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Mixed mode fracture analysis using extended maximum tangential strain criterion
Mixed mode fracture analysis using extended maximum tangential strain criterion M.M. Mirsayar PII: DOI: Reference:
S0264-1275(15)30204-5 doi: 10.1016/j.matdes.2015.07.135 JMADE 363
To appear in: Received date: Revised date: Accepted date:
5 January 2015 15 July 2015 24 July 2015
Please cite this article as: M.M. Mirsayar, Mixed mode fracture analysis using extended maximum tangential strain criterion, (2015), doi: 10.1016/j.matdes.2015.07.135
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Mixed mode fracture analysis using extended maximum tangential strain criterion
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* Corresponding author: M. M. Mirsayar, E-mail address: [email protected], Tel.: +1 (979) 7776056
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Abstract
The maximum tangential strain (MTSN) criterion is modified in this paper to predict fracture
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initiation in cracked specimens under mixed mode loading conditions. Theoretical investigations were performed to study the role of both first nonsingular strain term (T – strain) and Poisson’s
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ratio on prediction of the crack propagation direction and fracture toughness. The extended
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version of the MTSN (EMTSN) criterion is proposed by taking into account the T – strain as
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well as the singular strain field. The EMTSN criterion is then examined using the experimental data reported for angled cracked plate made from polymethylmethacrylate (PMMA). It is shown that the EMTSN criterion provides more accurate estimation of the experimental results than conventional MTSN criterion.
Keywords: Mixed mode fracture; maximum tangential strain criterion; Poisson’s ratio; Tstrain
1. Introduction Brittle fracture is a major concern in many engineering structures made from a wide range of materials such as ceramics, concrete, rocks, glasses and cold metals. Several fracture criteria are proposed so far to understand fracture mechanism and to predict onset of fracture in brittle materials. Among the stress based criteria, the maximum tensile stress (MTS) criterion proposed 1
ACCEPTED MANUSCRIPT by Erdogan and Sih [1] has widely been accepted by the researchers during the past decades for brittle materials. The maximum energy release rate (G) criterion developed by Hussain et al. [2]
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and minimum strain energy density (SED) introduced by Sih [3] are also among the popular
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energy based fracture criteria. One of the most important advantages of the stress – based criteria than energy based ones is the simplicity and representing a clear role for each stress field
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parameter in estimation of the onset of fracture. However, the energy – based criteria employ
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more realistic assumptions for the crack propagation phenomenon by taking into consideration the effect of energy dissipation during fracture. Nevertheless, both stress and energy based
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criteria were subjected to some modifications during the past decades to provide a better
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explanation of the onset of fracture in different loading and boundary conditions [4 – 7].
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A literature review reveals that, in some materials, the strain based criteria provides more realistic description of the failure mechanism than stress and energy based criteria [8 – 10]. For
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instance, based on the observation reported by Nalla et al. [8], the fracture mechanism in bone could properly be explained using a strain based criterion. In 1974, Wu [11], pointed out that a meaningful description of the fracture of concrete must be based on strain and developed a strain based fracture criterion. Based on his suggestion, fracture occurs when a scalar – valued function of strain tensor reaches a critical value. Inspired from this work and the maximum normal strain theory of St–Venant [12], Chang [13] developed the maximum tangential strain (MTSN) criterion to study mixed mode fracture initiation in angled crack problem. Based on the MTSN criterion, crack propagates in the direction where the tangential strain,
, reaches its maximum
value at a critical distance, rc, from the crack tip. According to linear elastic fracture mechanics (LEFM) assumption, the stress, strain and displacement fields around the crack tip could be represented in terms of a series expansion containing singular terms (corresponding to the stress 2
ACCEPTED MANUSCRIPT intensity factors) and higher order terms [14]. The criterion, suggested by Chang [13] only uses the singular strain field and does not take into account the effect of higher order terms. Although,
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the criterion provides a very simple and realistic description of the brittle fracture phenomenon,
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it was not successful to precisely estimate fracture test data in some geometries and boundary
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conditions.
In agreement with the pioneering work done by Irwin [15], recent studies on cracked and notches
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components reveals that higher order terms of the Williams [14] series expansion may
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significantly influence on stress/ strain distribution around the crack tip [16 – 25]. Among the higher order terms, the first nonsingular one was recognized as the most effective term in
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evaluation of the fracture initiation as well as stress/ strain distribution around the crack tip. In
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case of bi-material cracks and utilizing a stress based criterion, Mirsayar [16] showed that sometimes the T-stress (the first nonsingular stress term around crack tip) significantly
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influences on the stress distribution around the crack tip and neglecting T-stress may cause a considerable error in estimation of the fracture strength. However, the effect of first nonsingular term of the Williams series expansion on the fracture initiation has not been studied yet using a strain based fracture criterion.
This paper deals with the role of both Poisson’s ratio and the first nonsingular strain term (T – strain) on the fracture initiation using MTSN criterion. The effect of T – strain and Poisson’s ratio on the fracture propagation direction as well as the fracture strength is studied theoretically. The conventional MTSN criterion is extended by taking into account the T – strain term as well as the singular strain field. The extended MTSN (EMTSN) criterion is then employed to predict experimentally reported data for fracture propagation direction and fracture strength of some angled cracked plate made from polymethylmethacrylate (PMMA). The EMTSN predictions are 3
ACCEPTED MANUSCRIPT also compared with the conventional MTSN criterion. It is shown that the EMTSN criterion provides more accurate prediction of the experimental data than conventional MTSN criterion.
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2. Tangential strain based criteria
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2.1. Conventional MTSN criterion
, reaches its maximum value, at a critical distance, rc, from the crack tip. The
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strain,
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The MTSN criterion states that crack propagates in the direction, 0, where the tangential
criterion could simply be represented as:
(rc , 0 ) T T / E
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(1)
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0 r r ; c 0 2 0 2 r r ; c 0
(2)
Where, T is the tensile fracture strength and E is Young’s modulus of the material. The critical distance rc could be considered as damaged zone around the crack tip which is a material property [26, 27]. The linear elastic relationship between the tangential strain and stress components is:
1 ( rr ) E
(3)
4
ACCEPTED MANUSCRIPT Where is the Poisson’s ratio and and rr are respectively the tangential and radial components of the stress field in cylindrical coordinates with origin at the crack tip, shown in
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Fig. 1.
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The MTSN criterion claims that fracture initiation depends on not only the geometry and boundary conditions but also the Poisson’s ratio, , of the material. It is clear from Eq. (3) that
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the predictions based on MTSN criterion coincide with those based on MTS criterion
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proposed by Erdogan and Sih [1] when = 0. Based on the LEFM assumption, the stress field
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around the crack tip could be expressed in the form of a series expansion with infinite terms, given in Eqs. (4) – (6) [14].
KI K II f ,1 ( ) f , 2 ( ) T sin 2 ( ) O(r 1/ 2 ) ... 2r 2r
KI f r ,1 ( ) 2r KI rr f rr,1 ( ) 2r
r
K II f r , 2 ( ) T sin( ) cos( ) O(r 1 / 2 ) ... 2r K II f rr, 2 ( ) T cos 2 ( ) O(r 1 / 2 ) ... 2r
(4)
(5)
(6)
where (i, j) ≡ (r,) are the polar coordinates with the origin at the crack tip (see Fig. 1) and the functions fij,n() (ij ≡ r, ; n ≡1,2) are given in Appendix A. The coefficients KI and KII and T
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ACCEPTED MANUSCRIPT are the stress intensity factors corresponding to mode I (opening) and mode II (sliding) deformations and the T – stress, respectively. The first and the second terms represent the 1 ) and hence, the T – stress is the first r
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singular stress field (when r 0
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nonsingular stress term. Substituting Eq. (4) and (5) into the Eq. (3) and taking into account
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the only singular stress terms (associated with the KI and KII), the conventional MTSN could
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be represented as follows:
,1 ( 0 ) f rr ,1 ( 0 )
E 2rc
f
, 2 ( 0 ) f rr , 2 ( 0 ) T
T
(8)
E
d ( f ij,n ( )) d 0
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Where f 'ij,n ( 0 )
K II
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E 2rc
f
(7)
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KI
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KI f ' ,1 (0 ) f 'rr,1 (0 ) K II f ' ,2 (0 ) f 'rr,2 (0 ) 0 0 E 2rc E 2rc
Substituting fracture initiation angle, 0, calculated from Eq. (7), one can find the onset of fracture when the left hand side of the Eq. (8) reaches the tensile fracture strength.
2.2. Modified MTSN criterion The left hand side of the Eq. (7) and (8) are controlled by the values of the stress intensity factors. Retaining singular stress terms and adding the T – stress into the Eq. (3), the Eq. (7) and (8) could be rewritten as follows:
KI f ' ,1 (0 ) f 'rr,1 (0 ) K II f ' ,2 (0 ) f 'rr,2 (0 ) (1 ) T sin( 20 ) 0 0 E E 2rc E 2rc
(9)
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ACCEPTED MANUSCRIPT KI E 2rc
f
,1
( 0 ) f rr,1 ( 0 )
K II E 2rc
f
,2
( 0 ) f rr, 2 ( 0 )
T sin 2 ( 0 ) cos 2 ( 0 ) T E E
(10)
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In Eq. (10), the first nonsingular tangential strain term corresponding to parameter T is called
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tangential T – strain. Although the parameter T is identical with the well – known T – stress,
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the first nonsingular tangential strain term (or tangential T – strain) is the matter of concern in this paper. It needs to be mentioned that the radial and shear T – strains are associated with
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the first nonsingular radial and shear components of the strain, respectively.
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To simplify the Eqs. (9) and (10), the biaxiality ratio B, relative to the stress intensity factors,
T a K eff
(11)
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B
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could be represented as [28]:
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Where a is the crack length for edge cracks and semi-crack length for center cracks. The 2 2 effective stress intensity factor Keff is also defined K eff ( K I ) ( K II )
Defining the normalized critical distance 2rc / a , the direction of the crack propagation could be determined from the Eq. (12) KI f ' ,1 ( 0 ) f 'rr,1 ( 0 ) K II f ' ,2 ( 0 ) f 'rr,2 ( 0 ) (1 ) B sin( 2 0 ) 0 0 K eff K eff
(12)
The onset of the crack propagation could also be found from:
K I f ,1 (0 ) f rr,1 (0 ) K II f , 2 (0 ) f rr, 2 (0 ) T 2rc BKeff sin 2 (0 ) cos 2 (0 )
(13)
Equations (12) and (13) describe the extended MTSN criterion where the mixed mode fracture is predicted for any geometry and material when KI, KII, T and are known. The 7
ACCEPTED MANUSCRIPT effect of Poisson’s ratio, in absence of the tangential T – strain, on the fracture initiation angle is illustrated in Fig. 2 for different values of mixity parameter Me given by
tan 1 (
KI ) K II
(14)
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2
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Me
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The value of Me represents the participation of each fracture mode and varies from zero (pure mode II) to unity (pure mode I). Regardless of sign of the numbers on the vertical axis, the
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absolute values are considered as the fracture initiation angle.
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As shown in Fig. 2, all curves coincide together at Me = 0.6, 0 - 47o. For 0 < Me < 0.6 and
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0.6 < Me < 1, the MTSN criterion provides lower and higher predictions of the fracture
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initiation angle than MTS criterion ( = 0) [1] respectively. In pure mode I, the fracture
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initiation angle is predicted to be zero for all values of the Poisson’s ratio. Also, the fracture initiation angle decreases for Me less than 0.6 and conversely increases for Me > 0.6 by increasing the Poisson’s ratio. From Fig. 2, the highest effect of Poisson’s ratio on the fracture initiation angle is found to be in pure mode II and the no effect is observed for Me = 0.6 and pure mode I (Me = 1).
The effect of dimensionless parameter B on the fracture initiation angle is illustrated in Fig. 3. The parameter B presents the effects of both tangential T – strain and the critical distance rc. For a constant value of Poisson’s ratio ( = 0.35), the fracture initiation angle is plotted versus M e in different values of B ranging from -0.4 to 0.3. It is seen that the tangential T – strain significantly influences on the fracture initiation angle in different mixed mode conditions, except for pure mode I. Also, it is shown that for positive values of T – strain, the 8
ACCEPTED MANUSCRIPT modified MTSN criterion (solid line) provides higher estimates for fracture initiation angle and vice versa. From Figs. 2 and 3, it is concluded that both T – strain and Poisson’s ratio
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could significantly influence on the fracture initiation angle in pure mode II condition and no
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effect is observed for pure mode I.
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< Fig. 3. >
Figs. 4 and 5 illustrate the effects of Poisson ratio and T – strain on the fracture toughness
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respectively. The plots are presented in terms of the normalized stress intensity factors K I/
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K*IC and KII/ K*IC. The parameter K*IC, called generalized fracture toughness, is defined as
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Eq. (15) and takes into account the effect of both Poisson’s ratio and specimen geometry (B) in calculation of the mode I fracture toughness. (see Appendix B for more detail):
* K IC
T 2rc 1 (1 B )
(15)
As shown in Figs. 4 and 5, both Poisson’s ratio and T – strain remarkably affect on the mixed mode fracture loci. Most notable about Fig. 4 is that the MMTS criterion provides more conservative predictions of the onset of fracture than MTS criterion [1]. In other words, the mixed more fracture locus becomes more limited by increasing the value of Poisson’s ratio. The effect of tangential T – strain, on the fracture toughness is shown in Fig. 5 for a constant value of Poisson’s ratio ( = 0.35). The predicted fracture toughness increases for negative
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ACCEPTED MANUSCRIPT values of B and conversely decreases for positive values of B. Moreover, a significant
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< Fig. 4. >
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influence of the tangential T – strain could be seen on the mode II fracture toughness.
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< Fig. 5. >
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Figs. 3 and 5 represent the extended MTSN criterion for mixed mode fracture problems. In order to use these fracture diagrams, the crack tip parameters KI, KII and T must be known for
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a given specimen. The stress intensity factors as well as the T – strain could be obtained
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through finite element analysis or using handbooks [29] for simple geometries. The mixity
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parameter Me and the biaxiality ratio B are then calculated using equations 14 and 11. The normalized critical distance 2rc / a could also be determined for a specific crack length and material property [26, 27]. Calculating the parameters Me and B, one can find the corresponding fracture initiation angle from Fig. 3. Finally, by calculating the generalized fracture toughness K*IC, the onset of fracture could be determined using the same data from Fig. 5.
3. Results and discussion The extended MTSN criterion is examined here using the experimental data available in the literature for PMMA. Williams and Ewing [30], and Ueda et al. [31] studied the fracture initiation in PMMA using square plates containing an angled crack. The general configuration of the specimen is shown in Fig. 6 under biaxial loading. As shown, the plate is subjected to 10
ACCEPTED MANUSCRIPT uniform far-field stresses, and ( is the lateral load ratio) in the vertical and horizontal directions respectively. The angle between the crack line and the vertical direction, , is changed
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to make different mixed mode conditions from pure mode I to pure mode II. The specimens used
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by Williams and Ewing [30] were plates 152 mm wide, 305 mm long, 3.2 mm thick, containing
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angled cracks of length 2a varying from 14 to 50 mm. The plate employed by Ueda et al. [31] also were, 100 mm wide, 100 mm long,4 mm thick, containing angled cracks of length 2a equal
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to 40 mm.
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< Fig. 6. >
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Although the finite element method could be performed for any given geometry and boundary conditions to calculate crack tip parameters, it still would be easier to use closed form solution
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when dealing with the simple geometries. Smith et al. [25] suggested to use the following closed form solution, given by Sih et al. [32], to calculate stress intensity factors as well as the parameter T.
K I a ( cos 2 sin 2 )
(16)
K II a (1 ) cos sin
(17)
T (1 ) cos 2
(18)
It needs to be mentioned that although the equations 16 – 18 are only valid when the crack length is small compared with the size of the plate, Smith et al. [25] showed that these equations could successfully be employed for specimens tested by Williams and Ewing [30] and Ueda et al. [31]. 11
ACCEPTED MANUSCRIPT The specimens tested by Williams and Ewing [30] and Ueda et al. [31] were subjected to uniaxial loading ( = 0). It easily could be found from equations 16 – 18 that the pure mode I
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would be obtained when = 90°. For =0° the stress intensity factors approach zero while the
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parameter T reaches its maximum value. In other words, for small values of , the tangential T –
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strain plays the most important role in fracture event and there is no effect for the singular stress field.
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According to Eq. (18) the parameter T is positive within the range of and is negative
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when . Therefore, the fracture initiation angles predicted by EMTSN criterion are expected to be higher and lower than MTSN criterion for andrespectively.
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Also, it is expected from EMTSN criterion to provide lower and higher predictions of the
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fracture toughness than MTSN criterion for andrespectively.
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Fig. 7 illustrates the fracture initiation angles reported by Williams and Ewing [30] and Ueda et al. [31] for PMMA in different mode mixities. The predictions provided by the conventional MTSN and the EMTSN criteria are also presented. The non – dimensional characteristic distance, , of 0.2 is selected for the analysis according to Williams and Ewing [30] and Smith et al. [25]. As discussed before, the parameter T is positive for 0 < Me < 0.5 (0° < < 45°). Therefore, the fracture initiation angles predicted by EMTSN criterion is higher than those of estimated by the conventional MTSN criterion. Conversely, for 0.5 < Me < 1.0 (45° < < 90°), the fracture initiation angles predicted by EMTSN is less than those of estimated by the conventional MTSN criterion because of the negative values of parameter T. The significant effect of tangential T – strain on the fracture initiation angle is shown to exist under mixed mode conditions, especially in pure mode II.
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ACCEPTED MANUSCRIPT < Fig. 7. >
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The combined experimental data for the fracture toughness from Williams and Ewing [30] and Ueda et al. [31] are illustrated in Fig. 8 in comparison with the predictions provided by MTSN
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and EMTSN criteria. Same non – dimensional characteristic distance, , of 0.2 is assumed for fracture toughness assessments. The generalized fracture toughness is calculated for each crack
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angle, , by obtaining B as function of (equations 11 and 18), and the associated fracture load
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(T).
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< Fig. 8. >
As shown in Figs. 7 and 8, the substantial scatter is seen for the experimental data in all crack
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angles. These scatters may be explained if the higher order terms (such as order of r0.5 etc.) be considered in stress/ strain field when applying EMTSN criterion. Chao and Zhang [33] showed that such scatter under pure mode I , in fracture loci diagrams, are as a result of the effect of second nonsingular stress term (corresponding to the order of r0.5). However, it could be perceived from the literature that, for many laboratory specimens, only considering the effect of first nonsingular term of the Williams series expansion is enough to achieve a satisfactory accuracy in prediction of the fracture initiation and no higher order terms are needed [16, 18, 23 – 25]. It is also worth to compare the tangential strain based fracture criterion, developed in this paper, with the some popular tangential stress based fracture criteria. Among the tangential stress based fracture criteria, Ayatollahi and coworkers have employed the GMTS criterion [25] (using KI,
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ACCEPTED MANUSCRIPT KII, T and rc parameters) to investigate mixed mode brittle fracture behavior of materials [34 – 41]. Fig. 9 compares the predictions obtained from EMTSN criterion together with GMTS
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criterion for mixed mode fracture toughness of PMMA material, tested by Williams and Ewing [30] and Ueda et al. [31]. In order to represent a direct visual comparison, both criteria together
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with the experimental data are plotted in KI/KIC – KII/KIC space. It can be seen that EMTSN
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criterion presents better predictions than GMTS criterion for mixed mode fracture toughness,
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especially near mode II condition (for KI/KIC less than 0.3). Looking through the literature reveals that although the GMTS criterion successfully predicts the fracture toughness as well as
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the size effects under mixed mode conditions [34 – 36], it fails to provide an acceptable predictions in pure mode II (or predominantly pure mode II) condition for many brittle materials,
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such as ceramics, resins and rocks [37 – 39]. In many cases, the second nonsingular stress term and even first five terms of the tangential stress field are used to predict the test results and the
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GMTS criterion could not estimate the experimental data in a good accuracy [40, 41]. In these cases, considering a large number of tangential stress terms will lead to a complex formulation of the tangential stress based fracture criteria. Also, the higher order term could not be obtained as straightforward as stress intensity factors and the T – stress and sometimes, it is hard to obtain these parameters for a given specimen. Also, according to GMTS criterion, the T – stress has no effect on the fracture resistance for pure mode I condition. In other words, the GMTS states that the fracture toughness is identical for a specific material tested by different types of specimens. However, it is well – known that for a given material, different values of the mode I fracture toughness, KIC, could be obtained for different test specimens as a result of geometry effects. The effect of Poisson’s ratio is another important issue that is not considered in GMTS criterion. Using extensive 3-D finite element analyses, Aliha and Saghafi [42] recently showed that the
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ACCEPTED MANUSCRIPT stress intensity factors as well as the T – stress are significantly affected by the Poisson’s ratio of the material. As indicated previously, the effect of Poisson’s ratio as well as the specimen
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geometry is considered in EMTSN criterion. Indeed, the EMTSN criterion still needs to be
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examined for different brittle materials and specimen geometries. Further studies are needed to understand if the EMTSN criterion could cover the weakness of the stress based fracture criteria
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by taking into consideration of both Poisson’s ratio as well as the geometry effects.
4. Conclusion
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The extended version of the maximum tangential strain (EMTSN) criterion was introduced and discussed in this paper. The effects of Poisson’s ratio as well as the first nonsingular tangential
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strain term (tangential T – strain) were studied on the fracture initiation angle and the fracture toughness. The significant effects of both tangential T – strain and Poisson’s ratio on the fracture
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initiation angle as well as the fracture strength were shown to exist under mixed mode conditions. The EMTSN criterion was then developed by taking into account the effect of first nonsingular strain term of the tangential strain field well as the singular strain field. The proposed criterion was examined using the experimental data available in literature for the fracture toughness and the fracture initiation angles of the angled cracked plate made from PMMA. The EMTSN predictions were compared with the conventional MTSN criterion. It was shown that the EMTSN criterion provides more accurate prediction of experimental data than conventional MTSN criterion.
Appendix A. The stress field around the crack tip could be represented using the Eqs. (4 – 6) where frr,1(), frr,2(), f,1(), f,2(), fr,1(), fr,2() are given as follows: 15
ACCEPTED MANUSCRIPT (A – 1)
1 3 f rr, 2 ( ) [5 sin( ) 6 sin( )] 4 2 2
(A – 2)
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1 3 f rr,1 ( ) [5 cos( ) cos( )] 4 2 2
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1 3 f , 2 ( ) [3 sin( ) 3 sin( )] 4 2 2
(A – 4)
(A – 5)
(A – 6)
Appendix B.
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1 3 f r ,1 ( ) [sin( ) sin( )] 4 2 2
1 3 f r , 2 ( ) [cos( ) 3 cos( )] 4 2 2
(A – 3)
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1 3 f ,1 ( ) [3 cos( ) cos( )] 4 2 2
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Solution method for extended MTSN criterion Fracture initiation angle
Eq. (13) could be rewritten as:
KI K A II B K eff K eff
(B – 1)
where the coefficients A and B are
A
B
f ' f '
,2 ,1
( 0 ) f ' rr, 2 ( 0 )
(B – 2)
( 0 ) f ' rr,1 ( 0 )
B (1 ) sin( 2 0 ) f ' ,1 ( 0 ) f 'rr,1 ( 0 )
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ACCEPTED MANUSCRIPT 2 2 Recalling K eff ( K I ) ( K II ) and assuming = KII/KI, the Eq. (B1) can be written as:
1 2
(B – 3)
B
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1 2
A
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1
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or,
2A 1 B2 ( 2 ) 2 0 A B2 A B2
(B – 4)
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2
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The mixity parameter Me could then be calculated as follows:
1 2 A2 B 2 1 M tan tan 2 2 A B A B 1 2
1
(B – 5)
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e
Because the direction of fracture initiation 0 must be between −90° and 0°, only the negative sign in the numerator of Eq. (B – 5) is acceptable. Substituting A and B from Eq. (B – 2) into the Eq. (B – 5), a closed form solution would be achieved for the fracture initiation angle as a function of mode mixity.
Fracture loci Eq. (14) can be written as
1 1 2
C 1 (1 / ) 2
D / KI 1 2
H
(B – 6)
Where C, D and H are expressed as:
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ACCEPTED MANUSCRIPT C
f , 2 ( 0 ) f rr, 2 ( 0 )
(B – 7)
f ,1 ( 0 ) f rr,1 ( 0 ) (B – 8)
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sin 2 ( 0 ) cos 2 ( 0 ) f ,1 ( 0 ) f rr,1 ( 0 )
(B – 9)
T 2rc D gives K I K I [ f ,1 ( 0 ) f rr,1 ( 0 )]
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Solving the Eq. (B – 6) for , and recalling
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H B
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T 2rc D f ,1 ( 0 ) f rr,1 ( 0 )
(B – 10)
K II T 2rc [ f ,1 ( 0 ) f rr,1 ( 0 )][1 C H 1 2 ]
(B – 11)
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D
KI 1 T 2rc [ f ,1 ( 0 ) f rr,1 ( 0 )][1 C H 1 2 ]
According to the maximum tangential stress criterion [1, 24], for pure mode I when KII , T and 0 are equal to zero, KI at fracture can be replaced by the mode I fracture toughness KIC as follows:
K IC ,C 2rc
(B – 12)
Where ,C is the critical tangential stress which is assumed to be equal to T based on MTS criterion. According to MTSN criterion, the fracture occurs when (rr) reaches tensile fracture strength T. Based on an analogy with the MTS criterion and by substituting 0 = 0
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ACCEPTED MANUSCRIPT * and KII = 0 into the Eq. (10), the generalized fracture toughness, K IC , is defined as the
following for MTSN criterion;
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T 2rc K 1 (1 B )
(B – 13)
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* IC
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Eq. (B – 13) provides more general expression of the mode I fracture toughness for tangential strain based fracture criteria which takes into account the effect of both Poisson’s ratio and the
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specimen geometry. A comparison between Eqs. (B – 12) and (B – 13) reveals that these two
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equations are identical when = 0 or B . For tangential stress based fracture criteria (such as MTS [1] and GMTS [25]), there is no effect of first nonsingular stress term (B)
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when substituting 0 = 0 into the tangential stress component because the first nonsingular
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tangential stress is T sin 2 0 . However, from Eq. (13), it could be seen that because of
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dependency of the tangential strain component on Poisson’s ratio, the parameter B remains when 0 = 0 and as a result, the generalized fracture toughness should be used to normalize the fracture loci. It should be noted that the fracture loci could also be plotted in terms of the conventional fracture toughness, KIC. In that case, it is easy to show that the fracture loci do not coincide together at KI/KIC = 1. In other words, because of the effects of Poisson’s ratio and the specimen geometry (B), the EMTSN criterion provides different predictions of the mode I fracture toughness. While the conventional fracture toughness and stress based criteria (such as GMTS [25]) could not introduce the effect of specimen geometry in pure mode I condition. Substituting Eq. (B – 13) into the equations (B – 10) and (B – 11) leads to
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ACCEPTED MANUSCRIPT (B – 14)
K II 1 (1 B ) * K IC [ f ,1 ( 0 ) f rr,1 ( 0 )][1 C H 1 2 ]
(B – 15)
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KI 1 (1 B ) * K IC [ f ,1 ( 0 ) f rr,1 ( 0 )][1 C H 1 2 ]
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Equations (B – 14) and (B – 15) represent the fracture loci shown in Figs. 4 and 5 for different
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values of B and Poisson’s ratio.
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References
[1] Erdogan F, Sih GC. On the crack extension in plates under plane loading and transverse shear. J Basic Engng
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Trans ASME 1963; 85:525–7.
[2] Hussain MA, Pu SL, Underwood J. Strain energy release rate for a crack under combined mode I and Mode II.
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In: Fracture Analysis ASTM STP 560, American Society for Testing and Materials 1974; Philadelphia, PA, USA,
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2–28.
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[3] Sih GC. Strain energy density factor applied to mixed mode crack problems. Int J Fract 1974; 10:305–21. [4] Selvarathinam AS, Goree JG. T – stress based fracture model for cracks in isotropic materials. Eng Fract Mech 1998; 60(5-6):543-61.
[5] Lazzarin P, Campagnolo A, Berto F. A comparison among some recent energy- and stress-based criteria for the fracture assessment of sharp V-notched components under Mode I loading. Theoretical and Applied Fracture Mechanics 2014; 71:21–30. [6] Berto F, Lazzarin P. Recent developments in brittle and quasi-brittle failure assessment of engineering materials by means of local approaches. Materials Science and Engineering: R: Reports 2014;75:1–48. [7] Torabi AR, Pirhadi E. Stress-based criteria for brittle fracture in key-hole notches under mixed mode loading. European Journal of Mechanics - A/Solids 2015; 49:1–12. [8] Nalla RK, Kinney JH, Ritchie RO. Mechanistic fracture criteria for the failure of human cortical bone. Nature Materials 2003; 2:164 – 8. 20
ACCEPTED MANUSCRIPT [9] Maiti SK, Smith RA. Criteria for brittle fracture in biaxial tension. Eng Fract Mech 1984; 19(5): 793 – 804. [10] Wu X, Li X. Analysis and modification of fracture criteria for mixed – mode crack. Eng Fract Mech 1989;
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34(1): 55-64.
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[11] Wu HC. Dual failure criterion for plane concrete. J. Engng Mech. Div ASCE 1974;100(6):1167-81.
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[12] Timoshenko SP. History of Strength of Materials. New York: McGraw Hill, 1953. [13] Chang KJ. On the maximum strain criterion – a new approach to the angled crack problem. Eng Fract Mech
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1981; 14:107-24.
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[14] Williams ML. On the stress distribution at the base of a stationary crack. J Appl Mech 1957; 24:109–14. [15] Irwin GR. Analysis of stresses and strains near the end of a crack traversing a plate. J Appl Mech 1957; 24:
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361-364.
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[16] Mirsayar MM. On fracture of kinked interface cracks – The role of T-stress. Mater Des 2014; 61: 117–23.
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[17] Mirsayar MM, Aliha MRM, Samaei AT. On fracture initiation angle near bi-material notches – effect of first non-singular stress term. Eng Fract Mech 2014;119:124–31. [18] Mirsayar MM. A new mixed mode fracture test specimen covering positive and negative values of T-stress. Eng Solid Mech 2014;2(2):67–72.
[19] Ayatollahi MR, Mirsayar MM. Kinking angles for interface cracks. Procedia Eng 2011;10:325–9. [20] Ayatollahi MR, Dehghany M, Mirsayar MM. A comprehensive photoelastic study for mode I sharp V-notches. European J Mech – A/Solids 2013;37: 216–30. [21] Ayatollahi MR, Mirsayar MM, Nejati M. Evaluation of first non-singular stress term in bi-material notches. Comput Mater Sci 2010;50(2):752–60. [22] Ayatollahi MR, Mirsayar MM, Dehghany M. Experimental determination of stress field parameters in bimaterial notches using photoelasticity. Mater Des 2011;32(10):4901–8.
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ACCEPTED MANUSCRIPT [23] Shahani AR, Tabatabaei SA. Effect of T-stress on the fracture of a four point bend specimen. Mater Des 2009; 30: 2630–35
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[24] Li XF, Xu LR. T-stresses across static crack kinking. J Appl Mech 2007; 74(2):181-90. [25] Smith DJ, Ayatollahi MR, Pavier MJ. The role of T-stress in brittle fracture for linear elastic materials under
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mixed mode loading. Fatigue Fract Eng Mater Struct 2001; 24(2):137–50.
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[26] Taylor D. The theory of critical distances. Eng Fract Mech 2008;75(7):1696–705. [27] Taylor D, Merlo M, Pegley R, Cavatorta MP. The effect of stress concentrations on the fracture strength of
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polymethylmethacrylate. Mater Sci Eng A 2004;382(1): 288–94.
[28] Leevers PS, Radon JC. Inherent stress biaxiality in various fracture specimen geometries. Int J Fract 1982;
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19:311–25.
[29] Tada H, Paris PC, Irwin GR. The Stress Analysis of Cracks Handbook. 2nd ed. St. Louis, MO, USA: Paris
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Publications Inc.; 1985.
6.
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[30] Williams JG, Ewing PD. Fracture under complex stress—the angled crack problem. Int J Fract 1972; 8(4):441–
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[31] Ueda Y, Ikeda K, Yao T, Aoki M. Characteristics of brittle fracture under general combined modes including those under bi-axial tensile loads. Eng Fract Mech 1983; 18:1131–58. [32] Sih GC, Paris PC, Erdogan F. Crack-tip stress intensity factors for the plane extension and plate bending problem. J Appl Mech 1962; 29: 306–12. [33] Chao YJ, Zhang XH. Constraint effect in brittle fracture. In: Fatigue and Fracture Mechanics: 27th Symposium, ASTM STP 1296, American Society for Testing and Materials 1997, Philadelphia, PA, USA, 41–60. [34] Ayatollahi MR, Aliha MRM, Hassani MM. Mixed mode brittle fracture in PMMA—an experimental study using SCB specimens. Materials Science and Engineering: A 2006; 417 (1): 348-56. [35] Aliha MRM, Ayatollahi MR, Pakzad R. Brittle fracture analysis using a ring-shape specimen containing two angled cracks. International Journal of Fracture 2008; 153(1): 63-8. [36] Aliha MRM, Ayatollahi MR, Akbardoost J. Typical upper bound–lower bound mixed mode fracture resistance envelopes for rock material. Rock mechanics and rock engineering 2012; 45(1):65-74.
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ACCEPTED MANUSCRIPT [37] Ayatollahi MR, Aliha MRM. Fracture analysis of some ceramics under mixed mode loading. Journal of the American Ceramic Society 2011; 94(2):561-9.
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[38] Aliha MRM, Ayatollahi MR. On mixed-mode I/II crack growth in dental resin materials. Scripta Materialia 2008; 59(2): 258-61.
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[39] Ayatollahi MR, Aliha MRM. On the use of Brazilian disc specimen for calculating mixed mode I–II fracture
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toughness of rock materials. Engineering Fracture Mechanics 2008; 75(16): 4631-41.
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[40] Aliha MRM, Sistaninia M, Smith DJ, Pavier MJ, Ayatollahi MR. Geometry effects and statistical analysis of mode I fracture in guiting limestone. International Journal of Rock Mechanics and Mining Sciences 2012; 51: 128-
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[41] Saghafi H, Ayatollahi MR, Sistaninia M. A modified MTS criterion (MMTS) for mixed-mode fracture
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toughness assessment of brittle materials. Materials Science and Engineering: A 2010; 527(21): 5624-30.
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[42] Aliha MRM, Saghafi H. The effects of thickness and Poisson’s ratio on 3D mixed-mode fracture. Engineering
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Fracture Mechanics 2013; 98: 15-28.
Figure Captions:
Fig. 1. Crack tip stress components in polar coordinates. Fig. 2. Variation of the fracture initiation angle versus Me for different values of Poisson’s ratio and in absence of the T – stain (B = 0). Fig. 3. Variation of the fracture initiation angle versus Me for different values of T – strain in a constant value of Poisson’s ratio ( = 0.35).
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ACCEPTED MANUSCRIPT Fig. 4. Mixed mode fracture loci for different values of Poisson’s ratio and in absence of the effect of T – strain.
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Fig. 5. Mixed mode fracture loci for different values of B in a constant Poisson’s ratio ( =
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0.35).
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Fig. 6. General configuration of a biaxially loaded plate containing an angled internal crack.
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Fig. 7. Experimental data for the fracture initiation angle, in comparison with MTSN and
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EMTSN criterion.
Fig. 8. Experimental data for the fracture toughness, in comparison with MTSN and EMTSN
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criterion.
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fracture toughness.
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Fig. 9. Comparison between EMTSN and GMTS criteria in prediction of the mixed mode
Figures;
Fig.1.
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Fig. 1. Crack tip stress components in polar coordinates
Fig.2.
25
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-40
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,degrees
-20
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= 0, MTS [1] = 0.1 = 0.2 = 0.3 = 0.4 = 0.5
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0
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-60
0.2
0.4
0.6
0.8
1.0
Me
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0.0
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-80
Fig. 2. Variation of the fracture initiation angle versus Me for different values of Poisson’s ratio and in absence of the T – stain (B = 0).
Fig.3.
26
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B = - 0.4 B = - 0.2 B = 0, MTSN [13] B = 0.15 B = 0.3
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-40
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,degrees
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0
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-60
-80 0.2
0.4
0.6
0.8
1.0
Me
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0.0
Fig. 3. Variation of the fracture initiation angle versus Me for different values of T – strain in a
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constant value of Poisson’s ratio ( = 0.35).
Fig.4. 27
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= 0, MTS [1] = 0.1 = 0.2 = 0.3 = 0.4 = 0.5
1.0
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0.6
0.4
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KII/ K*IC
0.8
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0.2
0.2
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0.0
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0.0
0.4
0.6
0.8
1.0
KI/ K*IC
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Fig. 4. Mixed mode fracture loci for different values of Poisson’s ratio and in absence of the effect of T – strain.
Fig.5. 28
ACCEPTED MANUSCRIPT B = 0.3 B = 0.15 B = 0, MTSN [13] B = - 0.2 B = - 0.4
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0.4
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KII/ K*IC
0.6
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0.2
0.0 0.2
0.4
0.6
0.8
1.0
KI/ K*IC
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0.0
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Fig. 5. Mixed mode fracture loci for different values of B in a constant Poisson’s ratio ( = 0.35)
Fig.6.
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Fig. 6. General configuration of a biaxially loaded plate containing an angled internal crack
Fig.7. 30
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-20
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MTSN Criterion [13] Ueda et al [31] Williams & Ewing [30]
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EMTSN Criterion
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-60
-80
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Fracture initiation angle, 0 (degrees)
0
0.2
0.4
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0.0
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-100
Me
0.6
0.8
1.0
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Fig. 7. Experimental data for the fracture initiation angle, in comparison with MTSN and EMTSN criterion
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ACCEPTED MANUSCRIPT Fig.8.
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0.6 Williams & Ewing [30] Ueda et al [31] MTSN Criterion [13] EMTSN Criterion (present)
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0.5
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0.3
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KII/ K*IC
0.4
0.2
0.0 0.2
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0.4
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0.8
1.0
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KI/ K*IC
Fig. 8. Experimental data for the fracture toughness, in comparison with MTSN and EMTSN criterion
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Fig.9.
1.0
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Ueda et al. [31] Williams & Ewing [30] GMTS criterion [25] EMTSN criterion (present work)
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0.6
0.4
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KII / KIC
0.8
0.0
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0.2
0.0
0.2
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KI / KIC
Fig. 9. Comparison between EMTSN and GMTS criteria in prediction of the mixed mode fracture toughness.
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Graphical abstract
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ACCEPTED MANUSCRIPT Highlights:
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The MTSN criterion was extended to EMTSN criterion by considering effect of T-strain.
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T-strain remarkably affects on MTSN predictions for fracture initiation conditions.
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Poisson’s ratio influences on MTSN predictions for fracture initiation conditions.
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The EMTSN agrees with the experimental data better than MTSN and GMTS criteria.
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S0264-1275(15)30204-5 doi: 10.1016/j.matdes.2015.07.135 JMADE 363
To appear in: Received date: Revised date: Accepted date:
5 January 2015 15 July 2015 24 July 2015
Please cite this article as: M.M. Mirsayar, Mixed mode fracture analysis using extended maximum tangential strain criterion, (2015), doi: 10.1016/j.matdes.2015.07.135
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Mixed mode fracture analysis using extended maximum tangential strain criterion
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* Corresponding author: M. M. Mirsayar, E-mail address: [email protected], Tel.: +1 (979) 7776056
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Abstract
The maximum tangential strain (MTSN) criterion is modified in this paper to predict fracture
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initiation in cracked specimens under mixed mode loading conditions. Theoretical investigations were performed to study the role of both first nonsingular strain term (T – strain) and Poisson’s
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ratio on prediction of the crack propagation direction and fracture toughness. The extended
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version of the MTSN (EMTSN) criterion is proposed by taking into account the T – strain as
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well as the singular strain field. The EMTSN criterion is then examined using the experimental data reported for angled cracked plate made from polymethylmethacrylate (PMMA). It is shown that the EMTSN criterion provides more accurate estimation of the experimental results than conventional MTSN criterion.
Keywords: Mixed mode fracture; maximum tangential strain criterion; Poisson’s ratio; Tstrain
1. Introduction Brittle fracture is a major concern in many engineering structures made from a wide range of materials such as ceramics, concrete, rocks, glasses and cold metals. Several fracture criteria are proposed so far to understand fracture mechanism and to predict onset of fracture in brittle materials. Among the stress based criteria, the maximum tensile stress (MTS) criterion proposed 1
ACCEPTED MANUSCRIPT by Erdogan and Sih [1] has widely been accepted by the researchers during the past decades for brittle materials. The maximum energy release rate (G) criterion developed by Hussain et al. [2]
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and minimum strain energy density (SED) introduced by Sih [3] are also among the popular
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energy based fracture criteria. One of the most important advantages of the stress – based criteria than energy based ones is the simplicity and representing a clear role for each stress field
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parameter in estimation of the onset of fracture. However, the energy – based criteria employ
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more realistic assumptions for the crack propagation phenomenon by taking into consideration the effect of energy dissipation during fracture. Nevertheless, both stress and energy based
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criteria were subjected to some modifications during the past decades to provide a better
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explanation of the onset of fracture in different loading and boundary conditions [4 – 7].
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A literature review reveals that, in some materials, the strain based criteria provides more realistic description of the failure mechanism than stress and energy based criteria [8 – 10]. For
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instance, based on the observation reported by Nalla et al. [8], the fracture mechanism in bone could properly be explained using a strain based criterion. In 1974, Wu [11], pointed out that a meaningful description of the fracture of concrete must be based on strain and developed a strain based fracture criterion. Based on his suggestion, fracture occurs when a scalar – valued function of strain tensor reaches a critical value. Inspired from this work and the maximum normal strain theory of St–Venant [12], Chang [13] developed the maximum tangential strain (MTSN) criterion to study mixed mode fracture initiation in angled crack problem. Based on the MTSN criterion, crack propagates in the direction where the tangential strain,
, reaches its maximum
value at a critical distance, rc, from the crack tip. According to linear elastic fracture mechanics (LEFM) assumption, the stress, strain and displacement fields around the crack tip could be represented in terms of a series expansion containing singular terms (corresponding to the stress 2
ACCEPTED MANUSCRIPT intensity factors) and higher order terms [14]. The criterion, suggested by Chang [13] only uses the singular strain field and does not take into account the effect of higher order terms. Although,
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the criterion provides a very simple and realistic description of the brittle fracture phenomenon,
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it was not successful to precisely estimate fracture test data in some geometries and boundary
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conditions.
In agreement with the pioneering work done by Irwin [15], recent studies on cracked and notches
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components reveals that higher order terms of the Williams [14] series expansion may
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significantly influence on stress/ strain distribution around the crack tip [16 – 25]. Among the higher order terms, the first nonsingular one was recognized as the most effective term in
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evaluation of the fracture initiation as well as stress/ strain distribution around the crack tip. In
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case of bi-material cracks and utilizing a stress based criterion, Mirsayar [16] showed that sometimes the T-stress (the first nonsingular stress term around crack tip) significantly
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influences on the stress distribution around the crack tip and neglecting T-stress may cause a considerable error in estimation of the fracture strength. However, the effect of first nonsingular term of the Williams series expansion on the fracture initiation has not been studied yet using a strain based fracture criterion.
This paper deals with the role of both Poisson’s ratio and the first nonsingular strain term (T – strain) on the fracture initiation using MTSN criterion. The effect of T – strain and Poisson’s ratio on the fracture propagation direction as well as the fracture strength is studied theoretically. The conventional MTSN criterion is extended by taking into account the T – strain term as well as the singular strain field. The extended MTSN (EMTSN) criterion is then employed to predict experimentally reported data for fracture propagation direction and fracture strength of some angled cracked plate made from polymethylmethacrylate (PMMA). The EMTSN predictions are 3
ACCEPTED MANUSCRIPT also compared with the conventional MTSN criterion. It is shown that the EMTSN criterion provides more accurate prediction of the experimental data than conventional MTSN criterion.
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2. Tangential strain based criteria
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2.1. Conventional MTSN criterion
, reaches its maximum value, at a critical distance, rc, from the crack tip. The
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strain,
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The MTSN criterion states that crack propagates in the direction, 0, where the tangential
criterion could simply be represented as:
(rc , 0 ) T T / E
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(1)
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0 r r ; c 0 2 0 2 r r ; c 0
(2)
Where, T is the tensile fracture strength and E is Young’s modulus of the material. The critical distance rc could be considered as damaged zone around the crack tip which is a material property [26, 27]. The linear elastic relationship between the tangential strain and stress components is:
1 ( rr ) E
(3)
4
ACCEPTED MANUSCRIPT Where is the Poisson’s ratio and and rr are respectively the tangential and radial components of the stress field in cylindrical coordinates with origin at the crack tip, shown in
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Fig. 1.
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The MTSN criterion claims that fracture initiation depends on not only the geometry and boundary conditions but also the Poisson’s ratio, , of the material. It is clear from Eq. (3) that
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the predictions based on MTSN criterion coincide with those based on MTS criterion
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proposed by Erdogan and Sih [1] when = 0. Based on the LEFM assumption, the stress field
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around the crack tip could be expressed in the form of a series expansion with infinite terms, given in Eqs. (4) – (6) [14].
KI K II f ,1 ( ) f , 2 ( ) T sin 2 ( ) O(r 1/ 2 ) ... 2r 2r
KI f r ,1 ( ) 2r KI rr f rr,1 ( ) 2r
r
K II f r , 2 ( ) T sin( ) cos( ) O(r 1 / 2 ) ... 2r K II f rr, 2 ( ) T cos 2 ( ) O(r 1 / 2 ) ... 2r
(4)
(5)
(6)
where (i, j) ≡ (r,) are the polar coordinates with the origin at the crack tip (see Fig. 1) and the functions fij,n() (ij ≡ r, ; n ≡1,2) are given in Appendix A. The coefficients KI and KII and T
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singular stress field (when r 0
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nonsingular stress term. Substituting Eq. (4) and (5) into the Eq. (3) and taking into account
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the only singular stress terms (associated with the KI and KII), the conventional MTSN could
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be represented as follows:
,1 ( 0 ) f rr ,1 ( 0 )
E 2rc
f
, 2 ( 0 ) f rr , 2 ( 0 ) T
T
(8)
E
d ( f ij,n ( )) d 0
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Where f 'ij,n ( 0 )
K II
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E 2rc
f
(7)
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KI
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KI f ' ,1 (0 ) f 'rr,1 (0 ) K II f ' ,2 (0 ) f 'rr,2 (0 ) 0 0 E 2rc E 2rc
Substituting fracture initiation angle, 0, calculated from Eq. (7), one can find the onset of fracture when the left hand side of the Eq. (8) reaches the tensile fracture strength.
2.2. Modified MTSN criterion The left hand side of the Eq. (7) and (8) are controlled by the values of the stress intensity factors. Retaining singular stress terms and adding the T – stress into the Eq. (3), the Eq. (7) and (8) could be rewritten as follows:
KI f ' ,1 (0 ) f 'rr,1 (0 ) K II f ' ,2 (0 ) f 'rr,2 (0 ) (1 ) T sin( 20 ) 0 0 E E 2rc E 2rc
(9)
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f
,1
( 0 ) f rr,1 ( 0 )
K II E 2rc
f
,2
( 0 ) f rr, 2 ( 0 )
T sin 2 ( 0 ) cos 2 ( 0 ) T E E
(10)
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In Eq. (10), the first nonsingular tangential strain term corresponding to parameter T is called
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tangential T – strain. Although the parameter T is identical with the well – known T – stress,
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the first nonsingular tangential strain term (or tangential T – strain) is the matter of concern in this paper. It needs to be mentioned that the radial and shear T – strains are associated with
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the first nonsingular radial and shear components of the strain, respectively.
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To simplify the Eqs. (9) and (10), the biaxiality ratio B, relative to the stress intensity factors,
T a K eff
(11)
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B
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could be represented as [28]:
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Where a is the crack length for edge cracks and semi-crack length for center cracks. The 2 2 effective stress intensity factor Keff is also defined K eff ( K I ) ( K II )
Defining the normalized critical distance 2rc / a , the direction of the crack propagation could be determined from the Eq. (12) KI f ' ,1 ( 0 ) f 'rr,1 ( 0 ) K II f ' ,2 ( 0 ) f 'rr,2 ( 0 ) (1 ) B sin( 2 0 ) 0 0 K eff K eff
(12)
The onset of the crack propagation could also be found from:
K I f ,1 (0 ) f rr,1 (0 ) K II f , 2 (0 ) f rr, 2 (0 ) T 2rc BKeff sin 2 (0 ) cos 2 (0 )
(13)
Equations (12) and (13) describe the extended MTSN criterion where the mixed mode fracture is predicted for any geometry and material when KI, KII, T and are known. The 7
ACCEPTED MANUSCRIPT effect of Poisson’s ratio, in absence of the tangential T – strain, on the fracture initiation angle is illustrated in Fig. 2 for different values of mixity parameter Me given by
tan 1 (
KI ) K II
(14)
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2
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Me
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The value of Me represents the participation of each fracture mode and varies from zero (pure mode II) to unity (pure mode I). Regardless of sign of the numbers on the vertical axis, the
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absolute values are considered as the fracture initiation angle.
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As shown in Fig. 2, all curves coincide together at Me = 0.6, 0 - 47o. For 0 < Me < 0.6 and
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0.6 < Me < 1, the MTSN criterion provides lower and higher predictions of the fracture
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initiation angle than MTS criterion ( = 0) [1] respectively. In pure mode I, the fracture
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initiation angle is predicted to be zero for all values of the Poisson’s ratio. Also, the fracture initiation angle decreases for Me less than 0.6 and conversely increases for Me > 0.6 by increasing the Poisson’s ratio. From Fig. 2, the highest effect of Poisson’s ratio on the fracture initiation angle is found to be in pure mode II and the no effect is observed for Me = 0.6 and pure mode I (Me = 1).
The effect of dimensionless parameter B on the fracture initiation angle is illustrated in Fig. 3. The parameter B presents the effects of both tangential T – strain and the critical distance rc. For a constant value of Poisson’s ratio ( = 0.35), the fracture initiation angle is plotted versus M e in different values of B ranging from -0.4 to 0.3. It is seen that the tangential T – strain significantly influences on the fracture initiation angle in different mixed mode conditions, except for pure mode I. Also, it is shown that for positive values of T – strain, the 8
ACCEPTED MANUSCRIPT modified MTSN criterion (solid line) provides higher estimates for fracture initiation angle and vice versa. From Figs. 2 and 3, it is concluded that both T – strain and Poisson’s ratio
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could significantly influence on the fracture initiation angle in pure mode II condition and no
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effect is observed for pure mode I.
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< Fig. 3. >
Figs. 4 and 5 illustrate the effects of Poisson ratio and T – strain on the fracture toughness
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respectively. The plots are presented in terms of the normalized stress intensity factors K I/
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K*IC and KII/ K*IC. The parameter K*IC, called generalized fracture toughness, is defined as
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Eq. (15) and takes into account the effect of both Poisson’s ratio and specimen geometry (B) in calculation of the mode I fracture toughness. (see Appendix B for more detail):
* K IC
T 2rc 1 (1 B )
(15)
As shown in Figs. 4 and 5, both Poisson’s ratio and T – strain remarkably affect on the mixed mode fracture loci. Most notable about Fig. 4 is that the MMTS criterion provides more conservative predictions of the onset of fracture than MTS criterion [1]. In other words, the mixed more fracture locus becomes more limited by increasing the value of Poisson’s ratio. The effect of tangential T – strain, on the fracture toughness is shown in Fig. 5 for a constant value of Poisson’s ratio ( = 0.35). The predicted fracture toughness increases for negative
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< Fig. 4. >
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influence of the tangential T – strain could be seen on the mode II fracture toughness.
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< Fig. 5. >
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Figs. 3 and 5 represent the extended MTSN criterion for mixed mode fracture problems. In order to use these fracture diagrams, the crack tip parameters KI, KII and T must be known for
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a given specimen. The stress intensity factors as well as the T – strain could be obtained
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through finite element analysis or using handbooks [29] for simple geometries. The mixity
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parameter Me and the biaxiality ratio B are then calculated using equations 14 and 11. The normalized critical distance 2rc / a could also be determined for a specific crack length and material property [26, 27]. Calculating the parameters Me and B, one can find the corresponding fracture initiation angle from Fig. 3. Finally, by calculating the generalized fracture toughness K*IC, the onset of fracture could be determined using the same data from Fig. 5.
3. Results and discussion The extended MTSN criterion is examined here using the experimental data available in the literature for PMMA. Williams and Ewing [30], and Ueda et al. [31] studied the fracture initiation in PMMA using square plates containing an angled crack. The general configuration of the specimen is shown in Fig. 6 under biaxial loading. As shown, the plate is subjected to 10
ACCEPTED MANUSCRIPT uniform far-field stresses, and ( is the lateral load ratio) in the vertical and horizontal directions respectively. The angle between the crack line and the vertical direction, , is changed
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to make different mixed mode conditions from pure mode I to pure mode II. The specimens used
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by Williams and Ewing [30] were plates 152 mm wide, 305 mm long, 3.2 mm thick, containing
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angled cracks of length 2a varying from 14 to 50 mm. The plate employed by Ueda et al. [31] also were, 100 mm wide, 100 mm long,4 mm thick, containing angled cracks of length 2a equal
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to 40 mm.
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< Fig. 6. >
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Although the finite element method could be performed for any given geometry and boundary conditions to calculate crack tip parameters, it still would be easier to use closed form solution
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when dealing with the simple geometries. Smith et al. [25] suggested to use the following closed form solution, given by Sih et al. [32], to calculate stress intensity factors as well as the parameter T.
K I a ( cos 2 sin 2 )
(16)
K II a (1 ) cos sin
(17)
T (1 ) cos 2
(18)
It needs to be mentioned that although the equations 16 – 18 are only valid when the crack length is small compared with the size of the plate, Smith et al. [25] showed that these equations could successfully be employed for specimens tested by Williams and Ewing [30] and Ueda et al. [31]. 11
ACCEPTED MANUSCRIPT The specimens tested by Williams and Ewing [30] and Ueda et al. [31] were subjected to uniaxial loading ( = 0). It easily could be found from equations 16 – 18 that the pure mode I
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would be obtained when = 90°. For =0° the stress intensity factors approach zero while the
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parameter T reaches its maximum value. In other words, for small values of , the tangential T –
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strain plays the most important role in fracture event and there is no effect for the singular stress field.
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According to Eq. (18) the parameter T is positive within the range of and is negative
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when . Therefore, the fracture initiation angles predicted by EMTSN criterion are expected to be higher and lower than MTSN criterion for andrespectively.
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Also, it is expected from EMTSN criterion to provide lower and higher predictions of the
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fracture toughness than MTSN criterion for andrespectively.
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Fig. 7 illustrates the fracture initiation angles reported by Williams and Ewing [30] and Ueda et al. [31] for PMMA in different mode mixities. The predictions provided by the conventional MTSN and the EMTSN criteria are also presented. The non – dimensional characteristic distance, , of 0.2 is selected for the analysis according to Williams and Ewing [30] and Smith et al. [25]. As discussed before, the parameter T is positive for 0 < Me < 0.5 (0° < < 45°). Therefore, the fracture initiation angles predicted by EMTSN criterion is higher than those of estimated by the conventional MTSN criterion. Conversely, for 0.5 < Me < 1.0 (45° < < 90°), the fracture initiation angles predicted by EMTSN is less than those of estimated by the conventional MTSN criterion because of the negative values of parameter T. The significant effect of tangential T – strain on the fracture initiation angle is shown to exist under mixed mode conditions, especially in pure mode II.
12
ACCEPTED MANUSCRIPT < Fig. 7. >
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The combined experimental data for the fracture toughness from Williams and Ewing [30] and Ueda et al. [31] are illustrated in Fig. 8 in comparison with the predictions provided by MTSN
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and EMTSN criteria. Same non – dimensional characteristic distance, , of 0.2 is assumed for fracture toughness assessments. The generalized fracture toughness is calculated for each crack
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angle, , by obtaining B as function of (equations 11 and 18), and the associated fracture load
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(T).
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< Fig. 8. >
As shown in Figs. 7 and 8, the substantial scatter is seen for the experimental data in all crack
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angles. These scatters may be explained if the higher order terms (such as order of r0.5 etc.) be considered in stress/ strain field when applying EMTSN criterion. Chao and Zhang [33] showed that such scatter under pure mode I , in fracture loci diagrams, are as a result of the effect of second nonsingular stress term (corresponding to the order of r0.5). However, it could be perceived from the literature that, for many laboratory specimens, only considering the effect of first nonsingular term of the Williams series expansion is enough to achieve a satisfactory accuracy in prediction of the fracture initiation and no higher order terms are needed [16, 18, 23 – 25]. It is also worth to compare the tangential strain based fracture criterion, developed in this paper, with the some popular tangential stress based fracture criteria. Among the tangential stress based fracture criteria, Ayatollahi and coworkers have employed the GMTS criterion [25] (using KI,
13
ACCEPTED MANUSCRIPT KII, T and rc parameters) to investigate mixed mode brittle fracture behavior of materials [34 – 41]. Fig. 9 compares the predictions obtained from EMTSN criterion together with GMTS
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criterion for mixed mode fracture toughness of PMMA material, tested by Williams and Ewing [30] and Ueda et al. [31]. In order to represent a direct visual comparison, both criteria together
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with the experimental data are plotted in KI/KIC – KII/KIC space. It can be seen that EMTSN
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criterion presents better predictions than GMTS criterion for mixed mode fracture toughness,
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especially near mode II condition (for KI/KIC less than 0.3). Looking through the literature reveals that although the GMTS criterion successfully predicts the fracture toughness as well as
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the size effects under mixed mode conditions [34 – 36], it fails to provide an acceptable predictions in pure mode II (or predominantly pure mode II) condition for many brittle materials,
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D
such as ceramics, resins and rocks [37 – 39]. In many cases, the second nonsingular stress term and even first five terms of the tangential stress field are used to predict the test results and the
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GMTS criterion could not estimate the experimental data in a good accuracy [40, 41]. In these cases, considering a large number of tangential stress terms will lead to a complex formulation of the tangential stress based fracture criteria. Also, the higher order term could not be obtained as straightforward as stress intensity factors and the T – stress and sometimes, it is hard to obtain these parameters for a given specimen. Also, according to GMTS criterion, the T – stress has no effect on the fracture resistance for pure mode I condition. In other words, the GMTS states that the fracture toughness is identical for a specific material tested by different types of specimens. However, it is well – known that for a given material, different values of the mode I fracture toughness, KIC, could be obtained for different test specimens as a result of geometry effects. The effect of Poisson’s ratio is another important issue that is not considered in GMTS criterion. Using extensive 3-D finite element analyses, Aliha and Saghafi [42] recently showed that the
14
ACCEPTED MANUSCRIPT stress intensity factors as well as the T – stress are significantly affected by the Poisson’s ratio of the material. As indicated previously, the effect of Poisson’s ratio as well as the specimen
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geometry is considered in EMTSN criterion. Indeed, the EMTSN criterion still needs to be
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examined for different brittle materials and specimen geometries. Further studies are needed to understand if the EMTSN criterion could cover the weakness of the stress based fracture criteria
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by taking into consideration of both Poisson’s ratio as well as the geometry effects.
4. Conclusion
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The extended version of the maximum tangential strain (EMTSN) criterion was introduced and discussed in this paper. The effects of Poisson’s ratio as well as the first nonsingular tangential
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strain term (tangential T – strain) were studied on the fracture initiation angle and the fracture toughness. The significant effects of both tangential T – strain and Poisson’s ratio on the fracture
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initiation angle as well as the fracture strength were shown to exist under mixed mode conditions. The EMTSN criterion was then developed by taking into account the effect of first nonsingular strain term of the tangential strain field well as the singular strain field. The proposed criterion was examined using the experimental data available in literature for the fracture toughness and the fracture initiation angles of the angled cracked plate made from PMMA. The EMTSN predictions were compared with the conventional MTSN criterion. It was shown that the EMTSN criterion provides more accurate prediction of experimental data than conventional MTSN criterion.
Appendix A. The stress field around the crack tip could be represented using the Eqs. (4 – 6) where frr,1(), frr,2(), f,1(), f,2(), fr,1(), fr,2() are given as follows: 15
ACCEPTED MANUSCRIPT (A – 1)
1 3 f rr, 2 ( ) [5 sin( ) 6 sin( )] 4 2 2
(A – 2)
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1 3 f rr,1 ( ) [5 cos( ) cos( )] 4 2 2
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1 3 f , 2 ( ) [3 sin( ) 3 sin( )] 4 2 2
(A – 4)
(A – 5)
(A – 6)
Appendix B.
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D
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1 3 f r ,1 ( ) [sin( ) sin( )] 4 2 2
1 3 f r , 2 ( ) [cos( ) 3 cos( )] 4 2 2
(A – 3)
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1 3 f ,1 ( ) [3 cos( ) cos( )] 4 2 2
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Solution method for extended MTSN criterion Fracture initiation angle
Eq. (13) could be rewritten as:
KI K A II B K eff K eff
(B – 1)
where the coefficients A and B are
A
B
f ' f '
,2 ,1
( 0 ) f ' rr, 2 ( 0 )
(B – 2)
( 0 ) f ' rr,1 ( 0 )
B (1 ) sin( 2 0 ) f ' ,1 ( 0 ) f 'rr,1 ( 0 )
16
ACCEPTED MANUSCRIPT 2 2 Recalling K eff ( K I ) ( K II ) and assuming = KII/KI, the Eq. (B1) can be written as:
1 2
(B – 3)
B
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1 2
A
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1
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or,
2A 1 B2 ( 2 ) 2 0 A B2 A B2
(B – 4)
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2
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The mixity parameter Me could then be calculated as follows:
1 2 A2 B 2 1 M tan tan 2 2 A B A B 1 2
1
(B – 5)
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e
Because the direction of fracture initiation 0 must be between −90° and 0°, only the negative sign in the numerator of Eq. (B – 5) is acceptable. Substituting A and B from Eq. (B – 2) into the Eq. (B – 5), a closed form solution would be achieved for the fracture initiation angle as a function of mode mixity.
Fracture loci Eq. (14) can be written as
1 1 2
C 1 (1 / ) 2
D / KI 1 2
H
(B – 6)
Where C, D and H are expressed as:
17
ACCEPTED MANUSCRIPT C
f , 2 ( 0 ) f rr, 2 ( 0 )
(B – 7)
f ,1 ( 0 ) f rr,1 ( 0 ) (B – 8)
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sin 2 ( 0 ) cos 2 ( 0 ) f ,1 ( 0 ) f rr,1 ( 0 )
(B – 9)
T 2rc D gives K I K I [ f ,1 ( 0 ) f rr,1 ( 0 )]
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Solving the Eq. (B – 6) for , and recalling
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H B
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T 2rc D f ,1 ( 0 ) f rr,1 ( 0 )
(B – 10)
K II T 2rc [ f ,1 ( 0 ) f rr,1 ( 0 )][1 C H 1 2 ]
(B – 11)
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KI 1 T 2rc [ f ,1 ( 0 ) f rr,1 ( 0 )][1 C H 1 2 ]
According to the maximum tangential stress criterion [1, 24], for pure mode I when KII , T and 0 are equal to zero, KI at fracture can be replaced by the mode I fracture toughness KIC as follows:
K IC ,C 2rc
(B – 12)
Where ,C is the critical tangential stress which is assumed to be equal to T based on MTS criterion. According to MTSN criterion, the fracture occurs when (rr) reaches tensile fracture strength T. Based on an analogy with the MTS criterion and by substituting 0 = 0
18
ACCEPTED MANUSCRIPT * and KII = 0 into the Eq. (10), the generalized fracture toughness, K IC , is defined as the
following for MTSN criterion;
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T 2rc K 1 (1 B )
(B – 13)
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* IC
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Eq. (B – 13) provides more general expression of the mode I fracture toughness for tangential strain based fracture criteria which takes into account the effect of both Poisson’s ratio and the
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specimen geometry. A comparison between Eqs. (B – 12) and (B – 13) reveals that these two
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equations are identical when = 0 or B . For tangential stress based fracture criteria (such as MTS [1] and GMTS [25]), there is no effect of first nonsingular stress term (B)
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when substituting 0 = 0 into the tangential stress component because the first nonsingular
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tangential stress is T sin 2 0 . However, from Eq. (13), it could be seen that because of
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dependency of the tangential strain component on Poisson’s ratio, the parameter B remains when 0 = 0 and as a result, the generalized fracture toughness should be used to normalize the fracture loci. It should be noted that the fracture loci could also be plotted in terms of the conventional fracture toughness, KIC. In that case, it is easy to show that the fracture loci do not coincide together at KI/KIC = 1. In other words, because of the effects of Poisson’s ratio and the specimen geometry (B), the EMTSN criterion provides different predictions of the mode I fracture toughness. While the conventional fracture toughness and stress based criteria (such as GMTS [25]) could not introduce the effect of specimen geometry in pure mode I condition. Substituting Eq. (B – 13) into the equations (B – 10) and (B – 11) leads to
19
ACCEPTED MANUSCRIPT (B – 14)
K II 1 (1 B ) * K IC [ f ,1 ( 0 ) f rr,1 ( 0 )][1 C H 1 2 ]
(B – 15)
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KI 1 (1 B ) * K IC [ f ,1 ( 0 ) f rr,1 ( 0 )][1 C H 1 2 ]
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Equations (B – 14) and (B – 15) represent the fracture loci shown in Figs. 4 and 5 for different
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values of B and Poisson’s ratio.
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ACCEPTED MANUSCRIPT [9] Maiti SK, Smith RA. Criteria for brittle fracture in biaxial tension. Eng Fract Mech 1984; 19(5): 793 – 804. [10] Wu X, Li X. Analysis and modification of fracture criteria for mixed – mode crack. Eng Fract Mech 1989;
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ACCEPTED MANUSCRIPT [23] Shahani AR, Tabatabaei SA. Effect of T-stress on the fracture of a four point bend specimen. Mater Des 2009; 30: 2630–35
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[24] Li XF, Xu LR. T-stresses across static crack kinking. J Appl Mech 2007; 74(2):181-90. [25] Smith DJ, Ayatollahi MR, Pavier MJ. The role of T-stress in brittle fracture for linear elastic materials under
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[30] Williams JG, Ewing PD. Fracture under complex stress—the angled crack problem. Int J Fract 1972; 8(4):441–
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[31] Ueda Y, Ikeda K, Yao T, Aoki M. Characteristics of brittle fracture under general combined modes including those under bi-axial tensile loads. Eng Fract Mech 1983; 18:1131–58. [32] Sih GC, Paris PC, Erdogan F. Crack-tip stress intensity factors for the plane extension and plate bending problem. J Appl Mech 1962; 29: 306–12. [33] Chao YJ, Zhang XH. Constraint effect in brittle fracture. In: Fatigue and Fracture Mechanics: 27th Symposium, ASTM STP 1296, American Society for Testing and Materials 1997, Philadelphia, PA, USA, 41–60. [34] Ayatollahi MR, Aliha MRM, Hassani MM. Mixed mode brittle fracture in PMMA—an experimental study using SCB specimens. Materials Science and Engineering: A 2006; 417 (1): 348-56. [35] Aliha MRM, Ayatollahi MR, Pakzad R. Brittle fracture analysis using a ring-shape specimen containing two angled cracks. International Journal of Fracture 2008; 153(1): 63-8. [36] Aliha MRM, Ayatollahi MR, Akbardoost J. Typical upper bound–lower bound mixed mode fracture resistance envelopes for rock material. Rock mechanics and rock engineering 2012; 45(1):65-74.
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ACCEPTED MANUSCRIPT [37] Ayatollahi MR, Aliha MRM. Fracture analysis of some ceramics under mixed mode loading. Journal of the American Ceramic Society 2011; 94(2):561-9.
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[38] Aliha MRM, Ayatollahi MR. On mixed-mode I/II crack growth in dental resin materials. Scripta Materialia 2008; 59(2): 258-61.
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[39] Ayatollahi MR, Aliha MRM. On the use of Brazilian disc specimen for calculating mixed mode I–II fracture
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toughness of rock materials. Engineering Fracture Mechanics 2008; 75(16): 4631-41.
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[40] Aliha MRM, Sistaninia M, Smith DJ, Pavier MJ, Ayatollahi MR. Geometry effects and statistical analysis of mode I fracture in guiting limestone. International Journal of Rock Mechanics and Mining Sciences 2012; 51: 128-
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toughness assessment of brittle materials. Materials Science and Engineering: A 2010; 527(21): 5624-30.
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[42] Aliha MRM, Saghafi H. The effects of thickness and Poisson’s ratio on 3D mixed-mode fracture. Engineering
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Fracture Mechanics 2013; 98: 15-28.
Figure Captions:
Fig. 1. Crack tip stress components in polar coordinates. Fig. 2. Variation of the fracture initiation angle versus Me for different values of Poisson’s ratio and in absence of the T – stain (B = 0). Fig. 3. Variation of the fracture initiation angle versus Me for different values of T – strain in a constant value of Poisson’s ratio ( = 0.35).
23
ACCEPTED MANUSCRIPT Fig. 4. Mixed mode fracture loci for different values of Poisson’s ratio and in absence of the effect of T – strain.
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Fig. 5. Mixed mode fracture loci for different values of B in a constant Poisson’s ratio ( =
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0.35).
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Fig. 6. General configuration of a biaxially loaded plate containing an angled internal crack.
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Fig. 7. Experimental data for the fracture initiation angle, in comparison with MTSN and
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EMTSN criterion.
Fig. 8. Experimental data for the fracture toughness, in comparison with MTSN and EMTSN
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criterion.
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fracture toughness.
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Fig. 9. Comparison between EMTSN and GMTS criteria in prediction of the mixed mode
Figures;
Fig.1.
24
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D
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ACCEPTED MANUSCRIPT
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Fig. 1. Crack tip stress components in polar coordinates
Fig.2.
25
ACCEPTED MANUSCRIPT
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-40
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,degrees
-20
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= 0, MTS [1] = 0.1 = 0.2 = 0.3 = 0.4 = 0.5
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0
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-60
0.2
0.4
0.6
0.8
1.0
Me
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0.0
D
-80
Fig. 2. Variation of the fracture initiation angle versus Me for different values of Poisson’s ratio and in absence of the T – stain (B = 0).
Fig.3.
26
ACCEPTED MANUSCRIPT
B = - 0.4 B = - 0.2 B = 0, MTSN [13] B = 0.15 B = 0.3
RI
-40
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,degrees
-20
PT
0
MA
NU
-60
-80 0.2
0.4
0.6
0.8
1.0
Me
TE
D
0.0
Fig. 3. Variation of the fracture initiation angle versus Me for different values of T – strain in a
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constant value of Poisson’s ratio ( = 0.35).
Fig.4. 27
ACCEPTED MANUSCRIPT
= 0, MTS [1] = 0.1 = 0.2 = 0.3 = 0.4 = 0.5
1.0
PT RI SC
0.6
0.4
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KII/ K*IC
0.8
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0.2
0.2
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0.0
D
0.0
0.4
0.6
0.8
1.0
KI/ K*IC
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Fig. 4. Mixed mode fracture loci for different values of Poisson’s ratio and in absence of the effect of T – strain.
Fig.5. 28
ACCEPTED MANUSCRIPT B = 0.3 B = 0.15 B = 0, MTSN [13] B = - 0.2 B = - 0.4
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RI SC
0.4
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KII/ K*IC
0.6
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0.2
0.0 0.2
0.4
0.6
0.8
1.0
KI/ K*IC
TE
D
0.0
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Fig. 5. Mixed mode fracture loci for different values of B in a constant Poisson’s ratio ( = 0.35)
Fig.6.
29
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D
MA
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RI
PT
ACCEPTED MANUSCRIPT
Fig. 6. General configuration of a biaxially loaded plate containing an angled internal crack
Fig.7. 30
ACCEPTED MANUSCRIPT
-20
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MTSN Criterion [13] Ueda et al [31] Williams & Ewing [30]
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EMTSN Criterion
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-40
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-60
-80
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Fracture initiation angle, 0 (degrees)
0
0.2
0.4
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0.0
D
-100
Me
0.6
0.8
1.0
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Fig. 7. Experimental data for the fracture initiation angle, in comparison with MTSN and EMTSN criterion
31
ACCEPTED MANUSCRIPT Fig.8.
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0.6 Williams & Ewing [30] Ueda et al [31] MTSN Criterion [13] EMTSN Criterion (present)
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0.5
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0.3
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KII/ K*IC
0.4
0.2
0.0 0.2
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0.0
TE
D
0.1
0.4
0.6
0.8
1.0
1.2
KI/ K*IC
Fig. 8. Experimental data for the fracture toughness, in comparison with MTSN and EMTSN criterion
32
ACCEPTED MANUSCRIPT
RI
PT
Fig.9.
1.0
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Ueda et al. [31] Williams & Ewing [30] GMTS criterion [25] EMTSN criterion (present work)
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0.6
0.4
TE
D
KII / KIC
0.8
0.0
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0.2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
KI / KIC
Fig. 9. Comparison between EMTSN and GMTS criteria in prediction of the mixed mode fracture toughness.
33
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ACCEPTED MANUSCRIPT
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Graphical abstract
34
ACCEPTED MANUSCRIPT Highlights:
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The MTSN criterion was extended to EMTSN criterion by considering effect of T-strain.
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T-strain remarkably affects on MTSN predictions for fracture initiation conditions.
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Poisson’s ratio influences on MTSN predictions for fracture initiation conditions.
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The EMTSN agrees with the experimental data better than MTSN and GMTS criteria.
35