Comments on the mode II fracture from disk-type specimens for rock-type materials

Accepted Manuscript Comments on the mode II fracture from disk-type specimens for rock-type materials Qing Lin, Wei-Wei Ji, Peng-Zhi Pan, Siqi Wang, Yunhu Lu PII: DOI: Reference:

S0013-7944(18)31130-5 https://doi.org/10.1016/j.engfracmech.2019.02.024 EFM 6365

To appear in:

Engineering Fracture Mechanics

Received Date: Revised Date: Accepted Date:

14 October 2018 11 February 2019 15 February 2019

Please cite this article as: Lin, Q., Ji, W-W., Pan, P-Z., Wang, S., Lu, Y., Comments on the mode II fracture from disk-type specimens for rock-type materials, Engineering Fracture Mechanics (2019), doi: https://doi.org/10.1016/ j.engfracmech.2019.02.024

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Comments on the mode II fracture from disk-type specimens for rock-type materials Qing Lina&c, Wei-Wei Jib&d, Peng-Zhi Panb, Siqi Wanga&c, and Yunhu Lua&e a

State Key Laboratory of Petroleum Resources and Prospecting, China University of Petroleum, Beijing 102249, China b State Key Laboratory of Geomechanics and Geotechnical Engineering, Institute of Rock and Soil Mechanics, Chinese Academy of Sciences, Wuhan 430071, China c Department of Engineering Mechanics, College of Petroleum Engineering, China University of Petroleum, Beijing 102249, China d College of Geophysical Engineering, China University of Petroleum, Beijing 102249, China e Department of Drilling Engineering, College of Petroleum Engineering, China University of Petroleum, Beijing 102249, China Abstract: Laboratory determination of the mode II fracture toughness KIIC for rocks and rock-type materials remains a challenge task. One of the testing approaches is the compression of disk-type specimens, which is simple and convenient. However, although the notch in disk-type specimen is under mode II loading, the detailed studies reveal the drawbacks about interpretations of the disk-type specimens: (i) mode II fracture is only assumed without careful identifications about the actual fracture mode; or (ii) critical value of associated mode II loading is considered to be the fracture toughness KIIC regardless of the actual fracture mode. The widely accepted failure criterion for disk-type specimens is maximum tangential stress that actually is maximum tensile stress, such that the initiated fracture generally involves the opening displacements. Thus, it raises a serious doubt about the mode II fracture from the disk-type specimens for rock-type materials. In this study, a series of the disk-type specimens of the marble was tested, and digital image correlation (DIC) was used to obtain the displacements surrounding the notch where the fracture mode was identified. Experimental results show the initiated fractures from the specimens are the mode I fractures, suggesting that the assumption of the mode II fracture is not correct. In the discussion, the interpretation is provided for: (i) differences between the critical values of mode II loading from disk-type specimens without compression and from punch-through shear specimens under compression; (ii) application of the criterion of local symmetry; and (iii) characteristics for rocks or rock-type materials and other information. Keywords: Mode II fracture toughness, digital image correlation (DIC), self-similarity, maximum tangential stress, stress intensity factor, disk-type specimen.

1. Introduction When the principle of fracture mechanics is introduced to solid mechanics and material sciences, it fundamentally enhances the understanding of material failure mechanisms, helps to solve engineering problems, and improves engineering designs [1,2]. A parameter, fracture toughness, is also introduced from fracture mechanics as a basic property that represents fracture resistance of the material. Thus, the accurate determination of fracture toughness for different materials is an essential task for the society of fracture mechanics. For rocks and rock-type materials, because mode I fracture is relatively simple to be created, the testing procedure for mode I fracture toughness KIc has been suggested by ISRM (International Society for Rock Mechanics and Rock Engineering) on 1988 [3], and then modified on 2014 [4]. However, the mode II fracture toughness KIIc remains a significant challenge for society of rock mechanics and fracture mechanics. Although ISRM suggested a method for determination of mode II fracture toughness KIIc recently [5], different testing methods still have been used for rocks or rock-type materials due to their testing simplicity and convenience [1,6-9]. One of those methods is the compression of disk-type specimens, as shown in Fig.1, which mainly include two types of specimens: the angled edge crack semi-circular specimens under three-point bending (SCB), and the centrally cracked Brazilian disk specimens (CCBD). Before discussion about the disk-type specimens, it is necessary to clarify about fracture modes and related loadings. In the literatures, different concepts and names have been mixed, e.g., a mode II fracture and a fracture under mode II loading. Here only in-plane loadings and deformations are considered. A mode II fracture is a fracture with only sliding, without any opening; a fracture under mode II loading is a fracture under a pure shear loading, without any tensile loadings. They are actually not identical until two names represent a same phenomenon. For the disk-type specimens, the theoretical and numerical solutions [10,11] reveals a situation for a specified angle of the inclined pre-crack or notch, such that the loading of KI is zero and KII reaches its maximum value. That suggests a condition of the pure mode II loading can be achieved. The mode II fracture toughness KIIc can be determined by the maximum external load, if a mode II fracture is assumed. However, this statement has a conflict with the criterion of local symmetry, which implies that a crack would propagate in the opening mode, unless this mode is suppressed by sufficiently high compressive stresses [12,13]. In addition, the theoretical and experimental analysis for the rocks or brittle materials in compression also presented that the initiated cracks from an inclined pre-crack are the tensile cracks [14]. Thus, it poses a first fundamental question for the disk-type specimens: a mode II fracture is initiated?

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Recently, a different approach has been presented to extract information of the mode II fracture toughness KIIc from experimental results of the disk-type specimens [7,8]. Rather than to assume a mode II fracture, a failure criterion of maximum tangential stress is introduced to decide the fracture initiation. If a critical value of mode II loading satisfies the failure criterion and a fracture is created, this value is regarded as the mode II fracture toughness KIIc. However, as the theoretical and experimental analysis show, it is highly possible to initiate a mode I fracture or mixed-mode fracture under a mode II loading [12-15]. Thus, it poses a second fundamental question for the disk-type specimens: the critical value of mode II loading is the mode II fracture toughness while the actual fracture is not a mode II fracture? To answer those two questions, it is necessary to identify the fracture mode based on the information of local displacement fields surrounding the region of notch tip. Digital image correlation (DIC) serves as an optical approach for accurate measurements of full-field displacements on specimen surface. Fundamentally different to classic optical techniques based on wave interference, DIC is an optical technique based on the target tracking [16]. Using the natural or artificial feature as the target, DIC can determine the displacements by tracking the target movements. Due to its relatively technical simplicity, DIC has been widely used to study the fractures for rocks and rock-type materials [17-19]. In this paper, a series of disk-type specimens including SCB and CCBD specimens was tested for the marble, and DIC was used to obtain the full-field displacements. Note that rocks and rocktype materials always involve the development of a fracture process zone in front of crack tip [1,17,18], and thus the displacements along the fracture process zone can be used to decide the fracture mode. Only experimental results before the peak load are studied in this research. The results show both types of specimens do not generate any mode II fractures, and the initiated fractures are mode I fractures. On the base of detailed analyses about the disk-type specimens, an interpretation is presented in the discussion, such that it answers the question why the mode II fracture toughness cannot be estimated from the experimental results of disk-type specimens for most rocks and rock-type materials.

2. Digital image correlation and experimental setup 2.1 Digital image correlation Digital image correlation (DIC) was introduced to experimental solid mechanics on early 1980s [20]. It has not been popular in mechanical testing until two decades later, when the high resolution digital camera is affordable for the laboratory of rock mechanics and fracture mechanics. DIC extracts information of full-field displacements by comparisons of digital images

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of specimen surface between the un-deformed and deformed states respectively [16]. A subset that includes a group of random speckles is selected as a target. Searching the region of interest (ROI), the displacements can be accurately estimated by identifying the subset movement between the reference (un-deformed) and current (deformed) images [17,18]. Identification of a subset requires a DIC algorithm, whose principle is to match the intensities between the reference and current images. A correlation function in the DIC algorithm is used to calculate the correlation for intensities, and the maximum value of the correlation function is associated with the displacements. Generally, two different correlation criteria are used to find the initial guess and its subsequent refinement [21]. The initial guess is calculated by the normalized cross correlation (NCC) method

C cc 

i j

( , )s

i j

(f(x i ,y i )  fm )(g(x i ,y i )  g m )

( , )s

(1)

(f(x i ,y i )  fm ) (i ,j )s (g(x i ,y i )  g m )

2

2

where f and g are the reference and current image intensity function at location (x, y) respectively. fm and gm are the mean intensity value of reference image and current image in the subset respectively. A nonlinear optimizer is used to refine these results with sub-pixel resolution by finding the minimum of

C LS

    

f(x i ,y i )  fm

(i ,j )s (f(x i ,y i )  fm )2



 g(x i ,y i )  g m  2  (i ,j )s (g(x i ,y i )  g m ) 

2

(2)

After determination of the displacements from the current subset movement, DIC processing continues with a selection of a new subset, until the displacement fields with respect to the acquired image are obtained. 2.2 Experimental setup The testing materials are the marble from hydropower station at Jinping II, China. The material properties are: Young's Modulus under compression E is 45 – 55 GPa, Poisson ratio ν is 0.29 – 0.32, density ρ is 2780 kg/m3 and uniaxial compressive strength (UCS) is 110 – 160 MPa, tensile strength σt is 14 – 18 MPa. All specimens were cut at the same orientation from the same block of rock, as presented in Table 1. In order to produce the pure mode II loading, a series of requirements for SCB and CCBD specimens has to be satisfied [1,7,8,10,11,22-25]. Table 1 gives the ratio between notch length a and disk radius R and the inclined angle α of notch for 16 SCB and 10 CCBD specimens. For example, when a span ratio S/R is 0.8, normalized notch length a/R of 0.5 is produced for a SCB

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specimen and a notch inclination α is required to be 63° [1]. Since 1980s, a variety of approaches has been proposed for the requirements, although they are not fundamentally and significantly different [1,7,8,10,11,22-25]. Thus, it is not necessary to cover all the approaches. In this study, two main approaches are considered, one is based on the work of Atkinson et al [10], and the other is on the work of Ayatollahi et al [7,8,11]. The work of Atkinson et al [10] was one of early studies about the disk-type specimen, and their explicit solution laid a solid foundation for its future applications. Ayatollahi et al [7.8,11] improved the solution by consideration of T-stress, and have applied the disk-type specimens to study various brittle and quasi-brittle materials. The other approaches are either based on them, or modified from them. As shown in Table 1, the references are listed to demonstrate the approaches to fabricate the specimens. To ensure accurate measurements of the displacements, it is necessary to have a useful random pattern of speckles. A general procedure of preparation was followed: (1) lightly coat the specimen surface with white paint; (2) after the white paint was dry, overspray the coated surface with a dark mist from a spray paint; and (3) continue misting and re-misting until the speckle pattern characteristics were satisfied. The prepared speckles are shown in Fig. 1c and d. Generally, two basic features are required for the well-prepared speckle pattern: the distribution of speckles in the pattern and an estimate for the average size of the speckles [17,18]. The contrast of the speckle pattern depends on the distribution of speckles, which can help to decrease the measurement bias and noise. The average size of the speckles limits the image sampling and subset size selection. As shown in Fig. 1d, a typical speckle pattern has the size of speckles ranged from 100 – 200 μm, and a subset generally includes 3 – 10 speckles. A 2D DIC system mainly involves a high resolution CCD digital camera (3376×2704 effective square pixels, and a 35mm prime lens), a steady white light source, and an image acquisition computer. For the DIC system used in the study, the size of subset was 20 *20 pixels and ROI was 40 *40, 60*60 pixels, or even the whole specimen surface if necessary. Note that it is necessary for DIC measurements to transform from the image analysis in pixel units to the physical length on the specimen surface, and thus a magnification factor M has to be selected [17,18]. In the study, M is 30 μm/pixel, and the observation area covers the whole specimen surface (Fig. 1c). The measurement resolution of the DIC system is on the order of 0.3 μm, and the resolution of DIC grid is 0.15 mm. The experiment was performed in a closed-loop servo-hydraulic load frame (MTS815). For the SCB specimen, the experiment was controlled using crack mouth opening displacement (CMOD) as the feedback signal, such that crack propagated through a constant rate (0.2 μm/second) of CMOD. CMOD was measured by an extensometer called a clip gage, which

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detected displacement between two clips bonded to the bottom surface. For the CCBD specimen, the experiment was simply controlled by axial displacement with a constant rate of 0.5 µm/s, because it had difficulties on measurements of CMOD. A LVDT with ±1 mm linear range recorded the deflection of the specimens.

3. Disk-type specimens 3.1 The self-similarity of a mode II fracture For a mode II fracture, it only involves the sliding displacements along two fracture surfaces, without any opening displacements. Thus, one of intrinsic property, i.e., the self-similarity, is shared by a mode II fracture, which suggests that a mode II fracture should be co-planar with the pre-crack. As shown in Fig. 2, an example shows a SCB specimen with the notch of an inclined angle α. The initiation of a mode II fracture will follow the direction of notch, i.e., fracture path (1). If the fracture kinks and follows fracture path (2) with an angle θ, the self-similarity of fracture is broken and the fracture is not a mode II fracture. In contrast, it may be mixed-mode fracture or mode I fracture [26]. Thus, the principle of self-similarity is useful to decide whether a fracture is a mode II fracture, but it provides little information about the loading conditions. 3.2 The stress analysis for disk-type specimens According to [7,10], the stress intensity factors at the notch tip in disk-type specimens, KI and KII, can be expressed as:

KI  K II 

P a  a S  YI  , ,  2Rt  R R

(3)

 a S P a YII   , ,  2Rt  R R

(4)

where a is the notch length, t is the thickness of specimen, R is the radius of specimen, and S is the half span of three-point bending (Fig. 1a). The mode I and mode II geometry factors YI and YII are the functions of the inclined angle α, a/R, and S/R. Although the various expressions or names (e.g., some researchers call them non-dimensional coefficients) has been given for geometry factors YI and YII in the literature [1,7,8,10,11,22-25], their differences do not influence the stress intensity factors fundamentally. The analytical or numerical solutions for the factors Y I and YII are available in the research [7,11]. Note that eq. (3) and (4) are for SCB specimens, but actually they are similar for CCBD specimens. The geometry factors YI and YII for CCBD specimens do not include the parameter S/R [7,11]. 3.3 An assumed mode II fracture for disk-type specimens

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Based on those equations, when the angle α is zero, it always represents a mode I fracture. More importantly, a situation such that KI is zero and only KII is presented can be created for a special angle α, when the values of ratios a/R, and S/R are known. That suggests that a condition of the pure mode II loading is achieved at the notch tip with the angle α [7,10]. Thus, if a mode II fracture is assumed to occur under the pure mode II loading, the mode II fracture toughness can be determined by eq. (4) when the external load P reaches the peak [6,25]. However, a serious question arises whether a mode II fracture will be generated under a pure mode II loading. To answer this question, it is necessary to have an observation of the fracture path to know the fracture mode, as demonstrated in section 3.1. Previous experimental results clearly show that the similarity of fracture cannot be maintained for the tested specimens, and the fracture always kinks from the notch [6-8,26]. Thus, it does not seem correct to assume a mode II fracture under a pure mode II loading. More importantly, the mode II loading only represents a stress state at the notch tip, but the failure criterion is the key to decide the failure initiation. To initiate a mode II fracture, the failure criterion should be dominated by frictional strength, such as Coulomb friction criterion [27]. If the failure criterion involves the tensile stress, the initiated fracture will not be a mode II fracture. For the disk-type specimens, the broadly-accepted failure criterion is maximum tangential stress. 3.4 The failure criterion for disk-type specimens: maximum tangential stress (MTS) A failure criterion called the maximum tangential stress (MTS) has been introduced to decide fracture initiation for disk-type specimens [7,28]. As shown in Fig. 2, the elastic tangential stress in mixed-mode loading is:

1

  

2r

cos2

 2

 1    2 r 2  K cos  3 K sin  T sin   O II  I    2 2   

(5)

where T is a constant term independent of distance r, θ is the kinking angle. For the mode II loading, KI is zero, and then eq. (5) is modified:

  

 1  2  2 sin cos  T sin   O  r 2    2 2 2r  

 3K II

(6)

Indeed, the angle of maximum tangential stress θm can be determined:

  

 0

(7)

 m

By comparison of the critical value σθθc, the mode II fracture toughness KIIc is given:

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K IIc 

 2r T

c c

sin 2  m 

2rc  c



(8)

     3 sin m cos2 m  2 2  

where Tc is a critical value proportional to KIIc, and the ratio can be found [7]. Of course, T c can be ignored if only a rough KIIc needs to be estimated. Eq. (8) clearly shows the value of KIIc depends on σθθc, because Tc is proportional to KIIc. Now a fundamental question is: what is maximum tangential stress σθθc? From the analysis of Fig. 2, the stress σθθ is not shear stress, actually it is circumferential/normal stress. Note that Erdogen and Sih [28], who first introduced this failure criterion, clearly indicated: “The hypothesis that the crack will grow in the direction perpendicular to the largest tension at the crack tip seems to be verified also by cracked plates under combined tension and shear.” Thus, σθθc is the maximum tension and the initiated fracture will involve opening displacements based on the criterion. 3.5 Critical value of KII loading for disk-type specimens based on MTS The failure criterion confirms the situation of pure mode II loading. As the eq. 6 clearly indicates, once KII loading reaches a critical value, the material will fail because the fracture initiates and propagates from the notch. Thus, the critical value of KII loading is regarded as the mode II fracture toughness KIIc by some researchers [7,8]. Indeed, if only material failure is considered, that critical value of KII loading appears to be the KIIc. However, fracture is only one type of material failure, more information is required for understanding of fracture mechanism, because the fracture path will vary under different fracture mode. For disk-type specimens, the study of MTS criterion clearly demonstrates that fracture initiation actually dominated by maximum tangential/circumferential stress that is tensile stress, such that the initiated fracture involves the opening displacements [6,25]. Thus, the fracture is not a mode II fracture. A serious problem is posed, if the initiated fracture is a mixed-mode fracture, or a mode I fracture, is it appropriated to regard the critical value of KII loading as the KIIc? More importantly, as Liu et al. [15] pointed out, the initiation of mode I fracture under mode II loading represents a necessity to separate the “modes of loading” and “modes of fracture”. Generally, the mode II fracture includes both pure mode II (pure shear) and compressive shear modes [15]. Creation of a pure mode II fracture not only requires a state of pure shear loading, and a zero normal stress on potential fracture plane as well. However, a compressive shear fracture is also a sliding fracture that can be produced with the assistance from compressive stress. Thus, the application of compressive stress to a disk-type specimen (CCBD specimen) can create a different case, and their comparisons may provide some fundamental information for

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interpretation of associated critical value of KII loading when the initiated fracture is not a mode II fracture. Generally, the compressive stress can eliminate the mode I fracture, such that it is possible to create a mode II fracture (i.e., compressive shear fracture) for CCBD specimens [29,30]. For disk-type specimens, if a mode II fracture is created under a compressive stress that is large sufficient to suppress the mode I fracture, the associated critical value of K II loading will be larger than that without any confinements [31,32]. The difference between two critical values is due to the existence of compressive stress, which actually represents the difference of nonmode II (mode I or mixed-mode) and mode II fractures. The details about critical value of KII loading will be discussed in section 5. 3.6 Summary for disk-type specimens After review about the testing of disk-type specimens, it can be concluded: (1) stress analysis clearly shows the condition of mode II/shear loading can be achieved in the disk-type specimens, when some requirements are satisfied; (2) the previous researches simply assume that mode II fracture is created without any detailed measurements about the actual fracture mode. However, recent experiments demonstrate the initiated fracture is a mixed-mode or mode I fracture; (3) The well-accepted failure criterion to predict the fracture initiation in the disk-type specimens is the maximum tangential stress (MTS), which is maximum tensile stress. Thus, the criterion confirms that the initiated fracture involves the opening; (4) some recent researches regard the critical value of mode II loading as the fracture toughness KIIC for the tested material. However, because the initiated fracture is not a mode II fracture, it poses a challenge how to interpret the critical value of mode II loading from the disk-type specimens. In this study, based on the DIC obtained displacements surrounding the notch tip of a disktype specimen, the procedure to perform experimental analysis is: (1) the observation of the fracture path is used to determine whether the principle of similarity is maintained or broken; (2) the normal and tangential displacements along the fracture path are transformed from DIC measurements, and are used to identify the fracture mode; (3) an interpretation is provided for the critical value of mode II loading and some related information.

4. Experimental Results 4.1 SCB specimens When the loading is in the early stage (the loading of 0-70% of peak), the deformation of the specimen SCB-3 is elastic, as shown in Fig. 3. The inclined notch distorts the horizontal displacements u and breaks their symmetry (Fig. 3a). Although the horizontal displacements are still observed to be surround the notch, clearly the neutral axis (y = 30 mm) is tortuous due to the

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shear loading. The horizontal displacements are either tortuous or inclined. Because the specimen is under bending, it is reasonable to identify a region of horizontal compression under the loading position. For vertical displacements v (Fig. 3b), they remain more or less symmetry in the regions outside the notch, although the symmetry line has an inclination (14 o approximately) to the vertical. The notch also separates the vertical displacements and causes their offset. As the external load increases, a displacement discontinuity starts to occur at the notch tip when the load reaches 90% of the peak. Fig. 4a shows the incremental horizontal displacements ∆u between the loading of 70-90% peak. Again, the neutral axis is still distorted. The displacement discontinuity is observed to kink from the notch tip, which is the fracture process zone [17,18]. Fracture process zone is a zone of material damage that releases the singularity at crack tip physically. Of course, fracture process zone is not a concentration for current research, but it represents the fracture path during the fracture initiation. Note that there is no line or mathematical crack in reality, and thus the boundaries of fracture process zone are regarded as the sides of fracture path. From the incremental vertical displacements ∆v (Fig. 4b), the fracture process zone can also be clearly observed. Above it, the displacements display a symmetric pattern along an inclined symmetry line. However, to know the fracture mode, it is necessary to have the information about the opening and sliding displacements along the fracture process zone, rather than the horizontal and vertical displacements. Thus, a new coordinate (x1, y1) is established by rotating the coordinate (x, y) with an angle β, where the axis y1 is along the direction of the fracture, as shown in Fig. 4c. All the horizontal and vertical displacements (u, v) are transformed to the normal and tangential displacements (u1, v1) based on equations (9) and (10):

u1  u cos   v sin  (9)

v1  u sin   v cos  (10) Note that the opening displacements are the differences between normal displacements from two sides of fracture process zone, and sliding displacements are the differences between tangential displacements. The incremental normal and tangential displacements (∆u1, ∆v1) are plotted in Fig. 4c and d, based on the coordinates (x, y) and (x1, y1) together. The normal displacements show the length of fracture process zone is 8 mm approximately. Interestingly, it appears that there is also a “neutral axis” at y1 = 20 mm approximately. It is simply because the normal displacement gradient

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∂∆u1/∂x1 is zero along the zone of “neutral axis”, which suggests there is no loading on the materials. The normal displacement is about 4 μm inside that “neutral axis” zone, and it represents a motion associated with the loading point along x 1 axis (Fig. 4c). The incremental tangential displacements ∆v1 shows almost a symmetric pattern along a symmetry line (Fig. 4d), although they are distorted. Most interestingly, it has difficulties on identification of fracture process zone from tangential displacements, because there are no clear observations of the displacement discontinuity on the two sides of fracture process zone. It suggests that the differences of tangential displacements (i.e., the gradient ∂∆v 1/∂x1) along the fracture process zone are close to zero. In order to provide a detailed observation, the profiles of incremental normal and tangential displacements along fracture process zone are plotted to examine their distributions. Although it is a challenge task on accurate identification of the process zone boundaries, the zone width can be estimated to be 1 mm approximately. Thus, two cross sections (x1 = -23.04 mm and x1 = -24.04 mm) are selected to represent the zone boundaries, and the incremental normal and tangential displacements along them are plotted in Fig. 4e and f. The incremental normal displacements (Fig. 4e) confirm the length of fracture process zone is about 8 mm, and the opening displacement at the notch tip is 4 μm for loading interval (70-90% of peak). Fig. 4f shows the incremental tangential displacements almost overlap together from two sides of fracture process zone, which shows there is no sliding for the initiated fracture because the differences of tangential displacements are zero along fracture process zone. The observations indicate that the initiated fracture is a pure mode I/opening fracture, not a mode II fracture. Note that more normal displacements are observed on the left side of fracture process zone than the right. For example, at the notch tip, normal displacement is -7 μm on the left and -3 μm on the right. Because normal displacement is -4 μm in the center that is related to the “neutral axis”, the magnitude of normal displacement is 3 μm on the left and 1 μm on the right, which results in a difference of 2 μm. An ideal mode I fracture will involve a symmetric distribution of normal displacements, and thus the observed non-symmetric pattern suggests the possible existence of a shear loading, but interestingly, the tangential displacements do not support this suggestion. Fig. 5a-d shows the incremental displacements of (∆u, ∆v) and (∆u1, ∆v1), with respect to 90100% of peak. The horizontal and vertical displacements display a similar pattern to Fig. 5a and b, only the length of fracture process zone is observed to extend to 14 mm approximately. For the normal displacements (Fig. 5c), a symmetric pattern is identified with the “neutral” axis of y1 = 25 mm below the influence region of the loading position. For the tangential displacements (Fig. 5d), it appears that the fracture process zone cannot be clearly defined, but a series of

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displacement islands are observed to cover the process zone and the center of those islands coincides with the process zone. Thus, the tangential displacements along two sides of fracture process zone are almost same, suggesting the value of gradient ∂∆v1/∂x1 is zero. Furthermore, there exists a symmetry line from the loading position to the notch tip that also includes the fracture process zone (Fig. 5d), and the tangential displacements display a symmetric pattern. Those observations demonstrate that the crack growth is quite stable during the loading interval (90-100% of peak) and there is almost no shear loading. The incremental normal displacement profiles (Fig. 5e) show an excellent symmetry pattern to strengthen the visualization of DIC measurements from displacement contour in Fig. 5c. The opening displacement at the notch tip is 16 μm, and the length of fracture process zone is about 14 mm. The incremental tangential displacement profiles (Fig. 5f) show there are no differences, or little differences, between the displacements from two sides of fracture process zone. Again, it can be concluded that there is no mode II deformation and the fracture initiation is completely due to mode I deformation based on DIC measurements. Thus, experimental results show the fracture kinked in specimen SCB-3 is a mode I fracture, once the fracture initiates. Fig. 6 shows the incremental normal and tangential displacements for specimens SCB-1, SCB-7 and SCB-16 (the loading of 90-100% of peak) produced based on other approaches [1,7,8,10,11,22-25]. Obviously, the initiated fractures from those specimens are also the mode I fractures. The experimental results for all SCB specimens, including the kinking angle θ, the rotation angle β, etc., are listed in Table 1. Theoretically, initiation angle of mode I fracture under mode II loading is around 70.53° [15]. However, it is a challenge to obtain this ideal angle in the experiments (see Table 1), because of: (i) the influences of specimen boundary; (ii) the circular notch tip (it is very difficult for rock to produce a sharp notch tip as in the theoretical analysis); and (iii) material inhomogeneity and imperfection of loading alignment. It needs to note that an interesting observation for SCB specimens is: when the external loading is relatively small after the fracture initiates, the normal displacements show a nonsymmetric pattern along fracture process zone but the tangential displacements show little differences; when the loading is close to peak, the distribution of normal displacements is more or less symmetric and there are also no differences for tangential displacements. In one word, for most SCB specimens, the non-symmetric distribution of normal displacements is gradually eliminated, where the tangential displacements maintain little differences. 4.2 CCBD specimens Similarly, the specimen CCBD-2 is presented to study the fracture mode for initiated fracture. Fig. 7 shows the incremental horizontal and vertical displacements (∆u, ∆v) for early loading stage

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(40-60% of peak). A symmetric pattern is observed for them along the cross section y = 0 mm respectively, although it appears that a small rotation of the specimen results in an imperfect symmetry. For horizontal displacements, two neutral axes are identified at two parts of the specimen, i.e., along the sections of y = ±38 mm approximately, as shown in Fig. 7a. Both are distorted due to the existence of the inclined notch. For vertical displacements, the symmetric pattern of displacements is concentrated inside the region of notch (-10 mm < x < +10 mm), with the displacement values from +6 μm to -6 μm. Indeed, two zones of material compression are observed related to the two loading positions respectively. Two fracture process zone starts to develop at the corresponding tips of notch, when the external load increases (70-90% of peak in Fig. 8a and b). Indeed, both zones are kinked from the notch. The incremental horizontal and vertical displacements (∆u, ∆v) also display a symmetric pattern as Fig. 7, only it appears that the occurrences of fracture process zones make the specimen stable during the loading. Again, the coordinate (x 1, y1) is established to obtain the incremental normal and tangential displacements (∆u1, ∆v1), as shown in Fig. 8c and d. Because those displacements still remain a symmetric pattern along the x axis, only the upper part of specimen is presented to interpret the experiment. For normal displacements, the length of fracture process zone is identified to be about 8 mm, and the distributions of normal displacements at the two sides of the zone are not symmetric. For tangential displacements, fracture process zone cannot be clearly identified. The examination of the profiles of incremental normal and tangential displacements reveals that (i) again, more normal displacements are observed at the left side of the process zone than the right (Fig. 8e), which is 3 μm vs. 1 μm; (ii) the initiated fracture is a mode I fracture, since there are no sliding displacements along fracture process zone (Fig. 8f). When the external load reaches the peak, e.g., 90-100% of peak, the length of fracture process zone is observed to be 13 mm (Fig. 9a and b). The incremental horizontal and vertical displacements (∆u, ∆v) display an excellent symmetric pattern, which suggests the experiment is quite stable. After coordinate transformation, it appears that the non-symmetric distribution of incremental displacements ∆u1 along fracture process zone is even more severe. For incremental tangential displacements ∆v1, again identification of the process zone is difficult. The profiles of incremental normal displacements (Fig. 9e) show the normal displacement at the left side is more than 7 μm, but it is less than 3 μm on the right. Little differences of tangential displacements are observed along the process zone in Fig. 9f. For specimen CCBD-2, no matter how large are the magnitude differences of normal displacements at the two sides of fracture process zone, the differences of tangential displacements i.e., sliding displacements, are observed to be very small. Thus, it can be concluded

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that the kinked fracture is a mode I fracture. Similarly, the incremental normal and tangential displacements are shown in Fig. 10 for specimens CCBD-6 and CCBD-10 (90-100% of peak), and the initiated fractures are also mode I fractures. As listed in Table 1, the experimental results for all CCBD specimens are summarized. Interestingly, there is also an observation for CCBD specimens: (i) after the fracture initiates, a non-symmetric pattern is observed for the normal displacements along fracture process zone with respect to the small external loading. For example, Fig. 8c and e (specimen CCBD-2) shows the magnitude difference is 2 μm for normal displacements at the notch tip; (ii) when the loading reaches the peak, the differences of magnitudes of normal displacements are even larger. As shown in Fig. 9c and e (specimen CCBD-2), the magnitude difference changes to be 4 μm at the notch tip. During the two loading stages, little differences are observed for tangential displacements. However, it needs to emphasize that the opening displacements are also larger when magnitude differences of normal displacements are larger. At the notch tip of specimen CCBD-2, the opening displacement is 4 μm for 70-90% of peak, and the opening is 10 μm for 90100% of peak. It appears that their ratio remains more or less a constant, i.e., 0.5. Thus, for most CCBD specimens, the non-symmetric distribution of normal displacements is maintained, or possibly more severe, while the tangential displacements show almost no differences. 4.3 Summary of experimental results Displacements surrounding the region of notch tip from SCB and CCBD specimens clearly shows: (1) the initiated fracture is not a mode II/sliding fracture, because the fracture kinks from the tip of notch and the principle of similarity is broken; (2) a mode I fracture is identified for the disctype specimen of tested marble. Experimental results show that the differences of tangential displacements are small along two sides of fracture process zone for SCB and CCBD specimens after the fracture initiates. However, the distributions of normal displacements vary with the specimen type. For SCB specimens, non-symmetric distribution of normal displacements is gradually removed; for CCBD specimens, non-symmetric distribution of normal displacements maintains or becomes more severe.

5. Discussion 5.1 A fracture model to interpret critical mode II loading with and without the compression Imagine that an ideal two-dimensional machine can be built, such that a group of shear stresses (τ) are applied along the boundary of a rock specimen, which generates a condition of mode II stress intensity factor KII = τ(πa)0.5 at the crack tip, as shown in Fig. 11a. The stress state with respect to a kinking angle θ can be expressed as:

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  rr            r 

    2   sin 1  3 sin  2 2   K II    2    3 sin 2 cos 2  2r   cos  1  3 sin2     2  2    

(11)

Suppose that no any normal stresses are allowed to act the boundary of specimen (Fig. 11a). When the KII increases and reaches a critical value of KII(1), as shown of the loading path (a) in Fig. 11b, a mode I fracture initiates from the notch with a kinking angle θ, where the tensile stress σθθ reaches the tensile strength σt at the notch tip. Thus, this value KII(1) is what disk-types specimen obtained. Note that it appears that a couple of shear stresses τrθ is predicted along the fracture plane (Eq. 11), but experimental results do not observe them probably because they are too small to be detected. For this ideal machine, now suppose that a compressive stress σN can also be applied on the specimen boundary. Because the stress σN is used to suppress the tensile stress at the tip of notch , it is only applied perpendicular to the notch (Fig. 11c). Of course, the tensile stress at the tip of notch cannot be fully eliminated until the compressive stress σN is larger than a critical value σNc. Because the generated tensile stress is suppressed, a larger KII is required to initiate a fracture. For example, when KII reaches the value of KII(1), simultaneously a compressive stress σN is applied as an ideal manner. The fracture would not initiate until the KII is larger than KII(1). As shown of the loading path in Fig. 11b, the KII starts from KII(1) and increases with the compressive stress σN. Every value of KII in loading path actually is the critical stress to initiate a fracture with respect to the corresponding compressive stress σN. However, because the tensile stress is not fully eliminated when the compressive stress σN is smaller than σNc, the initiated fracture involves the opening displacements. Only when the compressive stress σNc is realized, a sliding fracture initiates, as a coplanar fracture with the notch. When the angle θ is zero, only the shear stresses τ rθ are presented along the fracture plane from the KII (Eq. 11), with the application of compressive stress σNc, the initiated fracture actually is a compressive shear fracture (Fig. 11c). Now the K II reaches a second critical value of mode II loading KII(2). The research [33] showed the value of KII(2) maintains a constant value once the compressive stress reaches σNc. The fracture model (Fig. 11) based on the ideal machine reveals the differences of the measured critical mode II loadings between the disk-type specimens [6-8] and the punch-through shear specimens [5]. The critical value KII(1) is associated with an opening fracture without the compressive stress; the critical value KII(2) is associated with a sliding fracture with the

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elimination of tensile stress by compression. A mode II fracture, according to definition of fracture mechanics, is an in-plane sliding fracture. Thus, it is not correct to regard the critical mode II loading KII(1) as the mode II fracture toughness, and the critical mode II loading K II(2) is the mode II fracture toughness KIIc. A recent publication tried to modify the definition of mode I and mode II fractures based on loading conditions [34], such that a mode II fracture is defined as a crack under a loading of only KII (zero KI) regardless of the crack displacements. It suggests that critical value KII(1) can be regarded as the mode II fracture toughness, even the initiated fracture is actually an opening fracture. However, most research and engineering problems in geophysics and geology involve the fracture process under compression. It is highly possible to have the sliding fractures during the earthquake. Then, it poses a serious problem how to interpret the physical meaning of critical value KII(2), if the KII(1) is regarded as the mode II fracture toughness. Again, the fracture mode should be decided based on the crack displacements, not the loadings although they are essential to create the fracture. Here it needs to note that mode II fracture can dominate over mode I for some materials, depending on the ratios of KIC/KIIC and KI /KII [15]. Thus, the fracture model in the study may only apply to most rocks and rock-type materials. 5.2 The principle of local symmetry The fracture model confirms application of the criterion of local symmetry [12]. Even fracture initiation under a pure mode II loading has to follow the criterion of local symmetry without suppression of the tensile stress. The criterion of local symmetry implies two situations: either k I is maximum or kII is zero [13], where k represents the local stress intensity factors at the kink angle.

kI  R11K I  R12K II (12)

kII  R 21K I  R 22K II

(13)

where the functions R11, R12, R21, and R22 are functions with respect to kink angle θ [13]. Two situations are not identical although their differences are not great, because the fracture kink angles are not same. For the tested marble, experimental observations clearly show the local kII is almost zero after the fracture initiates. It is perhaps because the sliding displacements, which are caused by the local mode II loading, are so small that they cannot be identified by DIC. The local mode I loading kI keeps increasing and eventually reaches the maximum, sincee the opening displacements are observed to increase till the external loading reaches the peak. Thus,

16

experimental observations suggest two situations (Eq. 12 and 13) need to be satisfied together to create a mode I fracture in the specimen for the tested marble. Thus, the criterion of local symmetry (eq. 12) represents a same physical meaning to the maximum tangential failure criterion (eq. 8), i.e., the fracture initiation is dominated by tension, although strictly speaking both criteria are not based on identical physical quantities. 5.3 Characteristics of rock-type materials and other information The fracture model presents the characteristics for rocks or rock-type materials, and actually it also represents the difficulties to create a mode II fracture for those materials. Without elimination of mode I fracture by the compressive/confining stresses, applications of the high shear stress do not necessity to create a mode II/sliding fracture. Some research also showed most rock-type materials are characterized by high value of ratio KIIc/KIc that is around unity or larger, so that mode I fracture prefers and mode II hardly occurs in the absence of a high confining [29,30]. Thus, it provides an interpretation why the attempts to create a mode II fracture fail by various approaches proposed in past a couple decades [1], including Iosipescu beam specimen, compact tension and shear specimen, compact shearing specimen, etc. The fracture model also provides an interpretation for the relation between the claimed mode II fracture toughness and tensile strength based on disk-type specimens [35]. Since the critical value of KII(1) is obtained when the tensile strength is reached at the notch tip, it is not surprised that a linear relation is observed for some rock materials. However, the relation does not present the actual correlation between the mode II fracture toughness KIIc (i.e., KII(2)) and tensile strength for the materials. The fracture model shows a possibility to create a mode II fracture for rocks or rock-type materials in disk-type specimens, i.e., the application of compressive/confining stresses [31,32]. For example, the compression tests [31] were performed for CCBD specimen with the confining pressure up to 28 MPa, but the fracture path in the specimen was not identified. Thus, it was actually unknown whether a mode II fracture was created in the specimen. However, according to the fracture model (Fig. 11), it is possible to produce a mode II fracture for CCBD specimen if the confining pressure is sufficiently large [33]. Of course, similar experiments [31] are suggested for different rocks or rock-type materials in the future research.

6. Conclusions Fracture toughness is a material property to evaluate fracture resistance, and its accurate determination in the laboratory is an essential task. For rocks and rock-type materials, it is the mode II fracture toughness that challenges the researchers in society of fracture and rock

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mechanics. Although a series of approaches has been proposed [1], the compression of disk-type specimens remain a simple and convenient approach. However, the detailed analysis about the disk-type specimens poses some serious questions on the determination of a mode II fracture toughness. Thus, a series of disk-type specimens were performed to study the fracture of the marble in this study. The full-field displacements surrounding the notch were obtained by DIC, and the fracture mode was identified. Based on the experimental results and related discussion, the conclusions are: (1) The fracture is observed to kink from the tip of notch such that the principle of similarity cannot be maintained. Thus, a sliding/mode II fracture cannot be initiated by the compression of disk-type specimens for most rocks and rock-type materials. (2) There are no observations of the sliding displacements, because the differences of tangential displacements are small along two sides of fracture process zone for SCB and CCBD specimens. Thus, it is an opening/mode I fracture that is initiated in the tested marble by the compression of disk-type specimens. Furthermore, the distributions of normal displacements vary with the specimen type. For SCB specimens, non-symmetric distribution of normal displacements is gradually removed; for CCBD specimens, non-symmetric distribution of normal displacements maintains or becomes more severe. (3) Experimental observations clearly show the sliding displacements are almost zero after the fracture initiates, and the opening displacements keep increasing and eventually reach the maximum when the external load reaches the peak. They suggest that the fracture initiation of tested marble follows the failure criterion of maximum tangential/circumferential stress or maximum tensile stress, and principle of local symmetry as well. (4) Based on the fracture model in the discussion, it is essential for application of the compression to suppress the tensile stress at the crack tip for most rocks and rock-type materials. Without compression, a mode I fracture initiates; under a sufficiently high compression, a mode II fracture initiates. Thus, it is not correct to regard critical mode II loading KII(1) associated with the initiation of a mode I fracture (Fig. 11) as mode II fracture toughness, because a larger mode II loading is required to initiate a mode II fracture if a sufficiently high compression is applied.

Acknowledgement The authors sincerely thank the financial support from the State Key Research Development Program of China (No. 2017YFC0804203), the National Natural Science Foundation of China (Grant No. 51304225, 51774305), and the Foundation of State Key Laboratory of Petroleum

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Resources and Prospecting, China University of Petroleum, Beijing (No. PRP/open-1402).

References [1] Whittaker BN, Singh RN, Sun G. Rock fracture mechanics principles, design and applications, New York, Elsevier; 1992. [2] Bažant ZP, Planas J. Fracture and size effect in concrete and other quasibrittle materials, Boca Raton Florida, CRC Press; 1997. [3] Ouchterlony F. Suggested methods for determining the fracture toughness of rock. Int J Rock Mech Min Sci Geomech Abstr 1988;25:71-96. [4] Kuruppu MD, Obara Y, Ayatollahi MR, Chong KP, Funatsu T. ISRM-suggested method for determining the mode I static fracture toughness using semi-circular bend specimen. Rock Mech Rock Eng 2014;47(1):267-274. [5] Backers T, Stephansson O. ISRM suggested method for the determination of mode II fracture toughness. Rock Mech Rock Eng 2012;45(6):1011-1022. [6] Chang SH, Lee CI, Jeon S. Measurement of rock fracture toughness under mode I and II and mixed-mode conditions by using disc-type specimens. Eng Geol 2002;66:79-97. [7] Ayatollahi M, Aliha M. On determination of mode II fracture toughness using semi-circular bend specimen. Int J Solid Struct 2006;43:5217-27. [8] Ayatollahi M, Sistaninia M. Mode ІІ fracture study of rocks using Brazilian disk specimens. Int J Rock Mech Min Sci 2011;48:819-26. [9] Burchell TD, Erdman D. The shear fracture toughness, KIIc, of graphite. Carbon 2016;98:267279. [10] Atkinson C, Smelser R, Sanchez J. Combined mode fracture via the cracked Brazilian disk test. Int J Fract 1982;18:279-291. [11] Ayatollahi MR, Aliha MRM. Wide range data for crack tip parameters in two disc-type specimens under mixed mode loading. Comp Mater Sci 2007;38:660-670. [12] Cotterell B, Rice JR. Slightly curved or kinked cracks. Int J Fract 1980;16:155-169. [13] Broberg KB. Cracks and Fracture, California, Academic Press; 1999. [14] Nemat-Nasser S, Horii H. Compression-induced nonplanar crack extension with application to splitting, exfoliation, and rockburst. J Geophy Res, Solid Earth 1982;87:6805-6821. [15] Liu HZ, Lin JS, He JD, Xie HQ. Dominant mode of planar fractures and the role of material properties. Eng Fract Mech 2018;195:57-79. [16] Sutton MA, Orteu JJ, Schreier H. Image correlation for shape, motion and deformation measurements: basic concepts, theory and applications, New York, Springer; 2009. [17] Lin Q, Labuz JF. Fracture of sandstone characterized by digital image correlation. Int J Rock Mech Min Sci 2013;60:235-245. [18] Lin Q, Yuan H, Biolzi L, Labuz JF. Opening and mixed-mode fracture processes in a quasibrittle material via digital imaging. Eng Fract Mech 2014;131:176-193.

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[19] Miao S, Pan PZ, Wu Z, Li S, Zhao S. Fracture analysis of sandstone with a single filled flaw under uniaxial compression. Eng Fract Mech 2018;204:319-343. [20] Chu TC, Ranson WF, Sutton MA. Applications of digital-image-correlation techniques to experimental mechanics. Exp Mech 1985;25(3):232-244. [21] Blaber J, Adair B, Antoniou A. Ncorr: Open-source 2D digital image correlation Matlab software. Exp Mech 2015;1-18. [22] Lim I, Johnston I, Choi S. Stress intensity factors for semi-circular specimens under threepoint bending. Eng Fract Mech 1993;44:363-82. [23] Lim I, Johnston I, Choi S, Boland J. Fracture testing of a soft rock with semi-circular specimens under three-point bending. Part 2—mixed-mode. International journal of rock mechanics and mining sciences & geomechanics abstracts: Elsevier; 1994. p. 199-212. [24] Kuruppu MD, Chong KP. Fracture toughness testing of brittle materials using semi-circular bend (SCB) specimen. Eng FractMech. 2012;91:133–150. [25] Awaji H, Sato S. Combined mode fracture toughness measurement by the disk test. J Eng Mater Technol 1978;100:175-82. [26] Ji WW, Pan PZ, Lin Q, Feng XT, Du MP. Do disk-type specimens generate a mode II fracture without confinements? Int J Rock Mech Min Sci 2016;87:48-54. [27] Zhang X, Jeffrey RG, Bunger AP, Thiercelin M. Initiation and growth of a hydraulic fracture from a circular wellbore. Int J Rock Mech Min Sci 2011;48:984-995. [28] Erdogan F, Sih GC. On the crack extension in plates under plane loading and transverse shear. J Basic Eng 1963;85:519-527. [29] Melin S. When does a crack grow under mode II conditions? Int J Fract 1986;30:103-114. [30] Melin S. Fracture from a straight crack subjected to mixed mode loading. Int J Fract 1987;32:257-263. [31] Al-Shayea NA, Khan K, Abduljauwad SN. Effects of confining pressure and temperature on mixed-mode (I-II) fracture toughness of a limestone rock. Int J Rock Mech Min Sci 2000;37:629-643. [32] Jin Y, Yuan J, Chen M, Chen KP, Lu Y, Wang H. Determination of rock fracture toughness KIIc and its relationship with tensile strength. Rock Mech Rock Eng 2011;44:621-633. [33] Backers T, Stephansson O, Rybacki E. Rock fracture toughness testing in mode II–punchthough shear test. Int J Rock Mech Min Sci 2002;39:755-769. [34] Ayatollahi MR, Zakeri M. An improved definition for mode I and mode II crack problems. Eng Fract Mech 2017;175:235-246. [35] Hua W, Dong S, Fan Y, Pan X, Wang Q. Investigation on the correlation of mode II fracture toughness of sandstone with tensile strength. Eng Fract Mech 2017;184:249-258.

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Figure captions: Figure 1. (a) SCB specimen; (b) CCBD specimen; (c) the acquired DIC digital image; (d) an enlarged digital image, 80*80 pixels, showing the speckles within a subset. Figure 2. An example of a SCB specimen for fracture path. Figure 3. Displacement fields for specimen SCB-3, 0-70% of peak, (a) horizontal displacement u; (b) vertical displacement v. Figure 4. Displacement fields for specimen SCB-3, 70-90% of peak, (a) incremental horizontal displacement ∆u1; (b) incremental vertical displacement ∆v; (c) incremental normal displacement ∆u1; (d) incremental tangential displacement ∆v1; (e) the incremental normal displacement profiles along fracture process zone; (f) the incremental tangential displacement profiles along fracture process zone. The zone boundary is at two cross sections x1 = -23.04 mm and x1 = -24.04 mm.

Figure 5. Displacement fields for specimen SCB-3, 90-100% of peak, (a) incremental horizontal displacement ∆u1; (b) incremental vertical displacement ∆v; (c) incremental normal displacement ∆u1; (d) incremental tangential displacement ∆v1; (e) the incremental normal displacement profiles along fracture process zone; (f) the incremental tangential displacement profiles along fracture process zone. The zone boundary is at two cross sections x1 = -23.04 mm and x1 = -24.04 mm.

Figure 6. Incremental normal and tangential displacement (∆u1, ∆v1), 90-100% of peak, (a) specimen SCB-1, (b) specimen SCB-7, (c) specimen SCB-16.

Figure 7. Displacement fields for specimen CCBD-2, 40-60% of peak, (a) incremental horizontal displacement ∆u1; (b) incremental vertical displacement ∆v.

Figure 8. Displacement fields for specimen CCBD-2, 70-90% of peak, (a) incremental horizontal displacement ∆u1; (b) incremental vertical displacement ∆v; (c) incremental normal displacement ∆u1; (d) incremental tangential displacement ∆v1; (e) the incremental normal displacement profiles along fracture process zone; (f) the incremental tangential displacement profiles along fracture process zone. The zone boundary is at two cross sections x1 = -11.30 mm and x1 = -10.30 mm.

Figure 9. Displacement fields for specimen CCBD-2, 90-100% of peak, (a) incremental horizontal displacement ∆u1; (b) incremental vertical displacement ∆v; (c) incremental normal displacement ∆u1; (d) incremental tangential displacement ∆v1; (e) the incremental normal displacement profiles along fracture process zone; (f) the incremental tangential displacement profiles along fracture process zone. The zone boundary is at two cross sections x1 = -11.30 mm and x1 = -10.30 mm.

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Figure 10. Incremental normal and tangential displacement ( ∆u1, ∆v1), 90-100% of peak, (a) specimen CCBD-6, (b) specimen CCBD-10. Figure 11. (a) A mode I fracture initiates from a notch in a specimen under pure shear loadings; (b) the relation of mode II stress intensity factors vs. compressive stress σN; (c) a mode II fracture (compressive shear fracture) initiates from a notch in a specimen under shear and compressive loadings.

Table title Table 1 Specimen dimensions and experimental results.

Nomenclature KIc: mode I fracture toughness KIIc: mode II fracture toughness Ccc: normalized cross correlation coefficient CLS: normalized sum of squared differences coefficient f: the reference image intensity function g: the current image intensity function fm: the mean intensity value of reference image in the subset gm: the mean intensity value of current image in the subset M: scalar magnification factor E: Young’s modulus ν: Poisson’s ratio ρ: density σt: tensile strength KI: mode I stress intensity factors KII: mode II stress intensity factors a: notch length R: disk radius α: inclined angle of notch t: specimen thickness P: external load S: half span of three-point bending YI: mode I geometry factor YII: mode II geometry factor σθθ: the tangential stress T: T-stress r: distance θ: kinking angle θm: the angle of maximum tangential stress u: horizontal displacement v: vertical displacement ∆u: incremental horizontal displacement ∆v: incremental vertical displacement β: rotation angle ∆u1: incremental normal displacement

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∆v1: incremental tangential displacement KII(1): a critical value of mode II loading associated with disk-type specimen without compression KII(2): a critical value of mode II loading associated with punch-through specimen under compression σN: compressive stress kI: local mode I stress intensity factor at the kink angle kII: local mode II stress intensity factor at the kink angle

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Fig. 1

(a)

(b)

𝒂 Notch

a

R

𝒂

R

a

Notch

2S (c)

(d)

Speckle

Subset 150μm

Fig. 2

∆u (micron)

y (mm)

external load

Fracture path (2)

Fracture path (1)

x (mm) line support

line support

Fig. 3 external load

external load

(b)

Horizontal compression

symmetry line

y (mm)

y (mm)

u (micron)

neutral axis

v (micron)

(a)

x (mm) line support

x (mm) line support

line support

line support

Fig. 4 external load

external load

(b)

(a)

y (mm)

∆u (micron)

y (mm)

∆v (micron)

symmetry line

neutral axis

fracture process zone

fracture process zone

x (mm)

x (mm) line support

line support external load

(c)

line support

line support external load

(d)

y (mm)

∆𝐮𝟏 (micron)

y (mm)

fracture process zone

∆𝒗𝟏 (micron)

symmetry line

“neutral axis”

x (mm)

x (mm) line support

line support

line support

line support

(f)

(e)

Notch tip (x1 = -23.04, y1 = -4.48)

y1 (mm)

y1 (mm)

Notch tip (x1 = -23.04, y1 = -4.48)

∆𝐮𝟏 (micron)

∆𝒗𝟏 (micron)

Fig. 5 external load

external load

(b)

(a)

∆v (micron)

y (mm)

y (mm)

∆u (micron)

neutral axis

x (mm)

x (mm) line support

line support

line support

line support

external load

external load

(c)

(d) “neutral axis”

line support

x (mm)

y (mm)

∆𝐮𝟏 (micron)

y (mm)

fracture process zone

∆𝒗𝟏 (micron)

symmetry line

line support

(e)

line support

x (mm)

line support

(f) Notch tip (x1 = -23.04, y1 =- 4.48)

y1 (mm)

y1 (mm)

Notch tip (x1 = -23.04, y1 = -4.48)

∆𝐮𝟏 (micron)

∆𝒗𝟏 (micron)

Fig. 6 external load

line support

external load

y (mm)

line support

x (mm)

x (mm)

line support

external load

(d)

∆𝐮𝟏 (micron)

(c)

∆𝒗𝟏 (micron) line support

∆𝒗𝟏 (micron)

x (mm)

y (mm)

line support

y (mm)

y (mm)

∆𝐮𝟏 (micron)

(a)

external load

(b)

line support

line support

external load

x (mm)

line support

external load

(e)

line support

x (mm)

line support

∆𝒗𝟏 (micron)

y (mm)

y (mm)

∆𝐮𝟏 (micron)

(f)

line support

x (mm)

line support

Fig. 7 external load

external load

y (mm)

∆u (micron)

∆v (micron)

(b)

neutral axis

y (mm)

(a)

neutral axis x (mm)

x (mm)

external load

Fig. 8

external load

(a)

x (mm)

∆v (micron)

fracture process zone

fracture process zone

y (mm)

∆u (micron)

y (mm)

(b)

x (mm)

external load

external load

(d)

(c)

∆𝒗𝟏 (micron)

y (mm)

y (mm)

∆𝒖𝟏 (micron)

fracture process zone

x (mm)

(e)

x (mm)

(f) Notch tip at (x1 = -11.44, y1 = 10.30)

y1 (mm)

y1 (mm)

Notch tip at (x1 = -11.44, y1 = 10.30)

∆𝒖𝟏 (micron)

∆𝒗𝟏 (micron)

external load

Fig. 9

external load

(c)

x (mm)

x (mm)

external load

external load

∆v (micron)

fracture process zone

fracture process zone

y (mm)

∆u (micron)

(b)

y (mm)

(a)

∆𝒗𝟏 (micron)

y (mm)

y (mm)

∆𝒖𝟏 (micron)

(d)

x (mm)

(e)

x (mm)

(f) Notch tip at (x1 = -11.44, y1 = 10.30)

y1 (mm)

y1 (mm)

Notch tip at (x1 = -11.44, y1 = 10.30)

∆𝒖𝟏 (micron)

∆𝒗𝟏 (micron)

Fig. 10

external load

external load

y (mm)

∆𝒖𝟏 (micron)

y (mm)

∆𝒗𝟏 (micron)

(b)

(a)

x (mm)

x (mm)

external load

external load

y (mm)

∆𝒖𝟏 (micron)

y (mm)

x (mm)

∆𝒗𝟏 (micron)

(d)

(c)

x (mm)

Fig. 11

(a)

(b)

(c)

𝝈𝑵𝒄

τ

τ

𝑲Ⅱ τ

Open crack crack

𝑲Ⅱ θ

σrr τrθ

σθθ

τrθ σrr

τ

Loading path

Closed crack

(2)

𝑲Ⅱ

θ

(1)

2a 𝑲Ⅱ

τrθ

𝑲Ⅱ

σNc

Mode Ⅰ Ⅰ crack kink σθθ Mode

τ

0

𝝈𝑵𝑪

𝝈𝑵

τrθ

2a Mode II growth, KI = 0

σNc

τ τ

τ

𝝈𝑵𝒄

SCB

CCBD

specimen #

a (mm)

S (mm)

R (mm)

t (mm)

α (deg.)

θ (deg.)

β (deg.)

peak load (N)

SCB-1

23.31

37.30

46.64

10

63

90

27

2054

SCB-2

23.54

37.30

46.67

10

63

90

27

2719

SCB-3

23.47

31.60

47.10

10

63

101

38

2515

SCB-4

23.89

31.70

47.24

10

63

101

38

2803

SCB-5

16.88

23.60

47.10

10

54

N/A

N/A

N/A

SCB-6

16.68

23.60

47.15

10

54

81

27

3232

SCB-7

23.85

23.70

47.30

10

40

75

35

2039

SCB-8

23.83

23.30

46.62

10

40

73

33

2134

SCB-9

23.68

23.60

47.10

10

40.5

84.5

44

2407

SCB-10

23.8

47.60

47.60

10

40.5

85.5

45

3126

SCB-11

23.71

28.10

46.78

10

52.1

N/A

N/A

N/A

SCB-12

23.63

46.53

46.53

10

52.1

90.1

38

2162

SCB-13

27.87

27.90

46.43

10

46.2

84.2

38

1933

SCB-14

28.64

47.54

47.54

10

46.2

82.2

36

2177

SCB-15

28.97

33.00

47.16

10

57.5

72.5

15

1771

SCB-16

28.5

46.68

46.68

10

57.5

101.5

44

1695

specimen #

a (mm)

R (mm)

t (mm)

α (deg.)

θ (deg.)

β (deg.)

peak load (N)

CCBD-1

15.42

50.15

10

27

46

19

6324

CCBD-2

15.4

50.15

10

27

48

21

4745

CCBD-3

20.15

50.15

10

25.2

50.2

25

4772

references 1 24

22,23

7,8,11

references 10,25

CCBD-4

20.03

50.15

10

25.2

48.2

23

3488

CCBD-5

25.13

50.15

10

23

50

27

3663

CCBD-6

25.23

50.15

10

23

48

25

2854

CCBD-7

30.05

50.15

10

20.3

62.3

42

2033

CCBD-8

30

50.15

10

20.3

65.3

45

1922

CCBD-9

30.22

50.15

10

21.3

59.3

38

2041

CCBD-10

30.21

50.15

10

21.3

61.3

40

2556

25

10

Highlights: Digital image correlation (DIC) is used to obtain the full-field displacements A mode II fracture cannot be initiated by the compression of disk-type specimen A mode I fracture is initiated by the compression of disk-type specimen. Fracture initiation of disk-type specimen follows failure criterion of maximum tensile stress Critical mode II load from disk-type specimen is not mode II fracture toughness