Three-dimensional numerical evaluation of the progressive fracture mechanism of cracked chevron notched semi-circular bend rock specimens

Engineering Fracture Mechanics 134 (2015) 286–303

Contents lists available at ScienceDirect

Engineering Fracture Mechanics journal homepage: www.elsevier.com/locate/engfracmech

Three-dimensional numerical evaluation of the progressive fracture mechanism of cracked chevron notched semi-circular bend rock specimens M.D. Wei a, F. Dai a,⇑, N.W. Xu a, Y. Xu a, K. Xia b a State Key Laboratory of Hydraulics and Mountain River Engineering, College of Water Resources and Hydropower, Sichuan University, Chengdu, Sichuan 610065, China b Department of Civil Engineering, University of Toronto, Toronto, ON M5S 1A4, Canada

a r t i c l e

i n f o

Article history: Received 31 May 2014 Received in revised form 22 November 2014 Accepted 25 November 2014 Available online 3 December 2014 Keywords: CCNSCB Progressive fracture Chevron notch Acoustic emission Heterogeneity

a b s t r a c t The cracked chevron notched semi-circular bending (CCNSCB) method for Mode-I fracture toughness measurement has adopted the straight-through crack propagation assumption but never being fully verified. In this study, three-dimensional progressive fracture processes of CCNSCB specimens are numerically evaluated for the first time considering different supporting spans and heterogeneities. Results show that the crack front of the CCNSCB specimen is not straight-through but considerably curved, which inevitably induces errors for fracture toughness measurement; and the damaged/fractured zone can be prominently confined in the chevron notched ligament for a CCNSCB specimen with a large supporting span. Ó 2014 Elsevier Ltd. All rights reserved.

1. Introduction Rock fracture mechanics has been widely adopted in many engineering applications related to rock breakage or failure such as rockbursts, rock mass slope stability, rock cutting, hydraulic fracturing, tunneling, underground excavation, oil exploration and deep burial of nuclear waste. As an intrinsic property of rocks to resist fracture, the fracture toughness is of great significance in the research of rock fracture mechanics. Among the three basic fracture modes (i.e. Mode-I, the tensile mode; Mode-II, the shear mode; Mode-III, the tear mode), Mode-I (opening mode) fracture is the most frequently encountered failure mode of rocks against fracture; and thus the Mode-I fracture toughness has been mostly studied and an accurate determination of the Mode-I fracture toughness has attracted broad interests in rock fracture mechanics community. Myriads of methods with varying sample configurations have been adopted in rock fracture toughness measurements including Brazilian disc (BD) method [1], notched semi-circular bend (NSCB) method [2] and [3], cracked chevron notched semi-circular bend (CCNSCB) method [4–6], cracked straight through Brazilian disc (CSTBD) method [7–11], cracked chevron notched Brazilian disc (CCNBD) method [12–16], diametric compression (DC) test [17], double edge cracked Brazilian disc (DECBD) method [18], edge crack triangular test [19], flattened Brazilian disc (FBD) method [20], hollow center cracked disc (HCCD) method [21], holed-cracked flattened Brazilian disc (HCFBD) method [22], holed-flattened Brazilian disc (HFBD) method [23], modified ring (MR) test [24], radial cracked ring [25] and [26], straight edge cracked round bar bend (SECRBB) ⇑ Corresponding author. Tel.: +86 28 8540 6701. E-mail address: [email protected] (F. Dai). http://dx.doi.org/10.1016/j.engfracmech.2014.11.012 0013-7944/Ó 2014 Elsevier Ltd. All rights reserved.

M.D. Wei et al. / Engineering Fracture Mechanics 134 (2015) 286–303

Nomenclature a a0 a1 am AE B BD CB CCNBD CCNSCB CSTBD D DC DECBD E E0 0 0 f c; f t FBD HCCD HCFBD HFBD ISRM k KI KIC m MR N ni NSCB P Pmax R RFPA RQD Rs S SECRBB SHPB SIF SNDB SR W(x) Y⁄ Y min

a0 a1 aB am as b

r01 ; r03

rc0, rt0 rrc, rrt e e et0 eut k

u

0

crack length initial crack length final crack length critical crack length acoustic emissions specimen thickness Brazilian disc chevron bending cracked chevron notched Brazilian disc cracked chevron notched semi-circular bend cracked straight through Brazilian disc damage variable diametric compression double edge cracked Brazilian disc elastic modulus of the damaged element elastic modulus of the undamaged element compressive and tensile failure strength of the element, respectively flattened Brazilian disc hollow center cracked disc holed-cracked flattened Brazilian disc holed-flattened Brazilian disc method International Society of Rock Mechanics current calculation steps Mode I stress intensity factor Mode I fracture toughness heterogeneity index modified ring total number of all elements in the model number of damaged elements in the ith step notched semi-circular bend load maximum load radius of the disc Rock Failure Process Analysis rock quality designation radius of rotary saw span of the two support rollers straight edge cracked round bar bend split Hopkinson pressure bar stress intensity factor straight notched disk bending short rod Weibull distribution dimensionless stress intensity factor minimum dimensionless stress intensity factor dimensionless initial crack length dimensionless final crack length dimensionless thickness dimensionless critical crack length dimensionless radius of rotary saw the ratio of span to diameter major and minor principal stress, respectively uniaxial compressive and tensile strength, respectively elemental residual compressive and tensile strength, respectively strain equivalent strain threshold strain at the elastic limit ultimate tensile strain residual strength coefficient angle of friction

287

288

M.D. Wei et al. / Engineering Fracture Mechanics 134 (2015) 286–303

method [27] and straight notched disk bending (SNDB) method [28]. At present, there are no standard methods of measuring Mode-I fracture toughness of rocks; only four suggested methods are proposed by International Society of Rock Mechanics (ISRM), namely chevron bending (CB) method and short rod (SR) method [29], CCNBD method [13] and NSCB method [30] and [31]. It is noted that all these documented methods on Mode-I fracture toughness measurements can be categorized into two groups (group I and group II). In group I, testing samples are fabricated with a sharp notch, which is the critical crack to determine the fracture toughness by virtue of the failure load. The ISRM suggested NSCB method [30] and [31] belongs to this group. While all samples in group II feature a chevron notch. The cracking initiates and grows first in a stable and then unstable manner as the load is applied; the critical crack occurs somewhere in the middle of the chevron notch ligament. The ISRM suggested CB method and SR method [29] and CCNBD method [13] fall into this group II. The cracked chevron notched semi-circular bending (CCNSCB) method, originally initiated by Kuruppu [4] and later developed by Dai et al. [5], appears to have retained the merits of the ISRM suggested CCNBD method [13] and the ISRM suggested NSCB method [30] and [31] such as larger number of core specimens can be prepared by halving the cores, simple specimen installation and loading fixture, adaptation of high strain rates, easier sample preparation, and availability of pure or mixed mode (Mode-I and Mode-II) fracture studies of rock materials. In addition, this CCNSCB method overcomes major weaknesses of the CCNBD method [13] and the NSCB method [30,31]. The CCNSCB sample geometry can be essentially considered as half CCNBD and it inherently avoids the symmetrical fracture propagation assumption of the CCNBD method; and the saw-cut chevron notch avoids the fabrication of a sharp crack for NSCB samples. Moreover, in the measurement of dynamic rock fracture parameters, the half CCNBD sample geometry (half disc) is especially favored with the shortened length than CCNBD sample (full disc), because a shorter sample facilitates the dynamic force balance in the sample, which is the prerequisite to employ the classic quasi-static data reduction method in SHPB test [30]. Indeed, the availabilities and advantages of CCNSCB method in dynamic tests are confirmed by Dai et al. [5]. Given its merits on both static and dynamic Mode-I fracture toughness measurements, the CCNSCB method can be a good candidate for more accurately determining toughness values, and thus it is chosen to be analyzed in this research. It is noted that for all chevron notched fracture samples involving both the CCNSCB and CCNBD methods, the assumed progressive fracture mechanism has rarely been fully assessed; but this assumption is crucial to determine the exact location and fracture shape of the critical crack length using which the fracture toughness can be accurately measured. In this study, a numerical code RFPA3D is adopted to systematically and realistically simulate the progressive fracture of the CCNSCB specimen considering the rock heterogeneity [32]; the acoustic emissions (AE) associated with microcracking are numerically demonstrated as well. This paper is organized as follows. In Section 2, the testing principle and postulation of CCNSCB method are introduced. The RFPA3D code and the model setup are described in Section 3. In Section 4, numerical simulation of progressive fractures of CCNSCB specimens with different heterogeneity indexes and different supporting spans are presented in detail; this is followed by a comprehensive discussion on the failure mechanisms in Section 5. The results are summarized in Section 6.

2. The principle and postulation of CCNSCB method Fig. 1 shows the geometric details of CCNSCB specimen, which can be fabricated from a CCNBD specimen by a diametrical cut. All the restrictions and geometrical relationships for CCNBD [13] are assumed to be applicable to CCNSCB. P is the applied load during the test. B and R are the thickness and radius of the specimen respectively. S is defined as the span of the two support rollers and b is the ratio of S to 2R in this paper. a is the crack length. a0 (=a0/R), a1 (=a1/R) and am (=a/R) are the dimensionless initial, final and critical crack length, respectively. as is equal to Rs (radius of rotary saw) divided by R. aB (=B/R) is the dimensionless thickness. In order to have extensive and valid tests, the dimensions a1 and aB of a specimen should satisfy the following restrictions [13]:

P

B

a1 R

a

Rs

b

S Fig. 1. Schematic of the CCNSCB sample geometry.

a0

a

R

M.D. Wei et al. / Engineering Fracture Mechanics 134 (2015) 286–303

8 a1 P 0:4 > > > > > a1 6 0:8 > > > < a P a =2 1 B > a B P 0:44 > > > > > aB 6 1:04 > > : aB P 1:1729  a1:6666 1

289

ð1Þ

Similar to the calculation method suggested by ISRM for CCNBD specimen [13], the stress intensity factor (SIF) of a crack (in CCNSCB with a certain span to diameter ratio b) at length a can be determined by:

P K I ¼ pffiffiffi Y  B R

ð2Þ

For determining the Mode-I fracture toughness, the minimum dimensionless SIF, denoted as Y min , is the most important value for the specimen. It corresponds to the failure load recorded during testing. Then, the fracture toughness can be calculated as:

Y  ¼ K I=



P pffiffiffi B R



Pmax K IC ¼ pffiffiffi Y min B R

ð3Þ

ð4Þ

where KI is the Mode-I SIF of the sample; Y⁄ is the dimensionless stress intensity factor; KIC is the Mode-I fracture toughness of rock material; Pmax is the recorded maximum load at failure and Y min is the critical dimensionless stress intensity factor of the semi-circular specimen. The testing principle of the CCNSCB method assumes that, cracks initiate from the tip of the jag notch, and then grow toward the top loading plate with perfect straight-through crack fronts. Additionally, cracks always propagate along center of the notch width h, as illustrated in Fig. 2. During the the whole process of this diametrical-compression test on the CCNSCB sample, the variation of Y  as the crack propagates is schematically shown in Fig. 2e. The value features a transition from decreasing trend to increasing trend with increasing crack length. The A, B and C in both Fig. 2d and e correspond to three progressive fracture processes shown in Fig. 2a–c, respectively. Theoretically, the fracture toughness value can be measured using Eq. (2) at any loading or unloading stage (e.g., point A or C in Fig. 2e), as long as the recorded force at given stage and corresponding dimensionless SIF of the fracture can be obtained. However, although the load can be recorded from experiments, it is impossible to determine the corresponding crack length and Y  simultaneously. Fortunately, when the loading force P reaches its maximum value of Pmax (point B in Fig. 2d), the dimensionless SIF Y  should meet its minimum of Y min (point B in Fig. 2e), i.e. the critical dimensionless SIF. At this moment, the dimensionless fracture length a is the critical value am. Thus, the fracture toughness can be calculated via Eq. (4) using the recorded peak force Pmax and the calibrated minimum dimensionless SIF Y min determined numerically before the experiments are performed. 3. Brief description of RFPA3D and model setup 3.1. The RFPA3D code The RFPA3D (Rock Failure Process Analysis-3D) code is developed by Tang et al. [33] based on finite element method. To simulate the progressive process, failure induced stress redistribution, seismic events, and initial random distribution of micro features are considered in the deformation of an elastic material. To take the mesoscopic properties of rocks into account, the elemental mechanical parameters (e.g., the elastic modulus and uniaxial compressive strength) are described to follow a certain statistical distribution such as the Weibull distribution [34]:

WðxÞ ¼

 m1   m  m x x exp  x0 x0 x0

ð5Þ

where x is a mechanical property such as the strength of elements, x0 is the mean value of the corresponding mechanical parameter, and m determines the shape of the distribution function and it can be deemed as the heterogeneity index of materials. According to the Weibull distribution, with a larger m, elemental mechanical properties get closer to the mean values; this depicts a more homogeneous rock specimen (see Fig. 3), and vice versa [35]. In this code, each element is supposed to be isotropic, linear elastic, and damage-free before it fails. As an element damages, its elastic modulus degrades gradually as damage develops according to the relation [36,37]:

E ¼ ð1  DÞE0

ð6Þ

290

M.D. Wei et al. / Engineering Fracture Mechanics 134 (2015) 286–303

∆a h

h

C

A

h

Pmax Y*

B Force

∆a

∆a

* Ymin

A

C B α(a/R)

Time

Fig. 2. Schematics of the ideal postulation of the chevron notched sample about fracture initiation and growth.

m=1.1

m=2

m=10

Fig. 3. Examples of rock models in RFPA3D with different heterogeneity.

where E0 and E represent the elastic modulus of the undamaged and damaged element, respectively. D denotes the damage variable. Tensile failure in an element occurs when the minimum principal stress reaches the uniaxial tensile strength, as described by:

M.D. Wei et al. / Engineering Fracture Mechanics 134 (2015) 286–303

r03 6 f 0t

ð7Þ

where r is the minor principal stress. summarized as follows: 0 3

8 > < 0; D ¼ 1  erErt0 ; > : 1;

291

0 ft

is the tensile failure strength of the element. The damage variable D can thus be

e < et0 et0 6 e 6 eut e > eut

ð8Þ

where rrt is the elemental residual tensile strength, and rrt = krt0. k and rt0 are the residual strength coefficient and uniaxial tensile strength, respectively. et0 is the threshold strain at the elastic limit, eut is the ultimate tensile strain when the element is completely damaged, and e is the strain. To judge the shear damage of an element, the Mohr–Coulomb failure criterion is employed:

r01  r03

1 þ sinu0 0 P fc 1  sinu0

ð9Þ 0

where r01 is the major principal stress, /0 is the angle of friction, and f c is the compressive failure strength of the element. The corresponding damage variable D can be determined as:

( D¼

0; 1  er1 rcE0 ;

e1 < ec0 e1 P ec0

ð10Þ

where rrc is the residual compressive strength, rrc = krc0 and rc0 is the uniaxial compressive strength and et0 is the threshold strain while shear damage occurs. Although only uniaxial tensile and compressive damage are introduced above, the constitutive law for the element subjected to multi-axial stress can also be obtained by replacing the strain in Eqs. (7) and (9) with the equivalent strain, defined as:

e¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi he1 i2 þ he2 i2 þ he3 i2

ð11Þ

where,

hei i ¼



ei ðei 6 0; i ¼ 1; 2; 3Þ 0 ðei > 0; i ¼ 1; 2; 3Þ

ð12Þ

An element releases its stored elastic energy as long as it fails or damages, thus the element can be regarded as an AE source [36]. By recording the number of damaged elements and the associated quantity of released energy, RFPA3D can simulate AE events including the AE occurrence rate, magnitude and location. The accumulative damage D can thus be expressed as:

D¼

k 1X ni N i¼1

ð13Þ

where k is the current calculation step and ni refers to the number of damaged elements in the ith step. N is the total number of elements in a typical model [37]. Compared with other numerical methods, the RFPA3D code has two distinctive features: (1) by introducing the heterogeneity of material properties into elements, it can simulate non-linear deformation of quasi-brittle behaviors with an ideal brittle constitutive law at the local scale; (2) by introducing reductions of mechanical parameters as element fails, it can simulate strain-softening and non-continuum mechanics problems with continuum methods. The robustness of the code in simulating rock failure and fracture has been evaluated by various studies, including the failure mechanism of rocks under static tension/compression loads [36,38], the research of three dimensional fracture initiation and growth [35,37], and engineering applications involving rock slope stability analysis [39,40]. 3.2. Model setup Several CCNSCB specimen configurations are simulated. To illustrate the progressive fracture mechanism under different configurations, a standard CCNSCB sample (denoted as SI) with the same sample geometry as the ISRM suggested CCNBD specimen [13] is demonstrated throughout this study. To compare the fracture mechanism between the CCNSCB and CCNBD method [42], specimen SII, which has a more slender notch ligament than SI, is also simulated. The dimension parameters of SI and SII are tabulated in Table 1, both of which satisfy the dimensional requirement of chevron notched samples described in Eq. (1). The two CCNSCB samples SI and SII are meshed with 592166 and 613324 elements, respectively. A typical numerical model composed of hexahedral elements is shown in Fig. 4. Mechanical parameters of the simulated marble are taken from tests on marble samples. Young’s modulus is 16.5 GPa; Poisson’s ratio is 0.25; uniaxial tensile and compressive strengths are 9 MPa and 93 MPa, respectively. The loading platen and support rollers are rigid. In this numerical study, displacement

292

M.D. Wei et al. / Engineering Fracture Mechanics 134 (2015) 286–303

Table 1 Dimension parameters of the CCNSCB specimen SI and SII. Parameters (mm)

Standard specimen (SI)

R B RS a1 a0

40 32 27.732 26 10.55

SII 40 B/R = 0.8 RS/R = 0.6933 a1/R = 0.65 a0/R = 0.2637

32 32 28 5.68

B/R = 0.8 RS/R = 0.8 a1/R = 0.7 a0/R = 0.142

Displacement control

Fig. 4. The numerical model for CCNSCB sample SI.

loading with an increment of 0.001 mm per step is applied vertically to the top loading plate while the support rollers are fixed. A low displacement increment facilitates a stable crack growth in physical experiments. 4. Numerical results 4.1. Progressive fracture of the CCNSCB sample Fig. 5a shows the minimum principal stress field of the diametric cut plane through the notch tip of the standard CCNSCB sample SI with b = 0.8 (ratio of the span to diameter) and m = 10 (heterogeneity index). Note that the numerical model here is rather homogeneous due to a relatively large value of m. The fracture processes at six typical loading stages (Fig. 5b) show that the main crack initiates from the notch tip due to the high stress concentration and then grows along the notch ligament. The associated AE distributions in simulating the fracture of the sample SI in the diametric cut plane through the notch tip and in the semi-circular face of the sample are illustrated in Figs. 6 and 7, respectively. The blue and red circles of AE distribution denote the tensile and shear failure of the elements respectively. It is evident that the representing AE circles (Figs. 6 and 7) are in blue, which reveals that the fracture of the CCNSCB specimen is induced by tension, consistent with the measurement theory of Mode-I fracture toughness. Both Figs. 5 and 6 show clearly the progressive cracking mechanism during the six typical loading stages (i.e. 20% peak force, 40% peak force, 60% peak force, 80% peak force, 100% peak force and post-50% peak force). At stage I (20% peak force), no fracture occurs at all. At stage II (40% peak force), visible cracks are seen to appear. Through stage III (60% peak force) to stage V (100% peak force), the fracture further develops not only from the notch tip but also from two chevron edges, forming a wing crack surface. As the crack grows within the chevron notched ligament, the crack front is observed to be considerably curved and not straight at all. At the post failure stage VI (post 50% peak force), the fracture has surpassed the whole notch ligament in a straight-through manner, propagating further toward the top loading plate. Special attentions should be paid to the fractured CCNSCB sample at the critical stage (i.e., 100% peak force), since it provides the necessary information (i.e., dimensionless crack length and the shape of crack front) to calibrate the critical SIF given in Eq. (2). Indeed, the crack front at the critical stage is considerably curved, which markedly violates the straightthrough crack assumption by Ouchterlony [41] for all chevron notched samples for fracture toughness measurements. A comparison of the minimum principal stress distributions and AE distributions between SI and SII is displayed in Fig. 8 as the recorded forces reach the peak. The progressive fracture of SII is similar to that of SI but a slightly more curved crack front can be observed for SII than that for SI.

M.D. Wei et al. / Engineering Fracture Mechanics 134 (2015) 286–303

293

(a)

20% peak force

40% peak force

60% peak force

80% peak force

100% peak force post 50% peak force

(b) Fig. 5. (a) Position of the selected cross section and (b) the minimum principal stress distribution during the whole fracture process.

4.2. Progressive fracture of CCNSCB specimens with different b Figs. 9 and 10 show the numerically simulated force–displacement curve and associated accumulative AE counts-displacement curve of the standard CCNSCB sample with the same heterogeneity index (m = 10) but different spans (b = 0.8, 0.5, 0.3). At initial stages, the force–displacement curves are nearly straight. When the force approaches the peak value, the corresponding accumulative AE counts quickly increase. After the peak, the force is quickly released, but the cracks propagate further in an unstable manner and the associated AE activity increase more sharply. The increasing and then decreasing nature of the loading force corresponds to the stable-unstable fracture transition point of the CCNSCB specimen test records according to the measurement principles described in Fig. 2. Comparing the three curves in Figs. 9 and 10, with decreasing b, we can find: (1) The recorded force increases more quickly at the linear stages. (2) The peak value is higher in the force–displacement curve.

294

M.D. Wei et al. / Engineering Fracture Mechanics 134 (2015) 286–303

20% peak force

40% peak force

60% peak force

80% peak force

100% peak force

post 50% peak force

Fig. 6. The AE activities in sample SI during the whole fracture process.

(3) The AE events increase sharply during the stable-unstable fracture transition and the accumulated AE events further increase after attaining the peak load. Fig. 11 shows the maximum principal stress distribution of the aforementioned diametric cut plane and AE distributions of CCNSCB specimen SI under three different supporting spans. Taking the dashed lines as references, it is evident that the critical crack length am increases as b decreases. Surprisingly, in case of b = 0.3, the crack front of the critical crack exceeds the root of the notch ligament. For chevron notched samples featuring the stable–unstable fracture propagation, the critical crack length is expected to occur somewhere in the middle of the chevron notched ligament according to the measuring principles illustrated in Fig. 2. The projection of AE events along the initial notch front is illustrated in Fig. 12. At this stage, the specimen fails completely. It is evident that for the case of b = 0.8 the AE activities are restricted extremely in the ligament; for b = 0.5, the AE activities somehow deviate from the ligament; and for b = 0.3, AE events remarkably appear to spread outside the ligament. This explains why the cumulative number of AE is greater for a smaller b, as shown in Fig. 10. The real fractures diverge from the ideal situations at different extents for CCNSCB specimens with different loading spans; and the main crack propagation is finely restricted in the notch ligament for a relatively larger span between the two supporting rollers (e.g., b = 0.8). The same circumstance of deviating damages outside the notch ligament occurs for the progressive fracture of CCNBD specimen [42]. Fig. 13 shows the stress field and AE projections along the initial notch front of CCNBD specimen. Herein, m is 10 and other dimensions are identical to those of CCNSCB sample SI. Compared with the case of b = 0.3 for CCNSCB specimen in Fig. 12, AE activities of the CCNBD specimen are similar. AE events in both specimens deviate substantially from their notch ligaments and this indicates their practical fractures deviate from the ideal case. Over all, the real fracture of CCNSCB with a small span to diameter ratio (e.g., b = 0.3) is similar to that of CCNBD, both of which differ significantly from the ideal fracture propagation (Fig. 2). 4.3. Fractures of CCNSCB specimens with different heterogeneity index m The numerically simulated force–displacement curve and the associated accumulative AE counts-displacement curve of the standard CCNSCB sample with the same span (b = 0.8) but different heterogeneity index (m = 1.1, 2, 5, 10) are illustrated

M.D. Wei et al. / Engineering Fracture Mechanics 134 (2015) 286–303

20% peak force

40% peak force

60% peak force

80% peak force

100% peak force

post 50% peak force

Fig. 7. AE projections of SI along their disc-rotating axis during the whole fracture process.

SI

S Fig. 8. A comparison of the minimum principal stress distributions and AE distributions between SI and SII.

295

296

M.D. Wei et al. / Engineering Fracture Mechanics 134 (2015) 286–303

4000

β =0.8 β =0.5 β =0.3

3500

Force (N)

3000 2500 2000 1500 1000 500 0 0.00

0.01

0.02

0.03

0.04

0.05

Displacement (mm) Fig. 9. The numerically simulated force–displacement curves of the CCNSCB specimens with different b.

Accumulative AE counts

20000

β=0.8 β=0.5 β=0.3

18000 16000 14000 12000 10000 8000 6000 4000 2000 0 0.00

0.01

0.02

0.03

0.04

0.05

Displacement(mm) Fig. 10. Simulated accumulative AE counts-displacement curves of the CCNSCB specimens with different b.

in Figs. 14 and 15 respectively. In the cases of m = 10, 5 and 2, the force–displacement curves are similar to those in Fig. 9. For m = 1.1, the force–displacement curve has a gentler post-peak behavior, consistent with that reported by Tang et al. [33]. In this case, the AE counts mostly increase as the load increases. Figs. 14 and 15 also show that the recorded force–displacement relationship and the peak force heavily depend on heterogeneity. The more homogeneous a sample is, the more significant the linearly elastic behavior that the sample shows; and the higher the peak force that is attained. In addition, fracture characteristics also depend on heterogeneity. For more heterogeneous sample, AE events increase more rapidly at initial loading stage and the accumulative AE activities are greater after the final failure of the specimen. For more homogeneous rocks, the accumulated AE counts increase quite rapidly beyond the peak force. Also note that there is negligible difference between curves of heterogeneity shown in Fig. 14 when m P 5, and the AE curves (Fig. 15) are almost the same in the cases of m = 5 and m = 10. Fig. 16 shows the minimum principal stress distribution of the central cross-section through the notch tip for the CCNSCB sample with m = 1.1. For this severe heterogeneous case, in the local scale, individual microcracks may not initiate from the notch tip, but scatter in the whole sample. In the global scale, the macro fracture forms in the vicinity of the notch tip and notch ligament and then propagates upward to the top loading point. It is observed that the cracking/fracturing seriously deviates from the ideal route in a discontinuous manner; and more micro-fractures appear compared to the cases with greater homogeneity. Fig. 17 shows the minimum principal stress distribution of the aforementioned diametric cutting plane at the peak force stage for the four cases with different heterogeneities, visually indicating the significant impact of heterogeneity. Obviously, in a more heterogeneous specimen, the stress distribution is more complicated; and the crack front is rougher and more discontinuously curved. The actual fracturing of a heterogeneous CCNCSB specimen is distinctly different from the ideal assumption. The local failure can first occur in a region with either high stress concentration or with lower assigned strength. In a very homogeneous CCNSCB specimen, the notch tip generates severe stress concentration due to the particular chevron notched sample configuration, thus initiating the crack and then it grows along the prescribed fracture route of the ligament. Note that, there are no micro-fractures in the remaining parts of the specimen. But in a heterogeneous specimen, micro fractures first occur in the parts where stress intensity might not be the highest. The reason is that due to heterogeneity, more elements are endowed with lower strength; these elements fail first under certain loads leading to increasing AE numbers.

M.D. Wei et al. / Engineering Fracture Mechanics 134 (2015) 286–303

β=0.8

β=0.5

297

β=0.3

(a)

β=0.8

β=0.5

β=0.3

(b) Fig. 11. (a) Maximum principal stress fields and (b) AE activities of CCNSCB specimens with different b.

5. Discussions For chevron notched samples, the measurement of Mode-I fracture toughness usually needs the critical dimensionless stress intensity factor, which is commonly calibrated by analytic or numerical means. According to the measuring principle, the CCNSCB specimen is assumed to fracture with an ideal straight crack front [41], a typical two-dimensional problem in fracture mechanics. Actually, the main fracture initiates not only from the notch tip but also from both sides of the notch due to local stress concentration. For either CCNSCB sample SI or SII, no matter the rock specimen is homogeneous or not, the crack front never follows the straight through crack assumption [41], but rather parabolic. Further, for the sample with a more slender chevron notch (SII), the curvature of crack front is more severe than that for SI, as shown in Fig. 18. That figure also illustrates the real crack front (in red) of SI and SII (m = 10, b = 0.8) at three typical stages during the progressive fractures of CCNSCB tests, namely the stages with 90% peak force, 100% peak force and post 90% peak force. The theoretical critical crack length (am) is marked with the blue dashed line, calibrated via finite element method adopting the straight-through crack assumption [13,41,43]; the difference between the ideal critical crack front and the real one is obvious. At the critical stage, the blue dashed lines are below the lowest points of the red curves, which denote that the equivalent critical crack length in the realistic fracture should be greater than that calibrated in the ideal situations. In other words, the critical dimensionless crack length (am), the minimal stress intensity factor (Y min ) as well as the Mode-I fracture toughness (KIC) determined with current straight-through crack propagation assumption are underestimated. This explains why the measured values of fracture toughness via chevron notched methods are generally smaller than those from other comparable methods in a certain degree [42,44–46]. Conventionally, for samples SI and SII, while b = 0.8, am are calibrated as 0.48 and 0.41; and Y min are 7.951 and 8.044 respectively, based on the straight-through crack assumption. A more accurate calibration of am and Y min for testing fracture toughness values should employ the crack front at curvy shape, which follows a more objective three-dimensional fracture response of the chevron notched samples; and these will be studied in future research.

298

M.D. Wei et al. / Engineering Fracture Mechanics 134 (2015) 286–303

β=0.8

β=0.5

β=0.3 Fig. 12. AE projections of the CCNSCB specimens with different b along the initial notch front.

100% peak force

100% peak force

post 50% peak force

Fig. 13. The maximum principal stress field and AE projections of the CCNBD specimen.

Available numerical methods for chevron notched samples usually simplify the crack growth into a two-dimensional problem. Cracks are assumed to initiate from tips of the jag notches and grow along half of the notch width. However, due to a relatively complex configuration and realistic fracture process of chevron notched samples, that problem is typically three-dimensional in fracture mechanics. Numerical results indicate that the failure mode of the chevron notched sample remain still Mode-I, while the very first crack never forms from the center of the notch tip, but mostly from two flanks of

M.D. Wei et al. / Engineering Fracture Mechanics 134 (2015) 286–303

900

m =10 m =5 m =2 m =1.1

800 700

Force (N)

299

600 500 400 300 200 100 0 0.00

0.01

0.02

0.03

0.04

0.05

Displacement (mm) Fig. 14. Force–displacement curves of the CCNSCB specimens with different m.

35000

Accumulative AE counts

30000 25000

m=10 m=5 m=2 m=1.1

20000 15000 10000 5000 0 0.00

0.01

0.02

0.03

0.04

0.05

Displacement (mm) Fig. 15. Accumulative AE counts-displacement curves of the four CCNSCB specimens with different m.

the notch tip. Crack propagation path can be found somewhat zigzag, not completely straight even though the selected rocks are quite homogeneous, as evidenced in a photo of recovered CCNSCB sample (Fig. 19a) by Ayatollahi and Alborzi [6]. Especially for the CCNSCB specimen with a smaller supporting span (b = 0.3), marked damage/fractures in the zones occur adjacent to the ligament (Fig. 12); in sharp contrast, the damaged/fractured zone in the simulation of a sample with a larger span (b = 0.8) is dominantly confined in the ligament region. Furthermore, as shown in Fig. 11, the larger the span b, the less the critical crack length is developed and the cracking zone is disturbed as the bearing load meets its maximum; and the better the realistic fracture approaches the assumed one. In the conventional three-point bending tests, the span usually ranges from 0.5 to 0.8; and our simulation reveals that when adopting CCNSCB method, a relative large span (e.g., 0.8) appear to be more reliable in the Mode-I fracture toughness determination because it causes less biased damages off the notch ligament. The AE activities for the CCNSCB specimen with small supporting span of b = 0.3 is quite similar to that of the CCNBD specimen. It is worth stressing that for CCNBD samples, the damaged/fractured zone deviates significantly from the notch ligament region at the critical stage (Fig. 13). These damages/fractures are not premeditated and may lead to errors for fracture toughness measurement. This is in distinct contrast with the CCNSCB sample with the same geometry but with relative large supporting span such as b = 0.8, where the damaged zone adjacent to the notch ligament is greatly mitigated. Dai et al. [42] reported that valid CCNBD sample dimensions in the ISRM suggested range of geometry should still be selected; and a CCNBD with rather slender chevron notch is more desirable. In sharp contrast, for CCNSCB samples with b = 0.8, both sample SI or SII shows favorable failure mechanism with damaged zone highly limited in the notch ligament, quite following the measuring principle. This suggests that with relative large supporting span (e.g., 0.8) the CCNSCB method can accommodate a much wider range of sample geometries than the CCNBD method and thus it could be more convenient for Mode-I fracture toughness measurements. It is also worth noting that the proposed RFPA code can potentially be used to delineate the fracture process zone (FPZ) in the rock fracturing process, especially during the progressive fracturing of heterogeneous rocks, where the micro-cracking region around the advancing crack front is featured. Researchers have reported that the FPZ can induce size effect of fracture toughness and result in the variation of measured toughness in a wide range for the specimens with different sizes [47–50]. The linear elastic fracture mechanics (LEFM), in which all the fracture process is assumed to occur in one point, might not be

300

M.D. Wei et al. / Engineering Fracture Mechanics 134 (2015) 286–303

20% peak force

40% peak force

60% peak force

80% peak force

100% peak force

post 50% peak force

Fig. 16. The minimum principal stress distribution of the central cross-section through the notch tip for the CCNSCB specimen with m = 1.1.

m=10

m=5

m =2

m=1.1

Fig. 17. Stress fields of the four CCNSCB specimens with different m at the stage of peak force.

M.D. Wei et al. / Engineering Fracture Mechanics 134 (2015) 286–303

90% peak force

100% peak force

301

post 90% peak force

(a)

90% peak force

100% peak force

post 90% peak force

(b) Fig. 18. Simulated realistic crack front (red curved line) and the straight-through crack front (blue dashed straight line) of (a) specimen SI and (b) specimen SII. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Asperous crack growth path

Deflected growth Secondary crack

(a)

(b)

Fig. 19. (a) Recovered sample of a CCNSCB test (reproduced after [6]) and (b) a CCNBD test (reproduced after [46]).

suitable for analyzing failure of brittle solids with a large FPZ. Indeed, the size of the FPZ is controlled by the local heterogeneity in the material as well as by the specimen geometry and the stress conditions [50], in accordance with our numerical simulation. In this study, the FPZ in CCNSCB sample with b = 0.8 is smaller than that of a sample with b = 0.5 or 0.3, comparing the AE activities. Thus, the CCNSCB method with a sample geometry of a large supporting span b is superior to the ones with lower b. Although it has been widely recognized that heterogeneity plays an important role on rock fracture, few people has studied the influence of heterogeneity on the three-dimensional rock fracture mechanism in general and for chevron notched

302

M.D. Wei et al. / Engineering Fracture Mechanics 134 (2015) 286–303

rock fracture in special. In a more heterogeneous specimen, although the stress distribution is more complicated and the micro cracks distribute more discretely, the main crack fronts are still parabolic but in a more discontinuous manner. With a finely homogeneous specimen, cracks apparently initiate from the notch tips, where the stress is highly concentrated. But this is not certainly the case for a heterogeneous rock because the weak elements might fail first even if the stress concentration is the severest. But even for the latter case, in general main cracks still propagate in the vicinity of the notch region and further grow up to the contacting points and their fracture process remain discrepant with the ideal one. Additionally, the fracture of heterogeneous CCNSCB specimen is accompanied by secondary micro-fractures. Although some findings about heterogeneity in CCNSCB specimens are given here, the significance of this simulation lies in introducing heterogeneity into the fracture of CCNSCB samples for the first time. The progressive fracture characteristics of differently heterogeneous specimens are presented in multi-perspectives. Stress distribution and internal fractures of the specimens, which are hard to be revealed in physical experiments, are intuitively displayed. The simulation results also fill the vacancy about numerical studies for the chevron notched methods and they can provide a reference for related physical tests and theoretical analyses. 6. Conclusions The chevron notched samples have been widely utilized to measure the rock fracture toughness due to many advantages, among which, the CCNSCB method has inherited the merits of two ISRM suggested methods, i.e., the CCNBD method and the NSCB method and thus received much attention recently. However, the fracture behavior of the CCNSCB specimen has not been fully examined. For the first time, this study numerically simulates the progressive fracture progress of the CCNSCB specimen; the internal fractures are revealed and some fundamental assumptions (e.g., straight-though crack assumption) are assessed considering the influence of heterogeneity and different supporting spans. For CCNSCB specimens with varying heterogeneity, not only the notch tips, but also the two saw-cut notch edges crack simultaneously. As a result, crack fronts are not straight-through, but significantly curved. The real fracture progresses of CCNSCB specimens are never consistent with the straight-though crack assumption; and thus a more reliable calibration for the critical dimensionless stress intensity factor should take into account of the curved crack front. Simulations with varying different supporting spans reveal that for the CCNSCB specimen with a smaller span, the crack deviates more severely from the prescript chevron notched ligament; while for a large span, the fracture initiates and develops within the ligament. Thus, a large span (e.g., b = 0.8) should be adopted if the CCNSCB method is employed. Furthermore, a comparison of the fracture details of CCNSCB with CCNBD reveals that, a much wide range of the suggested sample geometry by ISRM for CCNBD tests can be potentially employed in the CCNSCB tests, as long as the span of the three point bend configuration is large enough. The numerical simulation calls for a better understanding on the realistic progressive fracture mechanism of the chevron notched specimens (e.g., CCNSCB, CCNBD, SR, and CB) to more accurately determine the Mode-I fracture toughness of rocks considering sampling or testing configurations (i.e., sample geometries, supporting spans etc.). Acknowledgments The authors are grateful for the financial support from the National Program on Key basic Research Project (No. 2015CB057903), National Natural Science Foundation of China (No. 51374149), the Program for New Century Excellent Talents in University (NCET-13-0382), Youth Science and Technology Fund of Sichuan Province (2014JQ0004) and Doctoral Fund of Ministry of Education of China (20130181110044). References [1] Guo H, Aziz NI, Schmidt LC. Rock fracture toughness determination by the Brazilian test. Engng Geol 1993;33(3):177–88. [2] Dai F, Chen R, Xia K. A semi-circular bend technique for determining dynamic fracture toughness. Exp Mech 2010;50(6):783–91. [3] Dai F, Xia K. Laboratory measurements of the rate dependence of the fracture toughness anisotropy of Barre granite. Int J Rock Mech Min 2013;60:57–65. [4] Kuruppu MD. Fracture toughness measurement using chevron notched semi-circular bend specimen. Int J Fract 1997;86(4):L33–8. [5] Dai F, Xia K, Zheng H, Wang Y. Determination of dynamic rock mode-I fracture parameters using cracked chevron notched semi-circular bend specimen. Engng Fract Mech 2011;78(15):2633–44. [6] Ayatollahi MR, Alborzi M J. Rock fracture toughness testing using SCB specimen. In: 13th International Conference on Fracture. Beijing, China, 16–21 June; 2013. p. 1–7. [7] Awaji H, Sato S. Combined mode fracture toughness measurement by disk test. J Engng Mater Technol 1978;100(2):175–82. [8] Atkinson C, Smelser RE, Sanchez J. Combined mode fracture via the cracked Brazilian disk test. Int J Fract 1982;18(4):279–91. [9] Aliha MRM, Ayatollahi MR, Smith DJ, Pavier MJ. Geometry and size effects on fracture trajectory in a limestone rock under mixed mode loading. Engng Fract Mech 2010;77(11):2200–12. [10] Aliha MRM, Ayatollahi MR, Akbardoost J. Typical upper bound-lower bound mixed mode fracture resistance envelopes for rock material. Rock Mech Rock Engng 2012;45(1):65–74. [11] Aliha MRM, Sistaninia M, Smith DJ, Pavier MJ, Ayatollahi MR. Geometry effects and statistical analysis of mode I fracture in guiting limestone. Int J Rock Mech Min 2012;51:128–35. [12] Xu C, Fowell RJ. Stress intensityfactor evaluation for cracked chevron notched Brazilian disc specimen. Int J Rock Mech Min Sci Geomech Abstr 1994;31:157–62. [13] Fowell RJ. ISRM commission on testing methods: suggested method for determining mode I fracture toughness using cracked chevron notched Brazilian disc (CCNBD) specimens. Int J Rock Mech Min Sci Geomech Abstr 1995;32(1):57–64.

M.D. Wei et al. / Engineering Fracture Mechanics 134 (2015) 286–303

303

[14] Wang QZ, Jia XM, Kou SQ, Zhang ZX, Lindqvist PA. More accurate stress intensity factor derived by finite element analysis for the ISRM suggested rock fracture toughness specimen-CCNBD. Int J Rock Mech Min Sci 2003;4(2):233–41. [15] Wang QZ, Fan H, Gou XP, Zhang S. Recalibration and clarification of the formula applied to the ISRM-suggested CCNBD specimens for testing rock fracture toughness. Rock Mech Rock Engng 2013;46(2):303–13. [16] Dai F, Chen R, Iqbal MJ, Xia K. Dynamic cracked chevron notched Brazilian disc method for measuring rock fracture parameters. Int J Rock Mech Min 2010;47(4):606–13. [17] Szendi-Horvath G. Fracture toughness determination of brittle materials using small to extremely small specimens. Engng Fract Mech 1980;13:955–61. [18] Chen F, Sun Z, Xu J. Mode I fracture analysis of the double edge cracked Brazilian disk using a weight function method. Int J Rock Mech Min Sci 2001;38(3):475–9. [19] Aliha MRM, Hosseinpour GR, Ayatollahi MR. Application of cracked triangular specimen subjected to three-point bending for investigating fracture behavior of rock materials. Rock Mech Rock Engng 2013;46(5):1023–34. [20] Wang QZ, Xing L. Determination of fracture toughness KIc by using the flattened Brazilian disc specimen for rocks. Engng Fract Mech 1999;64(2):193–201. [21] Amrollahi H, Baghbanan A, Hashemolhosseini H. Measuring fracture toughness of crystalline marbles under modes I and II and mixed mode I–II loading conditions using CCNBD and HCCD specimens. Int J Rock Mech Min Sci 2011;48(7):1123–34. [22] Tang T, Bazˇant ZP, Yang S, Zollinger D. Variable-notch one-size test method for fracture energy and process zone length. Engng Fract Mech 1996;55(3):383–404. [23] Yang S, Tang TX, Zollinger D, Gurjar A. Splitting tension tests to determine concrete fracture parameters by peak-load method. Adv Cem Based Mater 1997;5:18–28. [24] Thiercelin M, Roegiers JC. Fracture toughness determination with the modified ring test. In: International symposium on engineering in complex rock formations. Beijing China; 1986. p. 1–8. [25] Shiryaev AM, Kotkis AM. Methods for determining fracture-toughness of brittle porous materials. Ind Lab 1982;48(9):917–8. [26] Chen CH, Chen CS, Wu JH. Fracture toughness analysis on cracked ring disks of anisotropic rock. Rock Mech Rock Engng 2008;41(4):539–62. [27] Ouchterlony F. A review of fracture toughness testing of rocks. Solid Mech Arch 1982;7:131–211. [28] Tutluoglu L, Keles C. Mode I fracture toughness determination with straight notched disk bending method. Int J Rock Mech Min Sci 2011;48(8):1248–61. [29] Ouchterlony F. ISRM commission on testing methods; suggested methods for determining fracture toughness of rock. Int J Rock Mech Min Sci Geomech Abstr 1988;25:71–96. [30] Zhou YX, Xia K, Li XB, Li HB, Ma GW, Zhao J, et al. Suggested methods for determining the dynamic strength parameters and mode-I fracture toughness of rock materials. Int J Rock Mech Min 2012;49:105–12. [31] Kuruppu MD, Obara Y, Ayatollahi MR, Chong KP, Funatsu T. ISRM-Suggested method for determining the mode I static fracture toughness using semicircular bend specimen. Rock Mech Rock Engng 2014;47(1):267–74. [32] Tang CA, Kaiser PK. Numerical simulation of cumulative damage and seismic energy release during brittle rock failure-Part I: Fundamentals. Int J Rock Mech Min Sci 1998;35(2):113–21. [33] Tang CA, Liu H, Lee PKK, Tsui Y, Tham LG. Numerical studies of the influence of microstructure on rock failure in uniaxial compression-part I: effect of heterogeneity. Int J Rock Mech Min Sci 2000;37:555–69. [34] Weibull W. A statistical theory of the strength of materials. Ing Vet Ak Handl 1939;151:5–44. [35] Liang ZZ, Xing H, Wang SY, Williams DJ, Tang CA. A three dimensional numerical investigation of fracture of rock specimen containing a pre-existing surface flaw. Comput Geotech 2012;45:19–33. [36] Tang CA, Tham LG, Wang SH, Liu H, Li WH. A numerical study of the influence of heterogeneity on the strength characterization of rock under uniaxial tension. Mech Mater 2007;39:326–39. [37] Wang SY, Sloan SW, Tang CA. Three-dimensional numerical investigations of the failure mechanism of a rock disc with a central or eccentric hole. Rock Mech Rock Engng 2013. http://dx.doi.org/10.1007/s00603-013-0512-6. [38] Wang SY, Sloan SW, Sheng DC, Yang SQ, Tang CA. Numerical study of failure behaviour of pre-cracked rock specimens under conventional triaxial compression. Int J Solids Struct 2014;51(5):1132–48. [39] Xu NW, Tang CA, Li LC, Zhou Z, Sha C, Liang ZZ, et al. Microseismic monitoring and stability analysis of the left bank slope in Jinping first stage hydropower station in southwestern China. Int J Rock Mech Min Sci 2011;48(6):950–63. [40] Xu NW, Dai F, Liang ZZ, Zhou Z, Sha C, Tang CA. The dynamic evaluation of rock slope stability considering the effects of microseismic damage. Rock Mech Rock Engng 2014;47:621–42. [41] Ouchterlony F. Compliance calibration of a round fracture toughness bend specimen with chevron edge notch. Stockholm: Swedish Detonic Research Foundation; 1984. [42] Dai F, Wei MD, Xu NW, Ma Y, Yang DS. Numerical assessment of the progressive rock fracture mechanism of cracked chevron notched brazilian disc specimens. Rock Mech Rock Engng 2014. http://dx.doi.org/10.1007/s00603-014-0587-8. [43] Chang SH, Lee CI, Jeon S. Measurement of rock fracture toughness under modes I and II and mixed-mode conditions by using disc-type specimens. Engng Geol 2002;66(1):79–97. [44] Dwivedi RD, Soni AK, Goel RK, Dube AK. Fracture toughness of rock under sub-zero temperature conditions. Int J Rock Mech Min Sci 2000;37:1267–75. [45] Iqbal MJ, Mohanty B. Experimental calibration of ISRM suggested fracture toughness measurement techniques in selected brittle rocks. Rock Mech Rock Engng 2007;40(5):453–75. [46] Cui ZD, Liu DA, An GM, Sun B, Zhou M, Cao FQ. A comparison of two ISRM suggested chevron notched specimens for testing mode-I rock fracture toughness. Int J Rock Mech Min Sci 2010;47:871–6. [47] Bazˇant ZP. Size effect on blunt fracture: concrete, rock, metal. J Engng Mech 1984;110:518–35. [48] Bazant ZP, Kazemi MT. Determination of fracture energy, process zone length and brittleness number from size effect, with application to rock and concrete. Int J Fract 1990;44(2):111–31. [49] Bazˇant ZP. Size effect. Int J Solid Struct 2000;37:69–80. [50] Gregoire D, Rojas-Solano L, Lefort V, Grassl P, Pijaudier-Cabot G. Size and boundary effects during failure in quasi-brittle materials: experimental and numerical investigations. Proc Mater Sci 2014;3:1269–78.