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a

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Materials Science and Engineering A 527 (2010) 5624–5630

Contents lists available at ScienceDirect

Materials Science and Engineering A

journa l homepage: www.e lsev ier .com/ locate /msea

modified MTS criterion (MMTS) for mixed-mode fracture toughnessssessment of brittle materials

. Saghafi ∗, M.R. Ayatollahi, M. Sistaniniaatigue and Fracture Lab., Department of Mechanical Engineering, Iran University of Science and Technology, Narmak, 16846, Tehran, Iran

r t i c l e i n f o

rticle history:eceived 3 April 2010eceived in revised form 6 May 2010

a b s t r a c t

A semi-circular specimen containing a vertical edge crack was used to assess mode I, mode II, and mixedmode fracture toughness of a marble rock. The specimen was subjected to three-point bending whilethe bottom supports were placed asymmetrically with respect to the crack position. Since the critical

ccepted 7 May 2010

eywords:racture criterionrittle fractureracture toughness

tangential stress should be the same in various fracture modes, it was shown that the first three termsof tangential stress infinite series in the vicinity of the crack tip should be used to satisfy this condition.Accordingly, a criterion was presented to predict the mixed mode fracture toughness of brittle materialswhich showed a good agreement with the experimental results.

© 2010 Elsevier B.V. All rights reserved.

ritical tangential stress

. Introduction

Fracture toughness, KIc, is an important parameter in fractureechanics, which indicates the resistance of materials against

he propagation of a pre-existing crack. The applications of thisarameter in many diverse areas, including blasting and fragmen-ation, hydraulic fracturing, rock slope analysis, rock excavation,utting process, and earthquake mechanics, have attracted manyesearchers to determine the fracture toughness for different brittleaterials.So far, numerous methods and test specimens have been intro-

uced for determining fracture toughness, where each specimenas practically its own advantages and disadvantages. For exam-le, some of them cannot be used in a wide range of mode I andode II mixities or require complicated test fixtures. The rect-

ngular plate containing an inclined center crack subjected touniform far-field tension [1–3], the centrally cracked Brazilian

isk specimen (BD) [4–7], the single-edge crack specimen sub-ected to asymmetric four-point bend loading [8–10], the angleddge crack specimen [11], the compact tension-shear specimen12–14], the diagonally loaded square plate (DLSP) specimen sub-ected to far-field pin loading [15], the cracked semi-circular bend

∗ Corresponding author at: Fatigue and Fracture Lab., Department of Mechanicalngineering, Iran University of Science and Technology, Narmak, 16846, Tehran,ran. Tel.: +982144410447.

E-mail address: h saghafi@mecheng.iust.ac.ir (H. Saghafi).

921-5093/$ – see front matter © 2010 Elsevier B.V. All rights reserved.oi:10.1016/j.msea.2010.05.014

specimen [16–18], and the edge cracked semi-circular specimensubjected to asymmetric three-point bend loading (ASCB) [19] aresome of the specimens used in the past for mixed mode fracturetests on brittle materials. Among them, ASCB is a new specimenrecently introduced by Ayatollahi et al. [19] which needs an in-depth research.

The critical tangential stress, ���c, in front of crack tip is consid-ered to be a constant material property [20]. According to Williams[21], this parameter can be written as an infinite series expansionin a linear elastic cracked body,

���c∼= 1√

2�rc

cos�0

2

[KIf cos2 �0

2− 3

2KIIf sin �0

]

+ Tf sin2(�) + O(r1/2) (1)

where KIf and KIIf represent the critical values of mode I and modeII stress intensity factors, respectively, T is a critical constant term

findependent of distance from the crack tip, rc is the critical distance,assumed to be a material constant, from the crack tip and �0 is thecrack initiation direction. So far, most researchers have used thefirst term of Eq. (1) and neglected the higher order terms. For somecases, however, this estimation is not accurate, such as the resultsreported by Williams and Ewing [1], Ueda et al. [2], and Lim et al.[16] shown in Fig. 1. In this figure, Me is defined as a mode mixtureparameter varying between 0 and 1:

nd En

M

Mbto�iasutv

trmtmtmo(

2

[

cos�

2

os�

2

sin

+ 54

− sin

�

2− 5

2

wcrica

A

wmu

lScp

H. Saghafi et al. / Materials Science a

e = 2�

tan−1(

KIf

KIIf

)(2)

e = 1 for pure mode I, and zero for pure mode II. The dispersaletween the test results is because of not utilizing the higher ordererms in the calculation of ���c (Eq. (1)). In other words, higherrder terms include high magnitude coefficients and rc which affect

��c, noticeably. For example, great values of rc and Tf for SCB spec-men manufactured from the Johnstone rock [22,23] in mode IInd mixed-mode loading affected ���c, considerably. However, ithould be mentioned that there are some other cases in which bysing only the first term of Eq. (1), ���c is the same in all mode mixi-ies [20,24,25]. For such crack problems, the higher order terms areery small relative to the magnitude of the first term.

In this paper, it is shown that utilizing only the first and seconderms of Eq. (1) for ASCB specimen manufactured from a marbleock, Neiriz, does not satisfy achieving the same ���c in mode I,ode II, and mixed mode I/II. Hence, the necessity of using the first

hree terms of Eq. (1) is discussed to reach this condition. Then, aodified brittle fracture criterion developed based on maximum

angential stress (MTS) [26] is introduced for estimating mixed-ode and mode II fracture toughness. While MTS criterion is based

n only the singular term of Eq. (1), the proposed modified MTSMMTS) criterion uses the first three terms.

. Crack tip parameters

According to the infinite series expansion offered by Williams21], the elastic stresses in the crack tip vicinity are written as,⎧⎪⎨⎪⎩

�rr

���

�r�

⎫⎪⎬⎪⎭ =

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

{r−0.5A1

(54

cos�

2− 1

4cos

3�

2

)+ 4A2(cos2 �) + 3r0.5A3

(34{

34

r−0.5A1

(cos

�

2+ 1

3cos

3�

2

)+ 4A2(sin2 �) + 15

4r0.5A3

(c

{12

r−0.5A1

(12

sin�

2+ 1

2sin

3�

2

)− 2A2(sin 2�) + 3

2r0.5A3

(12⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

{−r−0.5B1

(54

sin�

2− 3

4sin

3�

2

)− B2(0) + 3r0.5B3

(34

sin�

2{−3

4r−0.5B1

(sin

�

2+ sin

3�

2

)− B2(0) + 15

4r0.5B3

(sin

�

2{12

r−0.5B1

(12

cos�

2+ 3

2cos

3�

2

)− B2(0) − 3

2r0.5B3

(12

cos

here r and � are the conventional crack tip co-ordinates, and theoefficients An and Bn are mode I and mode II fracture parameters,espectively which are functions of specimen geometry and load-ng configuration. A1, A2, and B1 are prevalent fracture parameters,alculated by researchers for many common test specimens suchs SCB and BD [23] defined as,

1 = KI√2�

, B1 = KII√2�

, A2 = T

4(4)

here KI and KII are the stress intensity factors corresponding toode I and mode II, and T is a non-singular constant stress term

sually called the T-stress.

Other coefficients, i.e. A3 and B3, were often ignored but Ayatol-

ahi and Nejati [27] introduced recently a method to calculate them.ince B2 is multiplied by zero in all terms, it is not mandatory toompute it. In the next section, all these crack tip parameters areresented and discussed for the ASCB specimen.

gineering A 527 (2010) 5624–5630 5625

+ 14

cos5�

2

)+ O(r)

}

− 15

cos5�

2

)+ O(r)

}

�

2− 1

2sin

5�

2

)+ O(r)

}

⎫⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎭

+

sin5�

2

)+ O(r)

}

5�

2

)+ O(r)

}

cos5�

2

)+ O(r)

}

⎫⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎭

(3)

3. The asymmetric semi-circular bend (ASCB) specimen

In this part, first, the asymmetric semi-circular bend (ASCB)specimen is introduced and then the fracture parameters of thespecimen are described. This specimen has been recently suggestedby Ayatollahi et al. [19] to investigate mixed-mode fracture of brit-tle materials. The simple geometry and loading condition, littlemachining operations, and the ability of covering a wide range ofmode I and mode II mixities are some of the advantages in thisspecimen.

As shown in Fig. 2, this specimen is a semi-circular disk of radiusR containing a vertical edge crack of length a in its centre. The speci-men is located on two bottom supports which are asymmetric withrespect to the crack position. S1 and S2 are longer and shorter dis-tances between supports and crack, respectively. The specimen iscompressed by the vertical load P that is in the same direction ofthe crack line. Asymmetric supports give rise to mixed-mode load-ing, i.e. depending on the magnitude of S1/R, S2/R, and a/R variousmode mixities are obtained. When S1 is equal to S2, the specimenis under mode I, independent of a/R. For a specific position of S1and S2, the specimen is subjected to mode II.

The values of An and Bn in Eq. (3) are usually presented in nor-malized form as follow:

A1n(S1/R, S2/R, a/R) = Rt

P√

aA1

A2n(S1/R, S2/R, a/R) = Rt

PA2

A3n(S1/R, S2/R, a/R) = Rt√

a

PA3

B1n(S1/R, S2/R, a/R) = Rt

P√

aB1

B3n(S1/R, S2/R, a/R) = Rt√

a

PB3

(5)

where A1n, A2n, A3n, B1n, and B3n are mode I and mode II geome-try factors and t is the thickness of specimen. A1n, A2n, and B1n canbe extracted directly from ABAQUS software, and other parametersfor the ASCB specimen, i.e. A3n and B3n can be calculated by usingfinite element over deterministic (FEOD) method which is a numer-ical method for computing fracture parameters [27]. This methodutilizes the displacement/stress values of many nodes around thecrack tip obtained from finite element software and replaces themin the displacement/stress equations. By solving the acquired equa-tion set, the required parameters can be obtained.

Fig. 3 illustrates the typical mesh generated using eight-nodedplane strain elements for simulating the ASCB specimen. In the

models, the following geometry and loading configurations wereconsidered: R = 50 mm, t = 20 mm, a = 15 mm, and P = 1000 N. S1 wasset at a fixed value of 20 mm and S2 was varied from 6 mm to 20 mmin order to change the status of mode mixity. Since the Young’smodulus E and Poisson’s ratio � do not affect the values of fracture

5626 H. Saghafi et al. / Materials Science and Engineering A 527 (2010) 5624–5630

Fpm

p7sis

ig. 1. Experimental values of (���c at Me)/(���c at Me = 1) versus Me reported for: (a)olymethylmethacrylate (PMMA) [20], and (b) Johnstone rock tested under mixed-ode I/II loading [16].

arameters in 2D modeling [28], they are assumed to be equal to

0 GPa and 0.3, respectively. Also, a square root singularity in thetress/strain field was produced around the crack tip by consider-ng quarter point scaling between the circumferential rows of nodesurrounding the crack tip.

Fig. 3. A typical finite element mesh pattern u

Fig. 2. The asymmetric semi-circular bend (ASCB) specimen.

The combination of A1n and B1n determines the state of mixed-mode, i.e. pure mode I and pure mode II are obtained when B1n = 0and A1n = 0, respectively, and mixed-mode loading is obtainedwhen both of them are none-zero. As it can be seen from Fig. 4,when S2/R = 0.12 the specimen is subjected to pure mode II. By ris-ing S2/R, the effect of mode I increases and the effect of mode IIdecreases until S1 = S2 where pure mode I prevails. According toFig. 4, rising S2/R increases A2n coefficient but decreases A3n coef-ficient, and for B3n, firstly, there is a decrease until S2/R = 0.26 andafter that it increases.

4. Experimental program

In this section, the critical values of mode I and mode II stressintensity factors, KIc and KIIc, and tangential stress (���c) of Nei-riz rock are determined. This rock is a kind of marble which isexcavated from Fars province mines in Iran. Primary studies provethat the structure of this marble rock is relatively homogenousand isotropic. A total number of 11 ASCB specimens were preparedfor conducting the fracture tests. Geometrical dimensions of spec-

imens correspond with those mentioned in the last section, i.e.R = 50 mm, a = 15 mm, t = 20 mm, S1/R = 0.4, and S2/R = {0.12 (puremode II), 0.2, 0.26, 0.4 (pure mode I)}. The vertical crack generationin the semi-circular disk centre was performed as follows; firstly,

sed for simulating ASCB specimen [19].

H. Saghafi et al. / Materials Science and Engineering A 527 (2010) 5624–5630 5627

d B3n

antttcrt

f

�

Fig. 4. Variation of dimensionless parameters A1n , A2n , A3n , B1n , an

very thin fret saw blade of 0.4 mm width was used to generate aotch and then the notch tip was sharpened using a saw blade withhe thickness of 0.1 mm. Then, by utilizing a three-point bend fix-ure according to desired S1 and S2, the specimens were subjectedo a compressive load under a constant rate of 0.5 mm/min untilollapse due to fracture. The load–displacement curves for all testsesulted straight lines indicating the brittle fracture behavior of theested rock material. The loading setup is shown in Fig. 5.

According to Eq. (3), critical values of tangential stress in eachracture mode are obtained using:

��c ={

34

rc−0.5A1f

(cos

�0

2+ 1

3cos

3�0

2

)− 3

4r−0.5B1f

(sin

�0

2+ sin

3�0

2

)}

+ {4A2f(sin2 �0)} +{

154

r0.5A3f

(cos

�0

2− 1

5cos

5�0

2

)

+ 154

r0.5B3f

(sin

�0

2− sin

5�0

2

)}+ O(r) (6)

with S2/R for constant S1/R = 0.4 and a/R = 0.33 in ASCB specimen.

where Anf and Bnf can be computed by replacing the fracture load ofthe specimen (Pcr) in Eq. (5). A few theoretical methods have beensuggested previously to estimate the rc value. For instance, basedon the maximum normal stress theory [29], rc for rock materialscan be calculated as follows:

rc = 12�

(KIc

�t

)(7)

where �t is the tensile strength of rock acquired by using uncrackedBrazilian disk subjected to compressive load [17] and KIc is deter-mined by averaging the results obtained from the pure mode I

tests. In the case of Neiriz marble rock, the value of �t is obtained7.71 MPa, as an average of three uncracked BD specimen testresults.

The critical values of stress intensity factors KIf, KIIf and Tf cor-responding to the onset of fracture for the ASCB specimen, can be

5628 H. Saghafi et al. / Materials Science and Engineering A 527 (2010) 5624–5630

F

w

K

v(Iac

n

Ft

TS

ig. 5. Loading set-up utilized for conducting fracture tests in ASCB specimen.

ritten as:

If = Pcr

RtA1n

√2�a, KIIf = Pcr

RtB1n

√2�a, Tf = Pcr

RtA2n (8)

Table 1 illustrates the details of test parameters including thealues of S2, Pcr and the corresponding KIf, KIIf, and ���c using one�1

��c), two (�2

��c), and three (�3

��c) terms of Eq. (6) from pure mode

to pure mode II. These terms are separated from each other byccolade in Eq. (6). Using both Eq. (7) and Table 1, KIc and rc arealculated to be about 1.528 MPa

√m and 6.25 mm, respectively.

The critical tangential stress results are usually presented inormalized form. Therefore, all critical tangential stress results are

ig. 6. Critical tangential stress of Neiriz marble rock obtained by using differenterm numbers of Eq. (6).

able 1ummery of fracture tests conducted on ASCB specimens manufactured from Neiriz rock

Specimen no. Me S2 (mm) Pcr (kN) �◦0 KIf (MPa

√m) KIIf (MP

1 1 40 7.488 0 1.470 02 1 40 8.066 0 1.585 03 1 40 7.779 0 1.529 04 0.76 13 8.819 16 1.058 0.4145 0.76 13 9.880 19 1.185 0.4656 0.76 13 10.157 15 1.218 0.4787 0.46 10 10.218 17 0.760 0.8488 0.46 10 7.800 14 0.611 0.6829 0 6 12.607 11 0 1.936

10 0 6 10.400 9 0 1.59011 0 6 10.410 10 0 1.600

Fig. 7. Theoretical prediction of the MTS criterion for mixed-mode fracture for ASCBspecimen manufactured from Neiriz marble rock.

normalized to the average of �3��c

at pure mode I which is about5.01 MPa for Neiriz marble rock. These results are illustrated inFig. 6 for various combinations of modes I and II in the form of(���c at Me)/(�3

��cat Me = 1) versus Me.

5. Discussion

Until recently, researchers have only used one or two terms ofEq. (6) to study fracture behavior in laboratory specimens. But in

Fig. 8. Prediction of mixed-mode fracture resistance by MMTS criterion comparedto MTS for Neiriz marble rock.

.

a√

m) �1��c

(1 term) (MPa) �2��c

(2 terms) (MPa) �3��c

(3 terms) (MPa)

7.43 7.43 4.818.00 8.00 5.207.72 7.72 5.016.04 5.58 4.776.86 6.15 5.276.92 6.46 5.505.38 4.67 5.384.25 3.76 4.322.78 1.94 5.501.88 1.42 4.522.09 1.51 4.54

nd En

ttoasntr

weit

oirsMitsdscvmiottti

6

memcct�v

0

2+

3

(−

�0

2

)

3f

Ic

(

H. Saghafi et al. / Materials Science a

he case of the specimen and material studied in this work, thehird term should also be included. As shown in Fig. 6, utilizing oner two terms of Eq. (6) to compute ���c results in a high discrep-ncy between the theoretical and experimental data. Also, in thistate, ���c is not the same for all fracture modes. While, despite aatural scatter in the experimental results, by using three terms,his condition is satisfied. This is because: (1) rc for Neiriz rock iselatively great, so the effect of third term including r0.5 on ���c

ill be more obvious. (2) The coefficients of A3n and B3n are greatnough in ASCB specimen relative to the other common test spec-mens such as BD [30]. Accordingly, a new criterion is presentedhat is more accurate compared with other common criteria.

Many criteria have been suggested to predict mixed-mode I/IIr pure mode II brittle fracture in engineering materials. The max-mum tangential stress (MTS) criterion [26], the maximum energyelease rate (G) criterion [31] and the minimum strain energy den-ity (SED) criterion [32] are three popular criteria. Among them, theTS criterion has attracted more attention to researchers. The max-

mum tangential stress in this criterion is computed only by usinghe first term of Eq. (6). Fig. 7 depicts the experimental results pre-ented in Table 1 as KIf/KIc versus KIIf/KIc. As shown, there is a highifference between the experimental and the theoretical resultsuggested by the MTS criterion. In the case of mixed modes, thisriterion predicts greater values around the pure mode I and loweralues around pure mode II, relative to the experimental results. Asentioned above, by using three terms of Eq. (6), ���c is the same

n all mode mixities. Therefore, it can be concluded that insteadf using one term of Eq. (6) as in the case of MTS criterion, threeerms should be taken into account to improve the predictions ofhis criterion. In the next section, the modified MTS criterion abilityo predict the mixed-mode fracture toughness of brittle materialss assessed by using the ASCB specimen.

. Modified MTS criterion (MMTS)

In this section, the proposed theoretical estimation of mixed-ode I/II fracture toughness in ASCB specimens is outlined. The

lastic tangential stress distribution around a crack tip underixed-mode I/II loading is presented in Eq. (3). The modified MTS

riterion proposes that crack growth initiates radially from therack tip along the direction of maximum tangential stress �0. Also,he crack extension takes place when the tangential stress ��� along0 and at a critical distance rc from the crack tip attains a criticalalue ���c.

∂���

∂�| = 0

⇒ −34

r−0.5c

[A1

(12

sin�

+ 4A2 sin 2�0 + 154

r0.5c

[A

r−0.5c

1√2�

+ 3r0.5c

A3f

KIc=

{34

r−0.5c

A1f

KIc

(cos

�0

2+ 1

3cos

3

+{

4A2f

KIc(sin2 �0)

}+

{154

r0.5c

A

K

KIf

KIc=

r−0.5c + 3r0.5

c (A3n/aA1n) + (3/4)r−0.5c (B1n/A1n)(s

− (15/4)r0.5c (A3n/aA1n)(cos(�0/2) − (1/5) cos

(3/4)r−0.5c (cos(�

gineering A 527 (2010) 5624–5630 5629

While the conventional MTS criterion only uses the singularterm of stress in Eq. (3), the modified MTS criterion takes intoaccount the first three terms to calculate the direction of crackextension and also to determine the fracture resistance of crackedcomponents under mixed-mode I/II loading.

Based on the modified MTS criterion, the direction of fractureinitiation angle �0 will be realized from:

12

sin3�0

2

)+ B1

(12

cos�0

2+ 3

2cos

3�0

2

)]

12

sin�0

2+ 1

2sin

5�0

2

)+ B3

(12

cos�0

2− 5

2cos

5�0

2

)]= 0

(9)

The �0 angle determined from Eq. (9) is then used to predict theonset of mixed-mode fracture. According to the MMTS criterion,brittle fracture takes place when:

���(rc, �0) = ���c (10)

By replacing the �0 angle from Eq. (9) into Eq. (10), one canderive:

���c = 34

r−0.5c A1f

(cos

�0

2+ 1

3cos

3�0

2

)− 3

4r−0.5

c B1f

(sin

�0

2+ sin

3�0

2

)

+ 4A2f(sin2 �0) + 154

r0.5c A3f

(cos

�0

2− 1

5cos

5�0

2

)

+ 154

r0.5c B3f

(sin

�0

2− sin

5�0

2

)(11)

Eq. (11) can be used for pure mode I, pure mode II and any combi-nations of mode I and mode II. For mode I, brittle fracture in whichKIIf = 0, �0 = 0 and KIf = KIc, Eq. (11) yields to:

���c = r−0.5c A1f + 3r0.5

c A3f = r−0.5c

KIc√2�

+ 3r0.5c A3f (12)

Replacing Eq. (12) into Eq. (11) results in:

r−0.5c

KIc√2�

+ 3r0.5c A3f = 3

4r−0.5c A1f

(cos

�0

2+ 1

3cos

3�0

2

)

− 34

r−0.5c B1f

(sin

�0

2+ sin

3�0

2

)

+ 4A2f(sin2 �0)

+ 154

r0.5c A3f

(cos

�0

2− 1

5cos

5�0

2

)

+ 154

r0.5c B3f

(sin

�0

2− sin

5�0

2

)(13)

If both sides of Eq. (13) are divided by KIc, the outcome wouldbe:

− 34

r−0.5c

B1f

KIc

(sin

�0

2+ sin

3�0

2

)}

cos�0

2− 1

5cos

5�0

2

)+ 15

4r0.5c

B3f

KIc

(sin

�0

2− sin

5�0

2

)}(14)

By using Eqs. (5), (4) and (8), Eq. (14) simplifies to:

in(�0/2) + sin(3�0/2)) − 4(A2n/√

aA1n)(sin2 �0)(5�0/2)) − (15/4)r0.5

c (B3n/aA1n)(sin(�0/2) − sin(5�0/2))

0/2) + (1/3) cos(3�0/2))(15)

5 and En

(3�015/4

in(3

IupftdWtioMwia

7

mTuwecumista

A

af

[[[[[

[[

[[

[[[[[[[[[

[[

630 H. Saghafi et al. / Materials Science

Using a similar procedure, the ratio KIIf/KIc is derived as:

KIIf

KIc=

rc−0.5 + 3rc

0.5(A3n/aA1n) − (3/4)rc−0.5(cos(�0/2) + (1/3) cos

− (15/4)rc0.5(A3n/aA1n)(cos(�0/2) − (1/5) cos(5�0/2)) − (

−(3/4)rc−0.5(sin(�0/2) + s

Now, by conducting mode I fracture test and finding KIc, modeI and mixed-mode fracture toughness can be easily determined bysing Eqs. (15) and (16). Fig. 8 illustrates the theoretical resultsredicted by MMTS criterion for Neiriz marble rock. It is seenrom these curves that MMTS criterion predicts the experimen-al results better than the conventional MTS criterion. The mostiscrepancy between MTS and MMTS criteria is in pure mode II.hile the MTS criterion predicts the value of 0.87 for KIIf/KIc in

his loading state, which is 24% lower than the obtained exper-mental data, MMTS criterion predicts this value as 1.15 which isnly 1% lower. For mixed-mode loading, the curve corresponding toMTS criterion (the dashed green curve) shows better agreementith the test results. (For interpretation of the references to color

n this sentence, the reader is referred to the web version of therticle.)

. Conclusion

In this research, a new criterion was proposed to predict theode II and mixed-mode I/II fracture toughness of brittle materials.

he mixed-mode fracture test was conducted on Neiriz marble rocksing ASCB specimen. The values of maximum tangential stress ���cere calculated using different numbers of terms in the infinite

xpansion series representing the tangential stress distribution inrack tip vicinity in different mode mixities. It was found that bytilizing the first three terms, ���c will be the same in all fractureodes and this condition will be satisfied. Accordingly, a mod-

fied MTS (MMTS) criterion was presented which describes thetress field around the crack tip, more accurately. The experimen-al results obtained from mixed-mode fracture tests were in a goodgreement with the proposed criterion.

cknowledgements

The authors would like to thank Dr M.R.M. Aliha for his technicaldvice. The first author would also like to thank Mr. S.A. Monemianor his assistance in the technical writing of this paper.

[

[

[

gineering A 527 (2010) 5624–5630

/2)) − 4(A2n/√

aA1n)(sin2 �0))rc

0.5(B3n/aA1n)(sin(�0/2) − sin(5�0/2))

�0/2))(16)

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