular annular crack, the specimens are rst machined to the shape as shown in Fig. 7. Then,the notch is processed by the YAG laser for 0.3 s with the power 2.0 kW. During the process, the rotate speedof the specimen is 250 rpm. One of the processed notches is shown in Fig. 8. Finally, the mode I fatigue testsorder to demonstrate the validity of the aforementioned criterion, the failure experiments under the-mode mixed loadings have been carried out. We designed a loading xture shown in Fig. 6, and used4. Experimental examinationsan equivalent form can provide a concept of eective SIF for mixed three modes. That is, one can simplylate the eective SIF according to Eq. (26), and evaluate the fracture just by comparing it with the tough-Based on Eq. (25a), the branching fracture angles are shown in Fig. 4 (where j = 1.8). Fig. 5 shows there surface according to Eq. (26). An important observation from Fig. 4(b) is that the fracture angle de-es as KIII increases, i.e., KIII also has eect on the fracture direction. K2IC 262l 2 8 2 2

xpression equivalent to Eq. (25b) is

f K2eff 4K2I K2II K2III

j 1 cos2 hf2

j 18

cos2 w 3 4 sin hf2sin

hf2 2c

cos hf

sin2 w

f K2I K2II K2III cos2 hf j 1 cos2 w 3 4 sin hf sin hf 2c cos hf sin2 w GC 25bracture criterion given by Eq. (15) can be rewritten as

H j 18

cos2 w 2 sin3

2hf 2c

1 4 sin2 c sin hf

2 sin 3

2hf

sin2 w sin hf

2 0 25aThe second part can be rearranged into the following form:

KIKIC

2 KIII

KIIIC

2 1; KIIIC

j 1p

2KIC 22

That is, the elliptical fracture criterion has a theoretical basis under the mixed-mode I/III loadings.

3.3. Mixed-mode fracture curves

For convenience, the mode ratio angles are introduced as follows

c tan1KII=KI; w tan1 KIII=K2I K2II

q 23

Then the SIFs can be expressed in terms of c and w

KI K2I K2II K2III

qcos c cosw

KII K2I K2II K2III

qsin c cosw

KIII K2I K2II K2III

qsinw

24sides facing to experimenters in the experiments.

J. Chang et al. / Engineering Fracture Mechanics 73 (2006) 12491263 12570 15 30 45 6075

90

0

20

40

60

80

015

3045

6075

90

(Degree)

(Degre

e) f

(Deg

ree)

60

80

ree)

=0o =10o

=20o =30o

o

(a)Changing the xed angles of the specimens and the loading combination of tension and torsion, weachieved the purpose of fracture under various mode mixities. The experimental results under variousmixed-mode loading conditions are shown in Table 1, where the inner diameter of fracture face is the averagediameter of the fractured section.

Due to the complexity of the loading conditions, SIFs have been analyzed by ANSYS program. To make itsimple, the tensile and torsion loads applied on the clamp are transferred to the specimens by transforminginto the tensile force, torsion and bending moments, as depicted in Fig. 10. Thus, only the straight parts ofthe specimens (without two ends) are required to be analyzed. The FE-meshes of one specimen are shownin Fig. 11, it is noted that FE-meshes at the crack tip have been modied according to the digital photosof the cross-sections after the facture.

To predict the fracture loadings, we have to clarify the initial breakpoints. We rst worked out the SIFsalong various paths (that is, at various positions) (angle a) as shown in Fig. 11 by the extrapolation method.Fig. 12 is the extrapolation example of SIFs for the specimen N at the position of a = 3p/8. It can be seen thatthe extrapolation is of good linearity, which demonstrates the reliability of the numerical analysis. With thenumerical results of SIFs, the eective SIFs at various angles have been calculated by Eqs. (25a) and (26),as shown in Fig. 13(a). The position corresponding to the maximum eective SIF is regarded as the fractureinitiation point. For the specimen N, the predicted fracture initiation point is a = p/2, as indicated inFig. 13(b).

The similar analyses have been carried out for all the other specimens, and the results of the maximum eec-tive SIFs are shown in Fig. 14. The result of the specimen A, which is under the pure-mode I condition, can be

0 15 30 45 60 75 900

20

40

f(D

eg

(Degree)

=40

=50o

=60o =70o

=80o

(b)Fig. 4. Fracture angles under various mixed-mode loading conditions: (a) branching angles as a three dimensional surface, (b) branchingangles as two dimensional curves.

0.00.2

0.40.6

0.8

1.0

0.00.2

0.40.6

0.81.0

0.2

0.4

0.6

0.8

1.0

K I/K IC

KII

I/KIC

KII /KIC

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

KII/K

IC

KI/KIC

=0o

=20o

=40o

=60o =80o

(a)

(b)Fig. 5. The fracture surface and curves for three-mode mixities: (a) three dimensional fracture surface, (b) two dimensional fracturecurves.

Fig. 6. The loading xture for mixed-mode experiments.

1258 J. Chang et al. / Engineering Fracture Mechanics 73 (2006) 12491263

J. Chang et al. / Engineering Fracture Mechanics 73 (2006) 12491263 1259taken as the measured fracture toughness KIC 24:83 MPam

p. From Fig. 14, it can be seen that the maxi-

mum eective SIFs for specimens under various mode mixities approximately equal to the fracture toughnessKIC. For the specimen B, which is under the pure-mode III condition, the numerical resultKIII 19:89 MPa

m

pcorresponding to fracture, is very close to the toughness KIIIC 20:02 MPa

m

pobtained from Eq. (18). Therefore, we can draw a conclusion that the general fracture criterion given byEqs. (25a), (25b) or (26) matches the fracture experiment results for various mixed-mode loading conditions.

Fig. 7. The geometry of the specimen.

Fig. 8. The initial crack processed by the laser.

Fig. 9. The fatigue-propagated crack.

Table 1Experimental results

Test no. Breaking load Inner diameter (mm) Modes

Loading angle h () Tensile force (kN) Torsion moment (Nm)

A 0 24.61 0 8.2 IB 0 0 40.16 7.42 IIIC 0 20.51 11.15 7.66 I, IIID 0 19.75 20.44 7.8 I, IIIE 0 16.80 13.51 7.2 I, IIIF 0 16.53 22.22 7.4 I, IIIG 15 24.83 0 8.18 I, II, IIIH 30 20.71 0 7.94 I, II, IIII 45 10.99 0 9.32 I, II, IIIJ 60 11.95 0 6.0 I, II, IIIK 90 12.51 0 6.44 II, IIIL 30 10.33 11.26 6.2 I, II, IIIM 45 14.58 15.62 7.44 I, II, IIIN 60 14.1 13.16 7.4 I, II, III

ForceP

z=Pcos

Py=Psin

Bending momentM

x=PLsin

My=TsinTorsion momentT

z=Tcos

z

y

x

My=Tsin

Tz=Tcos

Mx=PLsin

P TP

2L

Fig. 10. The load analysis of the specimen.

Fig. 11. The FE-meshes of the specimen.

1260 J. Chang et al. / Engineering Fracture Mechanics 73 (2006) 12491263

J. Chang et al. / Engineering Fracture Mechanics 73 (2006) 12491263 12615

10

15

20

25

zz(2

r)1/2

(MPa

m1/

2 )

KI=18.8 MPa m1/2

0

2

4

6

8

rz(2

r)1

/2(M

Pa m

1/2 )

KII=4.45 MPa m1/2

-20

-15

-10

-5

z(2

r)1/2

(MPa

m1/

2 )

KIII=-11.44 MPa m1/25. Summary and conclusions

A general mixed-mode brittle fracture criterion was proposed based on the concept of MPERR by intro-ducing the approximate expression of Gh. Through the fracture experiments with aluminium alloy specimensunder various mixed-mode loading conditions, the validity of the general fracture criterion is veried. Themain results can be concluded as

(1) Fracture criteria for pure-mode or mixed I/II modes are just the degraded forms of the proposed generalfracture criterion, but the new criterion also contains the prediction of fracture direction even for pure-mode problems.

A B C D E F G H I J K L M N0

5

10

15

20

25

30

35

Keff

(MPa

m1/

2 )

Sequence number

KIC=24.83 MPa m1/2

Fig. 14. Experimental results of all specimens.

0.0 0.1 0.2 0.3 0.4r/a

0.0 0.1 0.2 0.3 0.4r/a

0.0 0.1 0.2 0.3 0.4r/a(a) (b) (c)

Fig. 12. The extrapolation examples of the SIFs for the specimen N: (a) KI, (b) KII, (c) KIII.

Fig. 13. Eective SIFs and the cross-section for the specimen N: (a) eective SIFs, (b) the cross-section.

[15] Tian CH. Mixed crack propagation principal stress factor model and I/III mode crack propagation. Chin J Appl Mech 2004;1:849.

1262 J. Chang et al. / Engineering Fracture Mechanics 73 (2006) 12491263In: Freitas MM, editor. Proceedings of the 6th international conference on biaxial/multiaxial fatigue and fracture, Lisboa, 2001. p.58996.

[20] Schollmann M, Richard HA, Kullmer G, Fulland M. A new criterion for the prediction of crack development in multiaxially loadedstructures. Int J Fract 2002;117(2):12941.

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[22] Richard HA. Eine Bruchmechanikprobe Zur Bestimmung Von KIIC-werten. Schweiaen Schneiden 1981;33:60612.[23] Shwalbe K-H, Cornec A. The engineering treatment model and its practical application. Fatigue Fract Engng Mater Struct

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[18] Sih GC. Mechanics of fracture initiation and propagation. New York: Kluwer Academic Publishers; 1991.[19] Schollmann M, Kullmer G, Fulland M, Richard HA. A new criterion for 3D crack growth under mixed-mode (I + II + III) loading.(2) The relationships among KIC, KIIC and KIIIC have been claried from the viewpoint of energy releaserate.

(3) The SIF KIII also has an eect on the fracture direction for mixed-mode problems containing mode II.The branch angle decreases with KIII increases.

(4) For mixed I/III modes, the elliptical empirical criterion can be deduced from the general fracturecriterion.

(5) The eective SIF concept has been proposed for arbitrary mixed-mode conditions, by which the fractureevaluation can be easily carried out.

(6) Mixed-mode fracture tests have been carried out, and the experimental results agree the general fracturecriterion very well.

Acknowledgment

This project is supported by National Natural Science Foundation of China (10372058).

References

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J. Chang et al. / Engineering Fracture Mechanics 73 (2006) 12491263 1263

A general mixed-mode brittle fracture criterion for cracked materialsIntroductionInvestigations on the existing mixed-mode criteriaGeneral remarksMaximum circumferential stress criterionMinimum strain energy density criterionThe empirical elliptical criterionMaximum energy release rate criterion

A general fracture criterionThe approximate expression of energy release rate and fracture criterionComparisons with the existing criteriaMixed-mode fracture curves

Experimental examinationsSummary and conclusionsAcknowledgmentReferences