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39
УДК 539.42
Оценка хрупкого разрушения типа II с использованием энергетического критерия
M. Rashidi Moghaddam1, M.R. Ayatollahi1, S.M.J. Razavi1, F. Berto2
1 Иранский университет науки и технологии, Тегеран, 16846, Иран 2 Норвежский университет естественных и технических наук, Тронхейм, 7491, Норвегия
Предложена модификация критерия минимальной плотности энергии деформации для предсказания значений сопротивления разрушению типа II, указанныж в литературе для некоторыж хрупких и квазихрупких материалов. Приведены результаты испытаний на сдвиговое разрушение полукруглыж образцов, подвергнутых изгибу. Модифицированный критерий разрушения типа II позволяет учитывать влияние Т-напряжений (наряду с сингулярными членами напряжений/деформаций) при выгшслении коэффициента плотности энергии деформации при малом значении расстояния от вершины трещины. Показано, что предлагаемый критерий более эффективен для предсказания сопротивления разрушению типа II по сравнению с классическим критерием минимальной плотности энергии деформации.
Ключевые слова: критерий плотности энергии деформации, разрушение типа II, сопротивление разрушению, Т-напряжения, хрупкое разрушение
Mode II brittle fracture assessment using an energy based criterion
M. Rashidi Moghaddam1, M.R. Ayatollahi1, S.M.J. Razavi1, and F. Berto2
1 Fatigue and Fracture Laboratory, Centre of Excellence in Experimental Solid Mechanic and Dynamics, School of Mechanical Engineering, Iran University of Science and Technology, Narmak, 16846, Tehran, Iran 2 Department of Engineering Design and Materials, Norwegian University of Science and Technology, Trondheim, 7491, Norway In this paper the minimum strain energy density criterion is modified to predict the values of mode II fracture toughness reported in literature for several brittle and quasi-brittle materials. The experimental results are all related to mode II fracture tests performed on the semi-circular bend specimen. The modified mode II fracture criterion takes into account the effect of T-stress (in addition to the singular terms of stresses/strains) when calculating the strain energy density factor at a very small critical distance from the crack tip. It is shown that the proposed criterion provides significantly better predictions for mode II fracture toughness compared with the classical minimum strain energy density criterion.
Keywords: strain energy density criterion, mode II fracture, fracture toughness, T-stress, brittle fracture
Nomenclature
a—crack length; B—biaxiality ratio;
dW/dA —strain energy density function; G—shear modulus of elasticity; K j —mode I stress intensity factor; K jc —mode I fracture toughness; Kjj —mode II stress intensity factor; Kjjc —mode II fracture toughness; L—half span in semi-circular bend specimen; P—applied load;
R—radius of semi-circular bend specimen; r, 6—polar coordinates with the origin located at the crack tip;
rc —critical distance; S—strain energy density factor; Scr —critical value of the strain energy density factor at the critical distance; T—T-stress;
t—thickness of semi-circular bend specimen; 6c —initial angle of fracture;
k—elastic constant introducing the stress state in the model;
Oy (i, j = r, 6, z)—stress components; ot —tensile strength; v—Poisson's ratio;
a—dimensionless form of the critical distance; P—crack inclination angle corresponding to pure mode II.
© Rashidi Moghaddam M., Ayatollahi M.R., Razavi S.M.J., Berto F., 2016
1. Introduction
There are three basic modes of deformation in classical fracture mechanics. In mode I or opening mode, the crack faces are displaced in a direction normal to the crack plane without any sliding. In mode II or shear mode, the crack faces slide normal to the crack front without any opening. In mode III or out-of-plane shear mode, the crack faces slide parallel to the crack front.
Mode II fracture is a failure mode in cracked specimens which has an important role in many practical applications such as rock engineering. For example, mode II fracture investigations can be used in hydraulic fracturing and rock cutting and also in other rock engineering applications such as the construction of concrete structures like dams, tunnels and undergrounds. Therefore, determination of mode II fracture toughness is an important subject for engineers and researchers.
A standard procedure for determination of mode II fracture toughness of brittle and quasi-brittle materials has not been yet established. However, several test specimens have been proposed by researchers for determination of the mode II fracture toughness. The edge cracked semi-circular specimen subjected to three point bending (SCB) [14], the centrally cracked Brazilian disk under radial compression [5-8], the single-edge crack specimen subjected to antisymmetric four point bend loading [9, 10] and the mixed-mode compact tension specimen [11-13] are some of the well-known test configurations for fracture tests on brittle and quasi-brittle materials. Pure mode II conditions can be achieved in these specimens by choosing appropriate crack locations or crack orientations with respect to the applied load. The SCB configuration has been frequently used in the past for obtaining the pure mode II fracture toughness KIIc for different materials. Hence, there are widespread experimental, numerical and theoretical research studies conducted on this test configuration. For example, Khan and Al-Shayea [1] and Ayatollahi et al. [4] have used the SCB specimen to perform mode II fracture experiments. Figure 1 illustrates a schematic view of SCB specimen in which P is the applied load, a is the crack length, P is the crack inclination angle corresponding to pure mode II, and R and L are the specimen radius and half of the loading span, respectively.
In addition to the experimental methods for obtaining the mode II fracture resistance of materials, there are various criteria for predicting the onset of brittle fracture under pure mode II loading. Three classical criteria for predicting mixed mode brittle fracture are the maximum tangential stress criterion [14], the minimum strain energy density (SED) criterion [15] and the maximum energy release rate criterion [16]. However, numerous experimental results have been reported to date for mode II fracture toughness that are not consistent with predictions of these classical criteria. Such inconsistencies between the theoretical and ex-
perimental results confine the validity of a fracture criterion like the conventional minimum strain energy density criterion to limited geometries and loading conditions.
The SED criterion considers only the effect of singular stress terms of William's series expansion and neglects the effect of non-singular stress terms. According to this criterion, mixed mode brittle fracture occurs when the value of strain energy density factor at the critical radius rc reaches a critical value named Scr. However, the classical SED criterion does not consider the influence of T-stress on fracture behavior of brittle materials. Ayatollahi et al. recently proposed a generalized form of the minimum strain energy density criterion (GSED) which allows taking into account the influence of the T-stress on the mode I fracture resistance of brittle and quasi-brittle materials [17, 18]. They showed that mode I brittle fracture is significantly influenced by T-stress values.
In the present manuscript, a generalized SED criterion is presented for predicting the experimental results obtained from mode II fracture tests on the semi-circular bend specimen. The modified criterion takes into account the effect of T-stress in addition to the stress intensity factors KI and Kjj. Using some experimental results reported by previous studies, the theoretical predictions calculated from the generalized SED criterion are verified. It is shown that the T-stress has a significant influence on mode II fracture of brittle materials.
2. Fracture theory
Elastic stresses around the crack tip under mode II loading can be written as a set of series expansions:
1 0
--cos—
V2nr 2
K i
0
—cos 0- 2 tan 2 2
+ T cos2 0 + O (r1/2), 1 0 V2nr 2
—K IIsin 0
+ Tsin2 0 + O(r1/2), (1)
10
cos—[[ II(3cos 0-1) ]-
r0 2^j2nr 2 - T sin 0 cos 0 + O( r 1/2);
Fig. 1. Schematic of a SCB specimen
where (r, 0) are the polar coordinates with the origin located at the crack tip, Kn is the mode II stress intensity factor (SIF) and arr, g00 and ar0 are the stress components in the polar coordinates. The first non-singular term is called the T-stress and the higher order terms O (r ) are often negligible at locations very close to the crack tip. By considering a= v (axx + ayy) for plane strain and a= 0 for plane stress conditions, the strain energy density factor S can be expressed in terms of the stress components for the plane elasticity problems under mode II loading as
dW dA
r
2G
к + 1
(аrr +а00)2 -аrrа0
2
)+а r0
, (2)
where d W / dA is the strain energy density function and G is the shear modulus, k is an elastic constant that depends on the Poisson's ratio v and takes the value of 3-4v for plane strain problems and (3 -v)/(1 + v) for plane stress ones. By replacing the stress components from Eqs. (1) into Eq. (2) and simplifying the obtained equation, one may write the strain energy density factor S in the following form under mode II loading:
1 -[(k(1 - cos6) + cos6x
S = -
16nG
x (1 + 3cos 6)) Kjj + (-2V2rcr sin(6/ 2)(cos(26)+
+ cos 6 + ( k +1))) KllT + (( k + 1)nr )T2 ]. (3)
If r is replaced by the critical distance rc, Eq. (3) can be rewritten as follows: 16nG
K2
- S = [(к(1 - cos 0) + cos 0(1 + 3 cos 0)) +
(-2sin( 6/ 2) (cos (26) + cos 6 + ( k + 1))) Ba +
+1/2(k + 1)( Ba )2]. (4)
The dimensionless parameters B and a in this equation are defined as:
а =
B =
Тл/па
K
(5)
a к jj
where a is the crack length for edge cracks and semi-crack length for internal cracks, and B is a dimensionless parameter, often called the biaxiality ratio [19], which shows the ratio of T-stress relative to the mode II stress intensity factor.
Sih [15] proposed the strain energy density criterion for brittle fracture in mixed mode I/II loading by considering only the singular term in Eq. (1). According to this criterion, the initial crack growth takes place in the direction 0c where the amount of the SED factor at the critical distance rc is minimum. Also the onset of fracture takes place when the strain energy density factor S along 0c and at the critical distance rc from the crack tip attains a critical value
Scr Both Scr
and rc are assumed to be constant material
properties. In this research, a generalized form of the SED criterion which contains both the singular term and the T-stress is used for investigating pure mode II brittle fracture. The angle of minimum strain energy density 6c is determined by solving Eq. (6):
dS
Э0
= - sin 0c (6 cos 0c -к + 1) - cos ( 0c /2) :
x (5(cos(20c) - cos 0c) + (к + 3))Ba = 0,
(6)
Э02
> 0.
According to Eq. (6), the crack growth angle 6c depends on the magnitude and the sign of T-stress and also on the material properties (v and rc). Figure 2 shows the angular variations of normalized strain energy density factor 16nGS/Kjj with respect to the angle 6 (Eq. (4)) under mode II loading for different values of Ba and under plane strain condition when v = 0.3. According to Fig. 2, Ba has a significant influence on the strain energy density factor. In general, the angular location of minimum strain energy density depends on the value of Ba and it is equal to -82° for Ba = = 0 when v = 0.3.
If the fracture initiation angle 6c (calculated from Eq. (6)) is replaced into Eq. (4), the onset of brittle fracture based on the generalized SED criterion can be found by using the following expression for pure mode II loading:
S„ =
K
IIc
16nG
[(к(1 - cos0c) + cos0(1+ 3cos0c))+
+(-2sin (0c/2)(cos (20c ) + cos0c + (k+ 1)))Ba + + (k + 1)/2 (Ba)2], (7)
where K IIc is the critical value of mode II stress intensity factor corresponding to the fracture load. For pure mode I brittle fracture, Sih [15] showed that there is a direct relation between S and mode I fracture toughness KIc as
1
8nG
( к-1) K2
(S)
By replacing Eq. (8) into Eq. (7), the fracture toughness ratio Kn„ /K,„ can be determined from
K2
-f- = 2(к - 1)[(к (1 - cos 0c ) +
KIc
+ cos 0c (1 + 3 cos 0c )) + (-2 sin (0c / 2)(cos (20c )+ + cos 0c + ( к +1))) Ba + ( к+1)/2 (Ba )2 ]-1.
(9)
Fig. 2. Angular variations of the normalized strain energy density factor versus 6 (degrees) under pure mode II loading for v = 0.3
0=0
0=0
Equation (9) shows that based on the generalized SED criterion the fracture toughness ratio depends on v and also on Ba (which is a function of T-stress and rc).
3. Results and discussion
The GSED criterion described in Sect. 2 is used here to predict the values of mode II fracture toughness obtained from the SCB specimens made of different materials. The theoretical predictions are then compared with the experimental results reported in the literature [20-26]. The experiments were performed on polymethylmethacrylate (PMMA) [20], Guiting limestone [21], Harsin marble [22] and Johnstone [2, 23]. Details of each test including the dimensions of specimens, the material properties (mode I fracture toughness KIc, tensile strength at and Poisson's ratio v) and the biaxiality ratio B are given in Table 1. According to the relatively large value of thickness t compared with other dimensions, plane strain condition was used in the computational investigations.
The values of mode II SIF and T-stress required for the analysis of mode II brittle fracture were extracted from the numerical results presented in [27]. By substituting the numerical values of KII and T-stress into Eq. (5), the biaxiality ratio B can be determined. The angle of P for pure mode II fracture test is dependent on the normalized values of crack length a/R and half span L/R. Based on the numerical results reported in [27], the mode II crack angle P is equal to 50o for (a/R = 0.3, L/R = 0.43) and 54° for (a/R = 0.35, L/R = 0.5).
Using a maximum normal stress criterion, Schmidt [28] proposed Eq. (10) to provide an estimate for the size of critical distance r around the crack tip:
2n
\2
(10)
Therefore, the value of critical distance rc was obtained for the tested materials by substituting the experimental values of tensile strength and mode I fracture toughness of the material into Eq. (10). The obtained values of critical distance rc given in Table 1 were then used for calculating Ba from Eq. (5).
Using Eq. (9), the normalized mode II fracture toughness can be calculated in terms of the values of T-stress and rc (represented by Ba). Figure 3 shows the variations of the normalized fracture toughness under mode II loading K IIc / K Ic versus B a for different values of Poisson's ratio and for plane strain conditions estimated based on the GSED criterion. The theoretical predictions suggest that the T-stress has a significant influence on the mode II fracture toughness of brittle materials when an energy-based criterion is employed.
For each curve in Fig. 3, there is a critical value of Ba which gives the highest fracture toughness ratio KIIc/ KIc. The critical Ba is generally about -1, but its exact value
Table 1
The specimen dimensions, biaxiality ratio B and Ba for the tested specimens
Material
t, mm
a, mm
R, mm
a/R
L/R
MPa-mm
,0.5
at, MPa
rc, mm
B
Ba
PMMA [20]
5
15.0
50.0
0.30
0.43
50°
0.38 [24]
67.35
0.07
4.58
0.44
Guiting limestone [21]
20
15.0
50.0
0.30
0.43
50°
0.20 [25]
9.49
3.40
1.25
4.58
1.87
Harsin marble [22]
25
16.5
55.0
0.30
0.43
50°
0.28 [26]
31.62
7.20
3.07
4.58
2.79
Johnstone [2, 23]
20
16.5
47.4
0.35
0.50
54°
0.30
2.20
0.44
4.07
3.04
2.14
depends on the material Poisson's ratio v. The fracture toughness ratio decreases when Ba becomes larger or smaller than its critical value.
Ayatollahi et al. [20] used the SCB specimen with a = = 15 mm, the crack length ratio a/R = 0.3 and the half span L/R = 0.43 to determine the fracture toughness of PMMA with Poisson's ratio v = 0.38. As reported in [20], the average experimental values for mode I and mode II fracture toughness obtained for this material are 67.35 and 36.37 MPa• mm0 5, respectively, and the critical distance rc is equal to 0.065 mm. Once the angle 0c is calculated using Eq. (6) for given values of Ba from Table 1, the onset of mode II brittle fracture can be predicted by use of Eq. (9). The theoretical fracture toughness ratio KIIc/KIc according to the GSED criterion was obtained as 0.61. It is seen that the theoretical prediction is in good agreement with the experimentally determined K IIc / K Ic = 0.54.
Fig. 3. Variations of the fracture toughness ratio KIIc/KIc for different values of Ba under plane strain conditions based on the GSED criterion
ß
v
For the Guiting limestone, Aliha [21] utilized the SCB specimens with a = 15 mm, a/R = 0.3, LjR = 0.43 and at = 3.4 MPa and reported the average values of mode I and mode II fracture toughness as 9.49 and 3.70 MPa-mm0'5 giving the experimental fracture toughness ratio as K IIc/ K Ic = 0.39. A typical value of v = 0.2 was considered for limestone from Ref. [25]. By using rc = 1.25 mm (determined from Eq. (10)), the fracture toughness ratio KIIc /KIc is calculated from Eq. (9) to be 0.42, which again is in very good agreement with the experimental value.
Fracture experiments were also performed by Aliha and Ayatollahi [22] on Harsin marble using the SCB specimens with a = 16.5 mm, a/R = 0.3 and L/R = 0.43. The average values of mode I and mode II fracture toughness were obtained as 31.62 and 11.38 MPa - mm0 5 and the fracture toughness ratio KIIc/KIc as 0.36. Since the Poisson's ratio v of Harsin marble was not provided in [22], it was taken from Whittaker et al. [26]. Using ctt = 7.2 MPa, rc is obtained as 3.7 mm. Now, according to Eq. (9) the fracture toughness ratio is determined as KIIc/KIc = 0.30, which again is consistent well with the experimental value of 0.36. In another research work, the fracture toughness of Johnstone was evaluated by Lim et al. [2, 23]. They used the SCB specimens with a = 16.5 mm, a/R = 0.35 and LjR = 0.5 and obtained the average values of mode I and mode II fracture toughness as 2.2 and 0.92 MPa-mm05 using some experiments, respectively. By using rc = 4.07 mm (calculated from Eq. (10)) and Poisson's ratio of 0.3, the theoretical fracture toughness ratio KIIc /KIc is estimated from Eq. (9) as 0.35. It is seen that the theoretical prediction is in good agreement with the experimentally determined K IIc/ K Ic = = 0.42. A detailed description of the experimental and theoretical results obtained for the fracture toughness ratio KIIc /KIc for the above-mentioned brittle and quasi-brittle materials are presented in Table 2.
According to the classical theories of linear elastic fracture mechanics, the elastic stresses in the vicinity of crack tip are characterized only by the singular stress terms, and hence, the higher order terms of stress series expansion in Eq. (1) are often neglected. However, the GSED criterion accounts for the effect of T-stress as a key parameter in calculating the fracture toughness of brittle materials under pure mode II loading. The mode II fracture toughness is dependent not only on the value of T-stress, but also on the
critical distance from the crack tip. The critical distance rc is a material property which typically represents the radius of fracture process zone around the crack tip in brittle and quasi-brittle materials. Along the boundary of this area, the singular stresses and strains, and their associated strain energy density tend to decrease and the effect of T-stress no longer becomes negligible. For this reason, the T-stress can play an important role, particularly in fracture behavior of materials with larger critical distances rc. Table 2 show that there are considerable differences between the experimental results and the fracture toughness ratios KIIc/KIc predicated by the classical SED criterion. The difference can be related to the effect of second stress term of the elastic stress field near the crack tip. As shown in Table 2, the differences between the results of generalized SED and classical SED criteria are much higher for the tests conducted on Guiting limestone, Harsin marble and Johnstone compared with those performed on PMMA. This can be related to the fact that the values of critical distance for rocks are known to be much larger than that of PMMA, as also shown in Table 1.
According to Eq. (2), the strain energy density factor is a function of elastic stresses. Therefore, more accurate stress components can improve the accuracy of strain energy density function around the crack tip. The stress components in the generalized SED criterion consist of two terms in the Williams series expansion. The two-term solution of Williams expansion is generally more accurate than the singular term alone. Therefore, one can expect more precise estimates of mode II fracture toughness when the generalized SED criterion is utilized instead of the classical SED criterion. This can be seen in Table 2 where good agreements exist between the experimental results and theoretical predications when the generalized SED criterion is used. Hence, the value of fracture toughness calculated by using the GSED criterion could provide more reliable fracture predictions for the investigated materials under mode II loading conditions.
It is worth mentioning that the modified SED criterion proposed in this paper for pure mode II, can be extended to predict fracture toughness under mixed modes I-II loading conditions. In the suggested mixed mode GSED criterion, in addition to KII and T-stress, the mode I stress intensity factor KI should also be taken into account in the related formulations.
Table 2
The values of experimental results and theoretical estimates for the fracture toughness ratio KIIc/KIc for four brittle/quasi-brittle materials
Material PMMA Guiting limestone Harsin marble Johnstone
K IIc/ K Ic (classical SED) 0.81 1.07 0.99 0.96
K IIc/ K Ic (generalized SED) 0.61 0.42 0.30 0.35
K IIc/ K Ic (experimental) 0.54 [20] 0.39 [21] 0.36 [22] 0.42 [2, 23]
4. Conclusions
In this paper, a generalized form of SED criterion was used to investigate the effect of T-stress on fracture toughness of four different brittle and quasi-brittle materials under pure mode II loading. By including both mode II SIF (Kn) and the non-singular T-stress, the generalized SED criterion could provide significantly improved estimates for the experimental data reported in literature for the SCB specimens made of different materials. Mode II fracture toughness of brittle materials was shown to be dependent on the value of T-stress. The fracture toughness values of different materials under mode II loading were predicted very well by employing the GSED criterion. Although, the related results were obtained for four specific brittle and quasi-brittle materials, the GSED criterion can be developed to estimate the mode II fracture toughness of other brittle/quasi-brittle materials and also the results obtained from other cracked specimens.
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Morteza Rashidi Moghaddam, PhD Student, Iran University of Science and Technology, [email protected] Majid R. Ayatollahi, PhD, Prof., Director, Iran University of Science and Technology, [email protected] S.M.J. Razavi, MSc., Iran University of Science and Technology, [email protected] Filippo Berto, PhD, Prof., NTNU, Norway, [email protected]
