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УДК 539.421
Влияние размера образца на траекторию трещины при нагружении смешанного типа в материалах горных пород
J. Akbardoost, А. Rastin
В статье предложен новый подход к предсказанию траектории распространения трещины в горных породах с учетом влияния размера образца. Данный подход представляет собой метод приращений, где угол зарождения трещины на каждом шаге определяется с помощью модифицированного критерия максимальных касательных напряжений. Модифицированные критерии максимальных касательных напряжений позволяют учитывать влияние членов высшего порядка в разложении напряжений вблизи вершины трещины наряду с сингулярными членами. Такой важный параметр предлагаемой модели, как критическое расстояние rc, также зависит от размера. Проведена оценка метода приращений на основе ранее полученных экспериментальных данных для образцов известняка и мрамора. Предлагаемый подход позволяет предсказывать траекторию трещин в образцах разного размера и дает хорошее соответствие с результатами эксперимента с учетом трех членов в разложении Уильямса, которые характеризуют поле напряжений вокруг вершины трещины.
Ключевые слова: влияние размера образца, траектория трещины, материалы горных пород, нагружение смешанного типа I/II, члены высшего порядка
Scaling effect on the mixed-mode fracture path of rock materials
J. Akbardoost and A. Rastin
Department of Mechanical Engineering, Faculty of Engineering, Kharazmi University, Karaj, 31979-37551, Iran
In this paper, a new approach is presented to predict the crack growth path in the rock materials by taking into account the size effect. The proposed approach is an incremental method in which the crack initiation angle for each step is determined from the modified forms of the maximum tangential stress criterion. These modified maximum tangential stress criteria take into account the influence of the higher order terms of the stress series at the crack tip in addition to the singular terms. As an important parameter in the proposed method, the critical distance rc is also assumed to be size dependent. Finally the incremental method is evaluated by experimental results obtained from Guiting limestone and Ghorveh marble specimens reported in the previous studies. It is shown that the proposed approach can predict the fracture trajectory of cracked specimens with different sizes in good agreement with the experimental results when three terms of Williams series expansion are considered for characterizing the stress field around the crack tip.
Keywords: size effect, fracture trajectory, rock materials, mixed mode I/II loading, higher order terms
Университет имени Хорезми, Кередж, 31979-37551, Иран
Nomenclature
a — half-crack length in CCCD specimen,
A3, B3 — constant coefficients of the third terms in the
Williams series expansion,
A3, B3 — non-dimensional forms of A3 and B3,
CCCD — center cracked circular disk specimen,
ft — tensile strength,
FEOD — finite element over-deterministic method,
FPZ — fracture process zone,
K j — mode I stress intensity factor,
K jj — mode II stress intensity factor,
K* , Kjj — non-dimensional forms of Kj and Kjj,
3 ' B3
I
II,
K If — mode I fracture resistance,
MTS — maximum tangential stress criterion,
P — applied load,
Pu — fracture load,
R — radius of CCCD specimens,
r, 6 — crack tip coordinate,
rc — critical distance around crack tip,
SCB — semi-circular bend specimen,
SED — strain energy density criterion,
T — T-stress,
T * — non-dimensional form of T, t — specimen thickness,
© Akbardoost J., Rastin A., 2016
XFEM — extended finite element method, a — crack inclination angle, CTee — tangential stress around the crack tip, 60 — fracture initiation angle.
1. Introduction
The fracture in rock materials often initiates from the natural cracks, flaws and inherent discontinuities and grows along a straight or curvilinear path. The investigation of the crack growth path in rock masses is one of the most important issues for mining and civil engineers and researches in practical rock related projects such as tunneling, gas and oil wellbores, underground mining, etc. The fracture trajectory is mainly predicted because of two aspects: (i) controlling the fracture path to accelerate excavation and rock cutting and then increasing the productivity of mines and oil wells, (ii) prevention of the undesirable fractures for higher safety and stability of mines and tunnels. When the crack flanks open without any sliding, i.e. mode I loading, the crack propagates usually along its initial direction. In the complex mode mixities such as opening-sliding condition (mixed mode I/II), the crack grows along a curvilinear path which should be determined for controlling the stability of rock structures or optimizing the rock pieces.
There are several studies in the literature investigated theoretically, experimentally and numerically the fracture path in the rock materials under mixed mode loading. From the viewpoint of theoretical studies, the fracture initiation angle and crack growth are usually predicted by using the conventional fracture criteria such as maximum tangential stress [1], minimum strain energy density [2] and the maximum energy release rate criterion [3]. In such studies, the stress components or/and strain energy are written only in term of the singular terms of Williams series expansion including the stress intensity factors. For example, Gdoutos and Zacharopoulos [4] employed the classical SED criterion for predicting the fracture trajectory of marble rock obtained from anti-symmetric three-point bend specimens under mixed mode loading. It is seen from these researches that the fracture trajectories predicted by the conventional fracture criteria have often discrepancy with those obtained from the experimental results. It is well-establish that the relatively large fracture process zone length at the crack tip is the main reason for this discrepancy in the rock materials [5]. In order to investigate experimentally the fracture path of rock materials under mixed mode, several test configurations have been suggested in the previous studies. The edge cracked semi-circular bend [6-10], the single edge notched beam specimen under three or four-point bend [11-13], the center-cracked circular disk with straight notch subjected to diametral compression [14-18], the compact tension-shear specimen [19, 20], the edge cracked triangular specimen under three-point bend loading [21], the angled edge crack specimen subjected to a uniform far-field tension [22]
and the double edge notch beam specimen [23, 24] are among the specimens used frequently in the past for assessing the mixed-mode fracture trajectory. The cohesive elements and extended finite element method are also two well-known methods in the numerical approaches which have been employed frequently for predicting the fracture path of rock and concrete specimens under mixed mode loading [25, 26].
The experimental results in the previous studies showed that the size of specimens affects not only on the fracture resistance, but also on the fracture path of rock materials under mixed mode loading. Therefore, the fracture trajectories in the real rock masses should be predicted by considering the effect of specimen size. There are enormous studies in the literature for investigating the size effect on the fracture resistance of rock materials under pure mode I, pure mode II and mixed mode loading such as studies carried out by Bazant [27], Karihaloo [28], Carpinteri [29], Jenq and Shah [30], Planas et al. [31] , Dyskin [32], Bazant and Pfeiffer [33], Khan and Al-Shayea [34] and Akbardoost et al. [35]. However, a few studies can be found to assess the dependency of the fracture trajectory on the specimen size. For instance, Aliha et al. [36] investigated experimentally the effect of specimen size on the fracture trajectory of Guit-ing limestone using SCB and CCCD specimens with different sizes. Therefore, an appropriate and applicable approach needs to be proposed for predicting the size dependent fracture path of cracked specimens as aimed to present in this paper. The proposed approach is an incremental method in which the fracture initiation angle for each step is determined from the modified forms of the MTS criterion. These modified forms of MTS criterion use the higher order terms of Williams series expansion for calculating the fracture initiation angle 60. The critical distance rc which is an important parameter in this approach, is considered to be dependent on the size of specimen. An empirical formula proposed recently by Ayatollahi and Akbardoost [37] is also used for explaining the size dependency of rc. In order to evaluate the proposed approach, the experimental results reported by Aliha et al. [36] for CCCD specimen of radius 50 mm under mixed mode loading is first utilized. Then, the incremental approach is employed for predicting the mixed mode fracture paths of Ghorveh Marble samples tested by Akbardoost and Ayatollahi [38] on the CCCD specimens with different radii. By taking into account the three terms of Williams series expansion for calculation of the initiation angle 60, the proposed approach predicts the fracture trajectories of Ghorveh specimens in good agreement with the experimental observations.
2. Fracture theory
Similar to the other quasi-brittle materials, rock is weak under tensile loading and fracture processes usually along the direction where the principal tensile stress around the
crack tip is a maximum [3 9-42]. The maximum tangential stress criterion [1] can take into account this inherent property of quasi-brittle materials and states that the fracture grows radially from the crack tip perpendicular to the direction of the maximum tangential stress. Accordingly, the fracture takes place when the maximum tangential stress at a critical distance rc from the crack tip reaches to a critical value ^66c. The tangential stress component around the crack tip can be written from the Williams series expansion [43] in form of
a66 (r, 6) = ^
6 1 36
,-1 cos — + — cos- | +
4V2nTi 2 3 2
, rr • 2 n , 15 0.5 A
+ Tsin 6+--r A3
6 1 56
cos---cos—
2 5 2
3 Ktt
4 V2nr
15
0.5
.6 . 36 , sin—+ sin— |-2 2
56
sin2 - siny |+O(rX
(1)
where r and 6 are the conventional crack tip coordinates, KI and KII are respectively the mode I and mode II stress intensity factors, Tis the T-stress parameter and A3 and B3 are the coefficients of the third terms in the series expansion. The higher order terms O(r) are the terms related to rn 2 (n = 4, 5, ...) which are assumed to be negligible near the crack tip. Based on the MTS criterion, the direction of fracture onset can be determined by differentiating Eq. (1) with respect to 6 and then setting the expression equal to zero: 3a6
J66
36
= 0.
(2)
6=60
It should be noted that those answers of Eq. (2) are acceptable which make the second differentiation of Eq. (1) positive, i.e.
3 2c
66
362
> 0.
6=60
The classical MTS criterion proposed by Erdogan and Sih [1] considers only the singular terms in Eq. (1) and ignores the higher order terms. Therefore, the fracture initiation angle 60 is determined from the classical MTS criterion by replacing the singular terms of Eq. (1) into Eq. (2):
KI sin 60 + KII(3cos 60 -1) = 0. (3)
It is recently established that the higher order terms have great influence on the fracture behavior of rock and concrete materials which have relatively large critical distance rc [15, 32, 35, 44-46]. For example, Smith et al. [46] took into account the T-stress term in addition to the singular terms for characterizing the tangential stress component and improved the estimates of the classical MTS criterion. They also proposed that the fracture initiation angle 60 can be obtained from [46]:
Kj sin 6 0+K n (3 cos 60 -1) -
16T /-- „ . 60 „
--\/2rtrc cos 60 sin—0 = 0,
3 V c 0 2
in which rc is the critical distance from the crack tip.
As another example, Ayatollahi and Sistaninia [47] demonstrated that the more accurate estimates for fracture behavior of rocks can be provided when the first three terms of Williams series expansion are used in calculating the tangential stress. Accordingly, the value of 60 can be calculated by ignoring the higher order terms O(r) in Eq. (1) and then replacing the expression into Eq. (2):
3K
. 60 . 360
sin—0 + sin-0
2 2
8V2
15 i-f . 60 . 560 ,
- — A3J rcl sin—0 - sin—0 |+ 0 ^ c 1 2 2
+ T (sin (260)) -
3 KII
cos60 + 3cos-36° |-2 2
8V2nTc
-15£3^fcos-5cos-6° |= 0,
(5)
The crack tip parameters KI, KII, T, A3 and B3 depend on the magnitude of the applied load and the size of specimens and their magnitudes may vary over a wide range for different test samples. Therefore, for the sake of convenience, these crack tip parameters are usually described in dimensionless forms as
KI = —•J2kRK *, I Rt
K ii =—V2nRK *I, Rt
4 P *
T =-T ,
Rt
A = —-L A* A3 = Rt4RA3'
(6) (7)
(8)
(9)
(10)
B3=—tRB;
where P is the applied load, t and R are the thickness and characteristic dimension of specimen, respectively. The di-mensionless parameters KI, Kn, T , A* and £3 depend only on the geometry ratios such as the crack length ratio a/R. They are also independent of the magnitude of load and the specimen dimensions and can be obtained by finite element analysis for each specimen configuration. By substituting Eqs. (6)—(10) into Eqs. (3)-(5), the relations for determining the fracture initiation angle 60 can be rewritten in terms of dimensionless parameters:
K* sin 60 + K*I (3 cos 60 -1) = 0, (11)
K* sin 6 0+K* (3 cos 60 -1) -
64T rc . 60 _
—J ^cos = 0,
~ K *
15
sin—0 + Sin-2
30o
+ 4T JRsin(20o) -
— A* rC
—A*
3 R
sin
sin
50n
+ * K *
15 r.
V
cos
0O 30n^ + 3cos—0
8 R
B*
cos-
■5 cos
50n
= 0.
(13)
* / *
According to Eq. (11), the ratio of Kn/ K* is only parameter required for determining the value of 60. Thus, the conventional MTS criterion predicts a unique value of 60 for different specimens with the same mode mixity. Experimental results showed that the directions of fracture initiation obtained from two specimens with a fixed value of * / *
Kn/ K* but various geometry are significantly different [36, 48]. This discrepancy were justified in the previous studies by taking into account the effect of second and third terms of Williams series expansion (i.e. Eqs. (12) and (13)) [35, 36, 47-53].
In order to use Eqs. (12) and (13) for calculating 60, the value of rc should be determined. There are several formulations in the literature for calculating the critical distance rc in rock materials [5, 28]. Ayatollahi and Akbardoost [37] proposed recently a modified form of Schmidt's formula [54] for determining the value of rc. The proposed equation can be written as
_ 2
/t ^ 2/2 -12( A3*/ K *)( K jf/R)
r =
6(A*/ K *)(K if /R )
(14)
In this formula, KIf is the mode I fracture resistance, K* and A* are the dimensionless parameters for the coefficients of first and third stress terms in pure mode I loading. It is seen from the previous studies (e.g. [47, 55]) that the magnitudes of rc can be independent of the mode mixities and are almost the same for modes I and mixed mode fracture experiments.
The experimental results have shown [5, 28, 37, 56] that the value of rc depends on the size of specimen and increases by increasing the specimen size. Since the major objective in this paper is investigation of the specimen size on the fracture trajectory of rock materials, the size dependent value of rc should be considered. In this paper, a simplified form of two well-established formulas proposed by Karihaloo [57] and Bazant et al. [56] is employed for explaining the size dependency of rc:
rc , (15)
c 1 + B/R
where the constant coefficients A and B are calculated by a linear regression on fracture resistance obtained from mode I tests conducted on specimens of different sizes. For calculating A and B, the value of rc corresponding to each
Initial crack
Fig. 1. The schematic of the incremental method for predicting the crack growth path
specimen size is first determined by replacing the mode I fracture resistance KIf obtained from the experiment into Eq. (14). Then a linear regression based on y = c0 + c1 x, in which
=1 = B = 1 = 1
y = r/ Cl = A 9 X = R 9 C0 = A
is used. One can find more details for determining parameters A and B in Ref. [37].
By substituting the value of rc corresponding to the specimen size into Eqs. (12) and (13) the fracture initiation angle 60 can be determined. In the next section, an incremental method will be described for predicting the fracture path in which the direction of fracture onset is determined from one of the Eqs. (11)—(13).
3. Incremental method
One of the semi-numerical methods for predicting the fracture trajectory is the incremental method. In this method, the direction of fracture initiation for a cracked specimen with initial length a0 is determined from the fracture criteria such as MTS. Afterward, a small crack is added to the initial crack along the fracture direction and then a speci-
Fig. 2. The schematic of the CCCD specimen
Table 1
Table 2
Specimen geometries of the tested CCCD specimens and their corresponding fracture loads for Guiting limestone [59]
Radius Thickness Crack length Crack inclina- Fracture
R, mm t, mm 2a, mm tion angle a load Pu, N
50 40 30 0° (pure mode I) 5261
50 40 30 5° 5272
50 40 30 10.5° 5300
50 40 30 18° 5380
50 40 30 27° (pure mode II) 4880
men with a new crack length is generated. The fracture initiation angle for this new sample is again determined from the fracture criteria. In the next step, a small crack is added to the new cracked sample. These increments are continued until the crack path reaches the back boundaries of specimens. Figure 1 shows a schematic of the incremental method. In this paper, the MTS criterion which is written in both the classical form (Eq. (11)) and the modified forms (Eqs. (12) and (13)) is used for determining the fracture initiation angle in each step. It should be noted that the value of rc in Eqs. (12) and (13) is assumed to be constant for each step and is substituted relative to the size of specimen. The acceptability of this assumption has been assessed recently by Ayatollahi and Akbardoost [58]. They calculated the values of rc for SCB specimens with different sizes under pure mode I and pure mode II loading. It has been shown that the differences between rc for pure mode I and pure mode II are less than 10% and therefore, one can consider the size-dependent curve of rc obtained from mode I
Specimen geometries of the tested CCCD specimens and their corresponding fracture loads for Ghorveh marble [38]
Radius R, mm Thickness t, mm Crack length 2a, mm Crack inclination angle a Fracture load Pu, N
0° (pure mode I) 4936
25 27 25 9° 4750
23° (pure mode II) 6073
0° (pure mode I) 8920
50 27 50 9° 8720
23° (pure mode II) 8971
0° (pure mode I) 14637
95 27 95 9° 15606
23° (pure mode II) 13905
0° (pure mode I) 24309
190 27 190 9° 24542
23° (pure mode II) 20532
(Eq. (15)) as an acceptable curve for rc in mixed mode and mode II. In the upcoming sections, the fracture paths of Guiting limestone and Ghorveh marble obtained from two experimental setups will be predicted by the incremental approach.
a= 10.5°
1 Fracture
path
^ 1/
i Initial
crack \
a = 27°
Fracture
__ path
/ V \
\ 1
Initial crack
Fig. 3. The fracture path in the Guiting limestone specimens
with R = 50 mm under different mode mixities [59]
a = 0° (pure mode I)
a = 9°
a = 23° (pure mode II)
Fracture path \
Initial crack '
b Fracture path \ t _^
— ------^ i I 1 I Initial__J crack
0 ■ ~ Fracture path
1. Initial crack
Fracture
\ i path
^ _ \ \ \ i V l \ \
Initial \ \ \ < \ 1
crack —1
Fig. 4. The broken specimens of Ghorveh marble with radii R = 25 (a), 50 (b), 95 (c) and 190 mm (d) and various mode mixities [38]
4. Experimental investigation
In order to evaluate the proposed approach, the fracture trajectories obtained from two sets of experiments are used. The first one is the experiments conducted by Aliha et al. [59] on the Guiting limestone and the second one is the experiments carried out by Akbardoost and Ayatollahi [38] on the Ghorveh marble specimens with different sizes. The test configuration in both studies was the center-cracked circular disk under diametral compression as shown schematically in Fig. 2. As seen from this figure, the CCCD specimen is a circular disk of radius R containing a central crack with length 2a. By changing the orientation of applied load relative to the direction of the crack, the combinations of mixed mode loading from pure mode I (a = 0°) to pure mode II (a = an) can be achieved. The dimensions and the fracture loads of the CCCD specimens for both Guiting limestone and Ghorveh marble are listed in Tables 1 and 2, respectively. Figure 3 shows the fracture path of the
specimens made of Guiting limestone under pure mode I, mixed mode I/II (Kj = K n) and pure mode II loading. Figure 4 also displays the path of crack growth in the CCCD specimen made of Ghorveh marble with various sizes subjected to the mixed mode loading.
5. Results and discussions
It is observed from Figs. 3 and 4 that the crack extends along its initial direction in the specimens under pure mode I, while its extension is along a curvilinear path from the crack tip for samples subjected to mixed mode loading. The MTS criterion states that the crack growth takes place along the direction where the tangential stress is a maximum. There is an important point raised here that which formula related to MTS criterion (i.e. Eqs. (11)—(13)) provides the more accurate estimates for fracture initiation angles 60. For this purpose, the experimental results obtained from the Guiting limestone samples are first used. In other words, the frac-
Fig. 5. The dimensionless parameters K*, KT* A*, B* versus the crack angles a for the CCCD specimen with a/R = 0.3 (a) and 0.5 (b) [61]
ture paths in Guiting limestone are predicted by using Eqs. (11)—(13) in the incremental method and it is shown which formula can provide more good estimates.
For determining the fracture initiation angles using Eqs. (11)-(13), the dimensionless parameters KI? Kn, T , A3, and B3 should be determined in each step. These dimensionless parameters are calculated from analytical methods for simple geometry and loading conditions or numerical methods for more complicated ones. Recently, Ayatollahi and Nejati [60] proposed a finite element over-deterministic method for determining not only the parameters KI , Kn, T , A3 , and B3 , but also the higher order terms in Williams series expansion as noted by O(r) in Eq. (1). In the FEOD method, the displacement components for a large number of nodes in a particular ring far enough from the crack tip are calculated from the finite element analysis. Employing the Williams series expansion for displacement field around the crack tip, an over-determined set of linear equations is obtained. Finally, the over-determined equations are solved in conjunction with the least-square method and the crack parameters are calculated. Akbardoost and Rastin [61 ] calculated comprehensively the
parameters KKH, T*, A*, and B* for two famous disktype specimens SCB and CCCD with an initial crack length ratio subjected to a wide range of mixed mode I/II loading. These dimensionless parameters for CCCD samples with initial crack length ratio a/R = 0.3 and 0.5 are extracted from Ref. [61] and are depicted in Fig. 5.
After calculating the dimensionless parameters, the value of rc should be determined when Eq. (12) or Eq. (13) is used. The parameter rc in this paper is obtained from Eq. (14). The mode I fracture resistance KIf for Guiting limestone is KIf = 0.206 MPa • m1/2 obtained from CCCD specimen of radius 50 mm under pure mode I [59]. The tensile strength ft of Guiting limestone is ft = 2 MPa as given in Ref. [59]. The dimensionless parameters KT and A** in Eq. (14) are also extracted from Fig. 5, a according to a = 0°. By substituting the values of KIf, ft, KT and A* in Eq. (14), the parameter rc for Guiting limestone is calculated as rc = 2.05 mm.
Now, the fracture path of Guiting limestone can be predicted by using incremental method. Figure 6 shows the comparison between the fracture trajectories of CCCD specimens predicted by the incremental method using
Eqs. (11) -(13)
. Experiment
r
Eq. (13)
Eq. (12) Eq. (11)
Eq. (11),
- Crack tip
; Experiment Crack tip
a = 5o
"Eq. (13)
Experiment
Eq. (12) Crack tip
Eq. (12)
Eq.(11)
Eq. (12)
Eq. (13)
Experiment Eq. (13)
Experiment Eq. (11^ Crack tip Crack tip
a = 18o
a = 0o a = 5° a = 10.5° a = 18° a = 27°
(pure mode I) (pure mode II)
Fig. 6. Comparison between the fracture path obtained from experiments [59] and those predicted by the incremental method using Eqs. (11)—(13)
Table 3
The values of KIf and rc for Ghorveh marble obtained from CCCD specimens of different sizes
Radius R, mm KIf, MPa • mI/2 rc, mm
25 0.681 3.26
50 0.887 5.29
95 I.0I5 6.46
195 I.206 I0.34
Eqs. (11)—(13) and those obtained from the broken samples. It is seen from this figure that the proposed technique predicts the crack extension in pure mode I loading along the direction of initial crack. In such a case, the predicted crack extension is close to the fracture path observed from the experimental results. Comparing the fracture path predicted
by incremental method with the experimental observations in the mixed mode loading, the estimates of the incremental method related to Eqs. (12) and (13) are more accurate than those obtained from Eq. (11) in comparison. The direction of the crack extension in pure mode II predicted by using Eq. (13) in the incremental method is found to be significantly compatible with the experimental results rather than two other equations. Therefore, one can conclude from Fig. 7 that the incremental approach by using the high order terms T, A3 and B3 for calculating the fracture initiation angle 60 can provide good estimates for crack growth trajectory.
To revalidate the proposed method, the experimental results of Ghorveh marble are utilized. It should be noted that the size of CCCD specimens is various and therefore the fracture path of Ghorveh marble is size dependent. In order to take into account the size dependency of fracture
a = 0°
Experiment
Eq. (13)
Crack tip
Eq. (13)
Experiment
Crack tip
Eq. (13)
\
(
Experiment
K 1
Crack tip
Eq. (13)
I \
\ r \
Experiment
Crack tip
a = 9°
a = 23°
Eq. (13)
Eq. (13)^i Experiment
Crack tip
Eq. (13)
Experiment
w
Experiment
\ , Eq. (13)
Crack tip
Eq. (13)
Eq. (13)
Experiment
Crack tip
Experiment
s \
Eq. (13)
Crack tip
Crack tip
Eq. (13)
Experiment
Crack tip
Fig. 7. The fracture trajectories of Ghorveh marble samples with different radii R = 25 (a), 50 (b), 95 (c) and 190 mm (d) and different mode mixities predicted by incremental method in comparison with the experimental results [38]
path, the parameter rc in the proposed method is assumed to be size dependent. The variations of rc with respect to the specimen size is described by using Eq. (15) in the present study. To utilize Eq. (15), the empirical parameters A and B should be determined from a linear regression on the values of rc for specimens with different sizes. Therefore, the values of rc should be first determined for each specimen size from Eq. (14). The parameters KI and A3 in Eq. (14) are respectively extracted from Fig. 5, b for a = 0° as K * = 0.221 and A3* = 0.072. The tensile strength of Ghorveh marble has been also given in Ref. [38] as ft = 5.4 MPa. The mode I fracture resistance KIf in Eq. (14) is determined for each size of specimen by substituting the fracture load Pu given in Table 2 and the dimen-sionless parameter K* corresponding to the pure mode I loading into Eq. (6). Table 3 shows the values of KIf and rc for each specimen size. Then, the linear regression fit to the size dependent values of rc gives the parameters A = 12.5 mm and B = 71.625 mm, respectively. By determining the parameters A and B, the variations of rc versus the specimen size can be described from Eq. (15) as
125 (16)
rc =-
- mm.
1 + 71.625/ R
After calculating the size dependent values of rc, the fracture path of Ghorveh marble samples are predicted by using the incremental method as shown in Fig. 7 for CCCD specimens with radii of R = 25, 50, 95 and 190 mm, respectively. Since the incremental method related to Eq. (13) is more accurate than the other formulas, only the fracture paths predicted by using Eq. (13) are depicted in these figures. Figure 7 demonstrate that the proposed approach can provide good estimates for the fracture trajectories of Ghorveh marble when the size of specimen is varied.
It is noteworthy that the incremental method does not need to the fracture loads for predicting the fracture path. In other words, the proposed method corresponding to Eq. (13) uses the dimensionless parameters (KI, Kn, T , A3 and B3) and the size-dependent value of critical distance rc. The dimensionless parameters are determined from FEOD method in conjunction with the finite element analyses. The value of rc is also obtained from mode I fracture tests for specimens with different sizes. Therefore, one can predict the fracture traj ectory of rock specimens under mixed mode loading without the need of the fracture force. This is the main advantage of the proposed approach relative to the other methods used for predicting the fracture path.
6. Conclusions
The fracture trajectory of rock materials under mixed mode loading was studied by taking into account the effect of specimen size. For this purpose, a new incremental approach based on the maximum tangential stress criterion was proposed. The proposed technique makes the use of
the higher order terms of Williams series expansion in addition to the singular terms for determining the fracture initiation angles 60 in each increment. The critical distance rc was also assumed to be size dependent and an empirical relation was used for characterizing the size dependency of rc. The proposed method was evaluated by the experimental results obtained from the Guiting limestone and Ghorveh marble samples. It was founded that the proposed method can provide good estimates for size-dependent fracture trajectories of cracked specimens under mixed mode loading by considering the first three terms of series expansion for describing the stress field at the vicinity of the crack tip.
References
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Поступила в редакцию 02.03.2016 г.
CeeoeHUM oo aemopax
Javad Akbardoost, PhD, Assist. Prof., Kharazmi University, [email protected] Amir Rastin, MSc. Stud., Kharazmi University, [email protected]
