Do disk-type specimens generate a mode II fracture without confinement?

International Journal of Rock Mechanics & Mining Sciences 87 (2016) 48–54

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International Journal of Rock Mechanics & Mining Sciences journal homepage: www.elsevier.com/locate/ijrmms

Do disk-type specimens generate a mode II fracture without confinement? Wei-Wei Ji a, Peng-Zhi Pan a,n, Qing Lin b, Xia-Ting Feng a, Meng-Ping Du a a b

State Key Laboratory of Geomechanics and Geotechnical Engineering, Institute of Rock and Soil Mechanics, Chinese Academy of Sciences, Wuhan, China Department of Engineering Mechanics, College of Petroleum Engineering, China University of Petroleum, Beijing, China

art ic l e i nf o Article history: Received 23 November 2015 Received in revised form 12 May 2016 Accepted 20 May 2016 Keyword: Mode II fracture Mixed mode fracture Digital image correlation (DIC) Centrally cracked Brazilian disc (CCBD) Semi-circular bend (SCB)

1. Introduction Fracture is of great importance for the behavior of rock engineering structures, such as rock excavation, tunneling, underground storage of oil, etc. From the perspective of fracture mechanics, a crack propagates under the three basic failure modes, mode I, tensile/opening mode; mode II, in-plane sliding mode; mode III, anti-plane sliding mode. Crack propagates if the stress intensity factor Ki (i¼I, II, III) reaches the critical value, i.e., fracture toughness Kic. Fracture toughness is the material property, and thus, it is important to develop appropriate experiment methods to determine its value for different rocks. The determination of mode I fracture toughness is relatively simple, and in 1988 the ISRM suggested a method to determine mode I fracture toughness.1 However, mode II fracture toughness poses a challenge for rock mechanics researchers. The ISRM did not recommend any suggested method until 2012.2 In the past three decades, a variety of testing methods have been proposed to determine mode II fracture toughness. Backers and Stephansson introduced a new testing method called punch-through shear test for measurement of Mode II fracture toughness in 2002,3 which was developed as an ISRM suggested method in 2012. Rao and Sun used a shear-box to provide a confinement to the specimen to generate a shear fracture.4 Chang used disk‐type specimens to n

Corresponding author. E-mail address: [email protected] (P.-Z. Pan).

http://dx.doi.org/10.1016/j.ijrmms.2016.05.010 1365-1609/& 2016 Elsevier Ltd. All rights reserved.

carry out test without confinement to measure rock fracture toughness under mode II.5 Ayatollahi used a cracked semi-circular specimen subjected to three-point bending without confinement to conduct mode II test.6 Azar used a Compact-Tension-Shear specimen to investigate mode II fracture without confinement.7 Generally, those methods can be divided into two groups. The first is with confinement,3,4 and the second is without confinement.5,6 The present research is concentrated on the second method, which is popular because it has relatively simple specimen geometry and loading device. The second approach usually utilizes the disc-type specimens, including the centrally cracked Brazilian disc (CCBD) specimen5,8– 13 and the angled edge crack semi-circular specimens under threepoint bending (SCB),5,6,8,14–21 as shown in Fig. 1(a) and (b). For disc-type specimens, one fundamental assumption is that a pure mode II loading will generate a mode II fracture, no matter it is confined or not. In other word, a mode II fracture can be created, under the conditions when KI is zero and KII is not zero. However, a serious drawback of CCBD and SCB is that they never provide a direct measured displacement field for the experiments. All the results are based on theoretical model. In particular, the loading configurations with respect to these methods do not provide any confinements, which seriously violate the ISRM suggestion.2 Thus, it is necessary to answer a fundamental question: do disc-type specimens such as CCBD and SCB generate a mode II (sliding) fracture without any confinement? The digital image correlation (DIC), which was first proposed in

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pure mode II loading, a series of requirements have to be satisfied. Table 1 gives the ratio of notch length a to disc radius R and the inclined angle of notch for CCBD and SCB specimens to create the pure mode II loading. For example, when a span ratio S/R of 0.8, normalized notch length a/R of 0.5 is employed for a semi-circular specimen and a notch inclination α of 63° is required.30 Ten CCBD specimens (Fig. 1(a)) and sixteen SCB specimens are prepared and tested. Specimen preparation also involves the preparation of speckles. Several procedures are conducted to achieve a random pattern of speckles. First, lightly coat the specimen surface with white paint. Second, after the white paint was dry, overspray the coated surface with a dark mist by a spray paint. Third, continue misting and remisting until the unique speckle pattern was produced. 2.2. Experimental setup

2. Specimen preparation and experimental setup

Experimental devices include image acquisition system and loading system. The image acquisition system consists of a CCD camera with 3376  2704 effective square pixels, a 35 mm prime lens, two digital white lights and a computer. The computer is used to control the camera and record images, note that the image acquisition is five frames per second. The loading system is MTS815 with the LVDT and CMOD (crack mouth opening displacement) gauge. The CCBD specimen is subjected to a diametrical compressive load at a constant axial displacement of 0.0005 mm/s. The SCB specimen is under three-point bending with CMOD control, 0.0002 mm/s. Being a 2D DIC system, the specimen surface is required to be flat and perpendicular to the image plane. Two white light sources are adjusted such that the intensities of captured image will not be influenced during the experiments. An external signal is used to synchronize MTS and digital camera. After the test, a DIC algorithm can be used to estimate the displacements. As a full-field non-contact optical method, Digital image correlation (DIC) directly provides full-field displacements by comparing the digital images of the specimen surface between the un-deformed and deformed states respectively.31,32 The basic principle of DIC is to match the intensities between the un-deformed (reference) and deformed (current) images, so that the displacements can be determined. Two concepts, subset and region of interest (ROI), are introduced in DIC processing, because it is impossible for the DIC algorithm to search for an individual pixel in an image.28,29 A subset contains a unique distribution of intensities, and the ROI is treated as a searching area for a subset since it is not necessary to search the whole image (Fig. 1(c)). The DIC algorithm determines the displacements of the subset movement, through locating the extreme value of a correlation function. In the DIC software we used, Ncorr,33 two different correlation criteria are used to find the initial guess and its subsequent refinement.33 The initial guess is calculated by the normalized cross correlation (NCC) method

2.1. Specimen preparation

Ccc =

Fig. 1. (a) CCBD specimen, (b) SCB specimen, (c) DIC system.

early 1980s,22,23 provides an effective tool to measure the full-field displacement and strain on a specimen surface. This technique has been widely used by many researchers in the study of rock fracture mechanism. For example, Zhang and Zhao present a detailed experimental procedure using DIC for determination of mechanical properties of rock material under dynamic loads.24 Le used DIC to study on the fracture process zone of Berea sandstone under cyclic loading.25 Zhang combined DIC with AE to investigate the damage and fracture of sandstone beams.26 Dautriat identified the localizations of damage and the local compaction mechanisms using DIC.27 Especially, DIC method can be used to determine the fracture mode. For example, Lin et al.28,29 used DIC to study on the fracture process of sandstone, the opening and shearing displacements were determined and the fracture mode was also identified. In this paper, a series of experiments based on DIC technique have been performed to examine whether a mode II fracture is created in CCBD and SCB specimens. The preparations of specimens and loading conditions, such as specimen sizes and notch orientations, all satisfy the requirements as the claimed in the literatures.5,6,8,9,11,12,14–19,21 DIC is used to measure the full-field displacement of the specimens, and a detailed analysis of displacement around the tip of crack can determine the fracture modes.

The testing materials are the marble from Jinping II hydropower station in Sichuan Province, China. The material properties are: Young’s Modulus 45–55 GPa, density 2780 kg/m3 and uniaxial compressive strength 110–160 MPa. All specimens were cut at the same orientation from the same block of rock, and all the dimensions are presented in Table 1. Both CCBD specimen (Fig. 1(a)) and SCB specimen (Fig. 1(b)) were prepared to study on the fracture mode. In order to produce

∑(i, j)∈ s (f (x˜ ref i , y˜ref i ) − fm )(g (x˜ cur i , y˜cur i ) − gm) ∑(i, j)∈ s (f (x˜ ref i , y˜ref i ) − fm )2 ∑(i, j)∈ s (g (x˜ cur i , y˜cur i ) − gm)2

(1)

where f and g are the reference and current image intensity functions at location (x,y), respectively, (x˜ ref i , y˜ref i ) and (x˜ cur i , y˜cur i ) are the local coordinates at reference and current images, respectively, and fm and gm are the mean intensity values of reference and current images in the subset respectively. A nonlinear optimizer is used to refine these results with sub-pixel resolution by finding the minimum of

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Table 1 Geometry and notch directions for specimen. Pioneer researches

Our experiments

Type

Reference

a/R

α (deg.)

Specimen number

a (mm)

R (mm)

Specimen number

a (mm)

R (mm)

Disc specimen (Fig. 3(a))

6,9,30

6

0.3 0.4 0.5 0.6 0.6

27 25.2 23 20.3 21.3

CCBD-1-1 CCBD-2-1 CCBD-3-1 CCBD-4-1 CCBD-5-1

15.42 20.15 25.13 30.05 30.22

50.15 50.15 50.15 50.15 50.15

CCBD-1-2 CCBD-2-2 CCBD-3-2 CCBD-4-2 CCBD-5-2

15.40 20.03 25.23 30.00 30.21

50.15 50.15 50.15 50.15 50.15

Type

Reference

a/R

Semi-circular specimen under symmetric loading (Fig. 3(b))

Whittaker 0.5 15 0.5 17,18 0.35 0.5 11,12 0.5 0.5 0.6 0.6

α (deg.) Specimen number 63 SCB-1-1 63 SCB-2-1 54 SCB-3-1 40 SCB-4-1 40.5 SCB-5-1 52.1 SCB-6-1 46.2 SCB-7-1 57.5 SCB-8-1

S/R 0.8 0.67 0.5 0.5 0.5 0.6 0.6 0.7

a (mm)

S (mm) R (mm)

23.31 23.47 16.88 23.85 23.68 23.71 27.87 28.97

37.30 31.60 23.60 23.70 23.60 28.10 27.90 33.00

⎡ f (x˜ ref i , y˜ref i ) − fm ⎢ CLS = ⎢ 2 ⎢⎣ ∑(i, j)∈ s (f (x˜ ref i , y˜ref i ) − fm )

S (mm) R (mm)

α (deg.)

23.54 23.89 16.68 23.83 23.80 23.63 28.64 28.50

37.30 31.70 23.60 23.30 23.80 27.90 28.60 32.70

63 63 54 40 40.5 52.1 46.2 57.5

46.67 47.24 47.15 46.62 47.60 46.53 47.54 46.68

3.1. Fracture mode determined from DIC data

⎤ ⎥ 2 ⎥ g ) − ) ⎦ i m ⎥

g (x˜ cur i , y˜cur i ) − gm ∑(i, j)∈ s (g (x˜ cur i , y˜cur

a (mm)

3. Experimental results

2

−

46.64 47.10 47.10 47.30 47.10 46.78 46.43 47.16

α (deg.) Specimen number 63 SCB-1-2 63 SCB-2-2 54 SCB-3-2 40 SCB-4-2 40.5 SCB-5-2 52.1 SCB-6-2 46.2 SCB-7-2 57.5 SCB-8-2

(2)

After determination of the displacement of the subset, DIC processing continues with a selection of a new subset until the complete horizontal and vertical displacement fields with respect to the image are obtained (Fig. 2).

According to the definition of fracture mode, it can be determined by analyzing the displacements along the crack. For instance, if there are the displacements that are perpendicular to the crack, i.e., opening displacements, the mode I fracture exists. Similarly, if there are the displacements parallel to the crack, i.e., sliding displacements, the mode II fracture exists. Thus, the measured DIC displacements help identify the fracture modes: (1) mode I fracture, only opening displacements exist; (2) mode II

Fig. 2. The typical matching process of DIC algorithm: (a) reference image, (b) current image, (c) horizontal displacement, (d) vertical displacement.

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Fig. 3. Specimen CCBD-3-1, 0–100% of peak load: (a) horizontal displacement Δu, (b) vertical displacement Δv, enlarged images of (c) horizontal displacement and (d) vertical displacement.

fracture, only sliding displacements exist; and (3) mixed-mode fracture, opening and sliding displacements exist simultaneously. It needs to emphasize that the DIC measurements are horizontal and vertical, which is defined as the global coordinate. However, the inclined notch on the specimen generates a crack that has an angle with the horizontal. Thus, it is necessary to establish a local coordinate along the initiated fracture. The displacements based on the global are transformed to the local, and the opening and sliding displacements are determined from horizontal and vertical displacements.29 The further details are displayed in following sections. 3.2. CCBD specimen The specimen CCBD-3-1 (Table 1) is presented to show how the fracture mode is determined. The horizontal displacement Δu and vertical displacement Δv at peak load (0–100% of peak) are displayed in Fig. 3(a) and (b). An area surrounding crack tip is selected as an observation window as shown in Fig. 3(a), since crack tip region is essential for fracture mode analysis. The enlarged image of window (Fig. 3(c) and (d)) clearly shows a displacement continuity in horizontal and vertical displacements. However, it may be the consequence, such that the global coordinate is not along the direction of crack propagation. Thus, it is necessary to establish a local coordinate to find the normal and tangential displacements along the crack.29 It needs to note that the normal displacements are the opening displacements, but the tangential displacements along the crack are generated by beam bending. Thus, only the differences between tangential displacements at two sides of crack can be regarded as sliding displacements. If there are no differences, there are no sliding, and indeed, no mode II fracture. As shown in Fig. 4, a local coordination system (X1, Y1) is established along the direction of crack propagation. Then all the displacements based on (X, Y) can be transformed to local coordinate (X1, Y1). The opening displacement Δu1 perpendicular the crack and tangential displacement field Δv1 along the crack are

obtained in Fig. 4. Two observe windows are used to further investigate the transformed displacements surrounding the crack (Fig. 4(a) and (b)). The enlarged images of those two windows are shown in Fig. 4(c) and (d). It’s clear to observe that the opening displacement Δu1 changes from left side of crack to the right side (Fig. 4(c)), which means the existence of an opening displacement discontinuity, and of course, a mode I crack. The observation of tangential displacements also reveals a small difference between two sides of the crack, i.e., sliding displacements. It implies that it also exists a mode II crack. The opening and sliding displacements clearly demonstrate that it is a mixed mode fracture, not mode II fracture. In order to obtain the crack opening and sliding displacement profiles, two cross-sections (X1 ¼ 70.2 mm) along the crack are selected to represent its two “sides”. It is because it is impossible to obtain the displacements from the mathematical crack sides in reality. In addition, the complicated and/or large displacements during the fracture process results a substantial measurement errors, when it is close to the crack. Nevertheless, the crack opening and sliding displacements Δu1 and Δv1 are plotted in Fig. 4(e) and (f). As shown in Fig. 4(e), a merged position of the opening displacements is observed above the notch. This position separates the opening displacements into two portions, (1) below the position, there is an opening displacement discontinuity; (2) above the position, no opening displacement discontinuity exists. Because the applied load only reaches the peak, it is reasonable to assume that the traction-free crack has not formed.25 Thus, the crack between the merged position and the notch is regarded as the cohesive crack, and the length is 14.6 mm approximately. The critical crack opening displacement can be estimated to be 25 mm at the notch tip. It needs to note that a small opening displacement, i.e., 3 mm, is observed at the cohesive crack tip, which is because the cross-sections are a little away from the crack sides. Fig. 4(f) shows the crack sliding displacements along the cohesive crack. The maximum value is 3 mm at the notch tip, which is only eighth of the opening displacement. Again, given the fact that both crack opening and sliding displacements exist, it is a

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Fig. 4. Specimen CCBD-3-1, 0–100% of peak load: (a) opening displacement Δu1, (b) sliding displacement Δv1, enlarged images of (c) opening and (d) sliding displacements, the displacement profiles along the crack (e) opening and (f) sliding displacements.

mixed mode fracture. 3.3. SCB specimen The procedures that determines fracture mode of SCB specimen are almost the same. The specimen SCB-2-1 is used as an example. Similarly, the crack opening and tangential displacements Δu1 and Δv1 (0–100% of peak load) can be transformed from horizontal and vertical displacements, as displayed in Fig. 5(a)–(d). The observation windows also can identify a cohesive crack whose tip is at the place where the opening displacement contour merged (Fig. 5(e) and (f)). Its length is 10.8 mm. Similarly, the crack opening and tangential displacements Δu1 and Δv1 can be selected from two cross-sections (X1 ¼ 70.2 mm). Fig. 5(g) shows the opening displacements Δu1, and critical value is determined to be 25 mm, which is same as CCBD specimen. However, the tangential displacements Δv1 shows little differences between the two sides along the crack, as shown in Fig. 5(h). Therefore, it can

be considered that no sliding occurs. Given the fact that only the crack opening displacement exists, it is a mode I fracture. The fracture modes of all the experiments are listed in Table 2. Experimental results show that no pure mode II fracture is observed under so-called theoretical mode II loading, i.e. no any confinement. Instead, they are mixed mode or mode I fractures. It is interesting to find that all the CCBD specimens generate a mixmode fracture and all the SCB specimens generate a mode I fracture.

4. Conclusions The pure mode II or sliding fracture cannot be created by socalled mode II loading in disc-type specimens including CCBD and SCB without the confinements, because there are opening displacements observed for all the specimens. The difference between CCBD and SCB specimens is the

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Fig. 5. Specimen SCB-2-1, 0–100% of peak load: (a) horizontal displacement Δu, (b) vertical displacement Δv, (c) opening displacement Δu1, (d) sliding displacement Δv1, enlarged images of (e) opening and (f) sliding displacements, the displacement profiles along the crack (g) opening and (h) sliding displacements.

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Table 2 Fracture mode in experiments. Specimen number

Fracture mode

Specimen number

Fracture mode

CCBD-1-1 CCBD-2-1 CCBD-3-1 CCBD-4-1 CCBD-5-1 SCB-1-1 SCB-2-1 SCB-3-1 SCB-4-1 SCB-5-1 SCB-6-1 SCB-7-1 SCB-8-1

Mix-mode Mix-mode Mix-mode Mix-mode Mix-mode Mode I Mode I Mode I Mode I Mode I Mode I Mode I Mode I

CCBD-1-2 CCBD-2-2 CCBD-3-2 CCBD-4-2 CCBD-5-2 SCB-1-2 SCB-2-2 SCB-3-2 SCB-4-2 SCB-5-2 SCB-6-2 SCB-7-2 SCB-8-2

Mix-mode Mix-mode Mix-mode Mix-mode Mix-mode Mode I Mode I Mode I Mode I Mode I Mode I Mode I Mode I

existence of sliding displacements, such that CCBD specimens generate a mix-mode fracture and the SCB specimens generate a mode I fracture.

Acknowledgements This work was financially supported by the National Natural Science Foundation of China under Grant nos. 51322906 and 41272349, and Youth Innovation Promotion Association CAS under Grant no. 2011240.

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