Determination of the stress field and crack initiation angle of an open flaw tip under uniaxial compression

Theoretical and Applied Fracture Mechanics 104 (2019) 102358

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Determination of the stress field and crack initiation angle of an open flaw tip under uniaxial compression ⁎

T

⁎

Hang Lina, Hengtao Yanga, , Yixian Wangb, , Yanlin Zhaoc, Rihong Caoa a

School of Resource Safety Engineering, Central South University, Changsha, Hunan 410083, China School of Civil Engineering, Hefei University of Technology, Hefei 230009, China c Work Safety Key Lab on Prevention and Control of Gas and Roof Disasters for Southern Coal Mines, Hunan Provincial Key Laboratory of Safe Mining Techniques of Coal Mines, Hunan University of Science and Technology, Xiangtan 411201, China b

A R T I C LE I N FO

A B S T R A C T

Keywords: Pre-existing flaws Open flaw Closed flaw Stress field formulae Crack initiation angle

In the uniaxial compression test of rock containing pre-existing flaws, the flaws are generally open, and the corresponding theoretical calculations are mostly based on the closed flaws. The suitable theoretical solution is needed for the crack initiation and extension of open flaws (Flaw: Prefabricated, inherent defects. Crack: Defects generated during the loading process). In this paper, first, the corresponding stress field formulae under compression are established by comparing and analyzing the stress field at the flaw tip under tensile condition. Then, ANSYS is used to simulate and confirm the stress field at the flaw tip under compression and shear conditions. Thus, the difference between the open flaw and the closed one is explained, and the stress field formulae of the open flaw tip under compression are determined without considering the flaw thickness. Second, based on the determined stress field formulae at the flaw tip, the relationship between the crack initiation angle of the open flaw tip and the inclination angle of the prefabricated flaw is derived. Finally, laboratory test and PFC simulation are conducted to verify the resulting crack initiation angle formulae. The experimental and simulation results are basically consistent with the theoretical formula.

1. Introduction Various discontinuities, such as cracks, joints, and faults, are found in the engineering rock mass, which remarkably impact the stability of a project [1–7]. Since the 1960s, many correlative studies have been performed using various aspects, such as laboratory experiments, numerical simulations, and theoretical analysis, and considerable results have been obtained [8–14]. The single-flaw test is the basis for studying other conditions. In the laboratory test on the single flaw, the specimens are usually fabricated by casting and cutting. For a real rock specimen, the flaws are formed only by mechanical or hydraulic cutting, and such flaws are almost always open [1]. For the cast-in-place method with materials, such as cement mortar, open or closed flaws can be made by inserting metal sheets or videotapes. During the test, the flaws in most specimens are open [8,15,16], or the flaws are open at least before the moment of the crack initiation. However, in the study of fracture problems, some mature theories are mostly based on closed flaws. The incipient sliding crack model [17–19] assumes that the flaw is closed, regardless of the type I stress intensity factor KI at the flaw tip, and the flaw is treated as a pure shear crack. Baud, Reuschlé [20], Steif [21],

⁎

Ashby and Hallam [18] studied the theory of crack initiation and propagation of wing cracks under the assumption that the main flaw is closed under compression. These studies used the sliding crack model. Li [22] believed that most of the compressed flaw are closed cracks. KI cannot be negative under this condition because of the mutual nonpenetration of the material between the closed crack faces. Moreover, a series of derivation processes are also premised on the fact that the flaw is closed under pressure, and, in this method, the open flaws are ignored accordingly. Liu and Cao [23] studied specimens containing flaws under uniaxial compression, and the results showed that pre-existing flaws are still in the open state after the initiation of the wing cracks. However, in the subsequent theoretical derivation, the authors still considered the flaw as in pure shear state and ignored the KI at the flaw tip. However, in the actual test, the opening of a flaw has a nonnegligible influence on the crack initiation and extension of the crack. Li and Yang [24] and Pu [19] believed that open and closed flaws should be treated differently. Pu [19] proved the difference between the two types through experiments and qualitatively analyzed the stress distribution at the flaw tip. However, theoretical analysis was not conducted, and the theoretical formula was not provided. Few

Corresponding authors. E-mail addresses: [email protected] (H. Lin), [email protected] (H. Yang), [email protected] (Y. Wang), [email protected] (Y. Zhao).

https://doi.org/10.1016/j.tafmec.2019.102358 Received 22 March 2019; Received in revised form 11 September 2019; Accepted 11 September 2019 Available online 13 September 2019 0167-8442/ © 2019 Elsevier Ltd. All rights reserved.

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compressive stress σ (σ only represents a positive value, while − σ represents compressive stress), the corresponding normal stress σn and shear stress τn on the closed flaw surface are easy to derive. The expressions of the two stresses can be obtained by elastic mechanics as follows: σ

⎧ σn = − 2 (1 + cos 2α ) ⎨ τn = − σ sin 2α 2 ⎩

(1)

When the friction coefficient μ on the closed flaw surface is introduced, the effective shear stress on the closed flaw surface is equal to the following:

τeff = τn + μ |σn|

(2)

In the sliding crack model [17–19], the normal stress on the closed flaw surface is generally considered to have no effect on the crack propagation, and the effective shear stress obtained in the Eq. (2) is the driving force that drive the crack propagation. Therefore, the flaw can be regarded as a pure type II crack under shearing, and the type II stress intensity factor KII at the flaw tip is given as follows:

Fig. 1. Schematic diagram of the closed flaw under uniaxial compression showing the resolved stress components σn and τn due to compressive load σ .

KII = τeff πa theoretical studies have focused on the open flaws, and no better theoretical solution is available on the crack initiation and extension for the open flaws. Accordingly, in the study of compression–shear test of the fractured rock mass, the flaw opening and closure should be distinguished, and open flaw should be considered separately to obtain a series of theoretical solutions about crack initiation and extension of the open flaw. Hence, this paper reports the differences between open and closed flaws under compression and shear conditions using numerical simulation and laboratory tests. The formulae of the stress field at the tip of the open flaw under compression and shear conditions are established. Moreover, the relationship between the crack initiation angle and the inclination angle of the pre-existing flaw is given and verified.

(3)

Under the combined action of effective shear stress on the flaw surface and far-field stress, the stress field at the flaw tip is written as follows:

⎧ σx = ⎪ ⎪ σy = ⎨ ⎪τ = ⎪ xy ⎩

−KII 2πr KII 2πr KII 2πr

θ

(

3θ 2

θ

sin 2 2 + cos 2 cos θ 2

θ 2

cos sin cos

3θ 2

θ 2

(1 − sin

θ 2

cos

)

−σ sin

3θ 2

)

(4)

where r is the distance from a certain point to the crack tip in the stress field. 2.2. Stress field at the open flaw tip

2. Stress field at the flaw tip under compression and shear conditions

For the open flaws under uniaxial compression, the flaw is internally hollow, and its surface is not in contact. So the upper and lower surfaces have no forces, and this condition is similar to the boundary conditions of the specimen with single flaw under tension. Therefore, the stress field at the open flaw tip under compression is assumed to be the same form as the tip stress field under tension. While the two cases are still distinct because of their different external forces. A single-center flaw specimen (with flaw thickness of 0) is under tension, at the tip of which the stress field is as follows:

2.1. Sliding crack model The closed flaw under uniaxial compression is shown in Fig. 1, in which the half-length of the flaw is a , and the inclination angle α is the angle between the pre-existing flaw and the horizontal direction. The coordinate system is established with the center of the flaw as the origin. The symbol convention is shown in Fig. 2. According to the elastic mechanics, the shear stress as shown in Fig. 2 is positive. And the crack initiation angle θ , which represents the angle between the initial crack and the plane of the flaw, is positive in the counterclockwise direction and negative in a clockwise direction. Under far-field

(

)

(

⎧ σx = KI cos θ2 1 − sin θ2 sin 32θ − KII sin θ2 2 + cos θ2 cos 32θ 2πr 2πr ⎪ ⎪ KI θ θ K θ θ 3θ 3θ σy = 2πr cos 2 1 + sin 2 sin 2 + 2IIπr cos 2 sin 2 cos 2 ⎨ ⎪ KI θ θ 3θ KII θ θ 3θ ⎪ σxy = 2πr cos 2 sin 2 cos 2 + 2πr cos 2 1 − sin 2 sin 2 ⎩

(

)

)

(

σ σ KI = (1 + cos 2α ) πa , KII = sin 2α πa 2 2

)

(5) (6)

In Eqs. (5) and (6), σ is a positive value, which represents the tensile stress. So if σ is changed to − σ , then Eqs. (5) can represents the stress field of the opening crack tip under uniaxial compression. 2.3. Verification of the stress field of flaw tip under compression and shear conditions The stress field formulae of the open flaw tip under uniaxial compression established in Section 2.2 are verified. The specimen models containing the open and closed flaws are generated by ANSYS, and the simulated stress fields are compared with the calculation result of Eqs. (4) and (5). First, a two-dimensional plane fractured model as shown in

Fig. 2. Schematic diagram of symbol convention around the flaw. 2

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(a) closed flaw model

(b) the open flaw Fig. 4. The final mesh at the flaw tip.

(b) open flaw model Fig. 3. Loading conditions for the closed and open flaw model in ANSYS.

Table 1 Flaw parameters in the ANSYS model.

Fig. 3 is established in ANSYS [(a) is a closed crack model and (b) is an open flaw model], the size of which is 200 × 200 mm2. For more accurate stress field results, the mesh of the crack tip is denser. The meshing size of the crack tip is only about 1/200 that of the sample edge. Besides, the degenerated quadrilateral element is used for the flaw tip element and the node of the tip element is densified at the same time. The final element mesh at the flaw tip is shown in Fig. 4. Only the linear elastic stage analysis is performed to maintain the consistency with the conditions of the calculation formula. Therefore, the PLANE183 element is selected for the two model materials, namely, whose properties only include the material elastic modulus E = 14 GPa, and Poisson’s ratio ν = 0.22 . For the closed flaw, the friction effect of the closed flaw surface should be simulated, and the contact elements: TARGE169 and CONTA172 elements are used. The contact elements satisfy the Mohr-Coulomb criterion, and its friction coefficient is 0.3. A fixed constraint is applied to the bottom of the two different models, and a horizontal constraint is applied to the lower half of the two sides. A horizontal leftward shear stress τ is imposed on the upper right half, and compressive stress σ is imposed on the top, in which the values of τ and σ are set to 3.2 MPa and 1 MPa respectively. Two different preexisting flaw parameters are shown in Table 1. The stress field results in ANSYS are not well displayed, because the stress contour of ANSYS can only be divided into 9 segments. Thus, the data of the node stress list, node coordinate list, and element information list in the simulation

Open flaw

Closed crack Flaw length 2a /mm Friction coefficient on the flaw surface/ μ

40 0.3

Flaw length 2a /mm Flaw thickness/mm

40 0.02

results are imported into Tecplot (Fig. 5). In addition, the vertical displacement of the nodes at the midpoint of the upper and lower flaw faces in the open flaw is monitored. The displacements of the upper and lower nodes are −1.16 × 10−5 m and −0.5025 × 10−5 m, respectively. This result indicates that the flaw is not closed at this time. In the numerical simulation test, due to the influence of some factors, such as finite boundary and fixed constraint, the stress near the crack is not equal to the value of external load. So the external load cannot be directly substituted into Eqs. (4) and (5) to calculate the stress field at the crack tip. For this reason, we applied the same load and constraint boundary conditions as those in the above test to the model without crack, and then obtained the stress values at the corresponding positions of crack tip, in which σy and σxy were about −0.9 MPa and 1.2 MPa. The theoretical stress field data of the tip is calculated and imported into Tecplot by considering into Eq. (4) and substituting KII = (−σxy + μσy ) πa KI = −σy πa , KII = −σxy πa into Eq. (5). The result is shown in Fig. 5. In order to display more visually the variation regularity of the stress field of the tip, the stress curve along the path of fixed ϕ and r is 3

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Close flaw-ANSYS

Close flaw-Formula

Open flaw-ANSYS

Open flaw-Formula

(a) Comparison of σ x contour from numerical and theoretical solution at flaw tip Fig. 5. Stress field at the flaw tip from ANSYS simulation and formula calculation under compression and shear conditions.

0, and notably less than that of the closed crack tip. Fig. 7(b) is the σyy stress distribution along the polar radius whose polar angle is −30°, which shows that σyy stress at the closed flaw tip varies from positive to negative and σyy stress at the closed flaw tip is always negative. As shown in Fig. 6(c), σxy at the closed crack tip generally have a symmetric distribution centered on 0° polar angle; However, for the open flaw, when the polar angle is negative, the stress value is much smaller; And the stress is visibly larger than that at the open flaw tip for the positive polar angle. This finding indicates that the equations in both cases are correct. Moreover, from the comparison in Figs. 5–7, it can be seen easily that the stress fields of the open and closed flaw, especially on σy, σxy , significantly differ. Therefore, in the study of cracks, open flaws are not suggested to be simplified into closed flaws for calculation. That is, the case that the closed flaw can be regarded as a pure type II shear crack, regardless of the influence of KI of the crack tip on the tip stress field. But as for the open flaw, similar to Eq. (5), in addition to the second half consistent with Eq. (4), the first half is also controlled by KI . Therefore, the influence of KI on the stress field of the open flaw tip cannot be ignored, where KI = −σ πa ; that is, KI is negative but may still be useful for the calculation. Eq. (5) can be used as the stress field formula

specially plot to compare the stress results in Figs. 6 and 7. The definition of ϕ here is very similar to θ mentioned above. However, for the purpose of avoiding textual ambiguity with θ , it is called ϕ , which means the polar angle in the polar coordinate system with the flaw tip as the origin. And the polar axis is pointing horizontally to the right, and the polar angle increases counterclockwise. For a fixed r path, circles with the flaw tip as the center of the circle and the radius of 0.0004 m are taken. For the fixed ϕ path, the polar radii that are with the polar angle of −30° are taken, and the length of the polar radius is 0.0016 m. The resulting curve is shown in Figs. 6 and 7. According to the stress contour map in Fig. 5 and the stress curve on the fixed path in Figs. 6 and 7, whether the crack is open or closed, the variation regularity reflected by the ANSYS simulation results is almost identical with that calculated by the formula. As shown in Figs. 6(a) and 7(a), no matter the crack is open or closed, the σxx have the same trend with the polar angle increasing, but the σxx at the open flaw tip is smaller than that at the closed crack tip on the same circular path. It can be seen from Fig. 6(b) that the overall σyy distribution of the closed crack tip on the annular path is an anti-symmetric about the 0° polar angle. Whereas at the open flaw tip, the stress distribution is obviously different. Along the circular path, the σyy stress value is mostly less than 4

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Close flaw-ANSYS

Close flaw-Formula

Open flaw-ANSYS

Open flaw-Formula

(b) Comparison of σ y contour from numerical and theoretical solution at flaw tip Fig. 5. (continued)

of the open flaw tip under compression and shear conditions (when the flaw thickness is not considered), in which the normal stress σ under the tension condition should be mathematically changed into − σ .

⎧

=0

⎨ ∂2σθθ < 0 ⎩ ∂θ2

(8)

The following equation can be obtained by considering the second formula in Eq. (7) into the first expression of Eq. (8):

3. Determination of crack initiation angle of open flaw Given that KI can be negative, the expression of stress field of the flaw tip under compression in the polar coordinate should be consistent with the tension case [22], as follows (ignoring high-order small quantities):

⎧ σrr = 1 ⎡KI (3 − cos θ)·cos θ2 + KII (3 cos θ − 1)·sin θ2 ⎤ 2 2πr ⎣ ⎦ ⎪ θ 1 σθθ = 2 2πr cos 2 [KI (1 + cos θ) − 3KII sin θ] ⎨ ⎪ τ = 1 cos θ [K sin θ + K (3 cos θ − 1)] rθ I II 2 2 2πr ⎩

∂σθθ ∂θ

cos

θ [KI sin θ + KII (3 cos θ − 1)] = 0 2

(9) θ

One solution of the Eq.(9) is cos 2 = 0 , θ = ± π . However, θ = ± π represents the cracked surface that is open, which is not realistic. Therefore, the initiation angle is determined as follows:

KI sin θ + KII (3 cos θ − 1) = 0

(10)

When neither KI nor KII is zero, the following quantity can be obtained from Eq. (10):

(7)

where KI , KII are the same as Eq. (6), but σ needs to be changed into − σ. According to the maximum hoop tensile stress criterion, the crack will initiate in the corresponding direction, which meets the following conditions:

θ0 = 2 arctan The

second

θ0 = 2 arctan 5

1±

1 + 8(KII / KI )2 4(KII / KI ) expression

1 − 1 + 8(KII / KI )2 4(KII / KI )

of

(11) Eq.

(8)

is

not

satisfied

if

. Thus, for the open flaw under uniaixial

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Close flaw-ANSYS

Close flaw-Formula

Open flaw-ANSYS

Open flaw-Formula

(c) Comparison of σ xy contour from numerical and theoretical solution at flaw tip Fig. 5. (continued)

uniaxial compression, as the inclination of pre-existing flaw increases, the crack initiation angle of the specimen containing the open flaw decreases continuously and gradually approaches 70.5°. This value is exactly the initiation angle of the pre-existing flaw under pure shear condition. Thus, the open pre-existing flaw cannot be calculated in accordance with the closed flaw which is under pure shear condition. The influence of negative KI on the whole stress field, the crack initiation, and a series of subsequent situations should be considered. Next, the crack initiation angle of the open flaw is verified by laboratory test and numerical simulation.

compression:

θ0 = 2 arctan

1+

1 + 8(KII / KI )2 4(KII / KI )

(12)

For the stress intensity factor of pre-existing flaw tip under uniaxial compression, Eq. (6) can be recasted as follows by replacing σ with − σ :

σ σ KI = − (1 + cos 2α ) πa , KII = − sin 2α πa 2 2

(13)

The ratio of KI , KII is as follows:

KII sin 2α = KI 1 + cos 2α

4. Experimental verification of the initiation angle of the open flaw under uniaxial compression

(14)

Substituting Eq. (14) into Eq. (12), the relationship between the crack initiation angle θ0 and the inclination of pre-existing flaw α can be obtained as follows:

1+

1+8

θ0 = 2 arctan 4

(

(

4.1. Laboratory test Rock-like specimens were fabricated from cement mortar materials with yield and failure characteristics similar to rocks. The mass ratio of each component of the rock-like material [25] (i.e., white cement: fine sand: water) is 8:4:3, in which the fine sand is sieved by a sieve with a hole diameter of 1 mm. The schematic diagram of the model is shown in Fig. 1. The size of the model is 150 × 100 × 30 mm3

2 sin 2α 1 + cos 2α

sin 2α 1 + cos 2α

)

)

(15)

Fig. 8 shows the relationship between θ0 and α in Eq. (15). Under 6

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Fig. 7. The comparison of σxx , σyy, σxy obtained by formula calculation and numerical simulation on the fixed radial path.

Fig. 9(a). The angle α of the pre-existing flaws changes from 0° to 90° with an interval of 15°, and four specimens are prepared for each angle. Speckles were made on the surface of the specimens to monitor the crack propagation. The speckled sample is shown in Fig. 9(b). The loading of the pre-cracked specimens adopts the displacement loading with a loading rate of 0.01 mm/s, and the grease is daubed on the end of the specimen to decrease the end friction and reduce the end effect. At the same time, the SLR camera is used to record the initiation and

Fig. 6. The comparison of σxx , σyy, σxy obtained by formula calculation and numerical simulation on the fixed circular path.

(length × width × height), and pre-existing open flaws are made by inserting PMMA sheets into the mortar. The size of the flaw is 30 × 25 × 1 mm3 (length × width × thickness). The sheets are pulled out after the initial setting, and the finished specimen is shown in 7

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50

Flaw inclination 0° 15° 30° 45° 60° 75° 90°

Stress (MPa)

40

30

20

10

0 0.0

Fig. 8. Relationship between crack initiation angle of the flaw tip and the inclination angle of pre-existing flaw.

1.0

1.5

2.0

2.5

3.0

Displacement (mm) Fig. 10. Stress-displacement curves of the specimens containing the prefabricated flaw of different inclination angles under uniaxial compression.

failure process. Thus, the test process can be subsequently processed by the digital image correlation technology, and the crack propagation of the specimen is analyzed. The test loading diagram is shown in Fig. 9(c). Fig. 10 shows the obtained stress–displacement loading curves after the specimens are loaded. Three stages exist before the peak strength. In the compaction stage, the axial displacement of the specimen is large, the stress growth is not obvious, and the curve is concaved downward. In the elastic stage, where the loading curve is basically kept linear, the strength of the specimen rises rapidly. In the crack initiation (stable extension) stage, the loading curve tends to be gentle, and the specimen still has a certain bearing capacity. However, the strength of the specimen is no longer rising greatly. The wing cracks are obviously generated and extended inside the specimen. When the wing crack is extended to the longest, the stress reaches the peak value. Then, with the generation of the secondary crack, failure quickly happens, the curve drops rapidly, and the brittleness after the peak is obvious. In addition, the comparison of the peak strengths in the curve shows that when the angle α increases, the peak strength first decreases and then increases. The minimum value of approximately 30° is achieved, which is similar to the many previously reported results [8,15,16,25,26]. After the loading test, the test video recorded by the SLR camera is

(a) specimen with 60° open flaw

0.5

decomposed frame by frame to obtain a large number of pictures. Then, each frame image is analyzed by GOM Correlate software, which can well display the strain field on the surface of the specimen. The obtained principal strain field results are shown in Fig. 11. Since this study is aimed to investigate the initiation angle of the wing crack, the generation of secondary cracks in the specimen and the subsequent stage of failure are not evaluated. Fig. 11 shows the crack propagation at a certain moment when the wing cracks develop fully before the occurrence of secondary cracks during the loading of specimens containing different flaw inclination angles. The obtained principal strain field is depicted by AutoCAD to obtain an intuitive sketch of crack propagation to facilitate the calculation of the crack initiation angle. Fig. 11 shows that, due to the opening of the flaw, the initiation angle of the crack is not kept at 70.5° as the closed crack but was obviously greater than 70.5°, as shown by previous experiments [1,3,6,14]. In all the results, the cracks in several specimens for α = 15° − 75° emanate from the flaw tips. While α = 0° , in addition to the cracks generating from the tip, a crack is also formed in the middle of the flaw. Due to the flaw opening, the loading condition on the flaw surface is just like the simply supported beam, which causes the tensile stress concentration in the

(b) specimen with 60° open speckled flaw (c) Test loading diagram Fig. 9. Open flaw specimen and the test loading diagram. 8

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(a) 0°

(b) 15°

(c) 30°

(d) 45°

(e) 60°

(f) 75°

(g) 90°

Fig. 11. Principal strain field of crack propagation and crack propagating sketch.

inclination angles were loaded, and the flaw initiation angle values statistically obtained are listed in Table 2. According to the statistical results, as α increases, the crack initiation angle decreases constantly. When α is 75°, the initiation stress of the wing crack is large. Thus, before the emergence of wing cracks, and the failure directly occurs along the coplanar surface of the flaw [Fig. 11(f)]. While at 90°, the preexisting flaw has little effect on the overall failure of the specimen, and the abrupt rupture happens at the edge of the specimen or in the intact rock, rather than around the flaw. Therefore, the initiation angle of the wing crack was not obtained at 75° and 90°.

Table 2 Crack initiation angle at the flaw tip for different flaw inclination angles in the laboratory tests. Flaw inclination angle, α /° Crack initiation angle, θ0 /°

0 126

15 110

30 99

45 89

60 87

75 –

90 –

middle of flaw surface and leading to the occurrence of the crack at the same place. Moreover, the crack initiation angle of 180° in the theoretical solution represents the open flaw surface itself, which is inconsistent with reality and does not exist. Therefore, when α = 0° , the crack initiation angle of the left end of the flaw is taken as the crack initiation angle, which is 126°. After several specimens for different flaw 9

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4.2. PFC numerical simulation Given the influence of various uncertain factors, several shortcomings still exist in the laboratory test results. Numerical simulation software is also used to simulate the test process which can be better verify the initiation angle curve plotted in Fig. 8. In order to verify the crack initiation angle, it is necessary to obtain the crack propagation path without calculating SIF, stress field and other data. Considering the advantages of the discrete element, this section will simulate the test process based on the PFC2D numerical software, which is based on the mesoscopic discrete element theory [27–30]. The mechanical behavior of the mesoscopic particles is considered in constructing the model. The internal fracture of the model occurs after the contact failure between the particles, consequently realizing the macroscopic crack propagation. First, a particle model with the same size as the laboratory specimen was established in the PFC. After generating particles with a given radius, several calculations was performed to ensure a uniform density between the particles. The sidewall was subsequently removed, and the displacement loading was simulated by controlling the top wall. Then, the prefabricated flaw was added by the DFN module of the PFC, the length and thickness of which were also 25 × 1 mm2. Next, based on the sample with a flaw inclination of 30°, the microscopic parameters of particles in PFC were adjusted for many times, so that the experimental results were consistent with the simulation results, and the microscopic parameters of particles in the parallel bond model in Table 3 were determined. The final failure pattern of the PFC simulation at α= 15° and 60° is shown in Fig. 12, and compared with the physical test results, the two phenomena remain highly consistent in small-angle and large-angle specimens. The peak strengths of the specimens for different flaw inclination angles obtained by numerical simulation and laboratory tests are plotted in Fig. 13, where the solid scatter is the peak strength of several specimens in the lab test. The hollow dotted line represents the PFC simulation result. In both phenomena, the crack propagation process, the sample failure mode, and the final peak strength, the simulation is in good agreement with the laboratory test. These characteristics indicated the validity of the simulation results. Similar to the laboratory test, the strength of the specimen reaches its maximum before the secondary cracks occur. For the pre-cracked specimens with different flaw inclination angles, the crack propagation at any time when the wing cracks extend sufficiently and before the peak strength is listed in Fig. 14. The crack propagation path is also drawn by AutoCAD for comparison. The data listed in Table 4 are obtained by counting the initiation angle results of the PFC simulation in Fig. 14. Fig. 8 is redrawn by adding the test and simulation data from Tables 2 and 4, and the initiation angle data without considering the negative Mode I SIF was added, finally Fig. 15 was obtained. Given that failure usually does not happen around the pre-existing flaw for most of

(a) 15°

(b) 60° Fig. 12. Comparison of the failure pattern from the lab test and the simulation when α is 15° and 60°.

Table 3 Microscopic parameters of model particles in PFC simulation. Keyword linear group Deformability

Parallel bond group pb-deformability

Description

Value

emod kratio fric

Effective modulus Normal-to-shear stiffness ratio Friction coefficient

1.5 × 107 2.0 0.6

emod kratio coh fa ten

Effective modulus Normal-to-shear stiffness ratio Cohesion Friction angle Tensile strength

6.25 × 107 2.0 0.71 × 106 55.0 0.48 × 106

gap

Gap interval

3.5 × 10-4

Fig. 13. Peak strength in the lab test and PFC simulation.

the specimens when α is 90°, only the other six angles were compared. Fig. 15 shows that the experimental and simulation results are very close to the derived theoretical curve in the range of 15°–60°, while the difference is obvious for 0° and 75°. Atα = 0°, the crack initiation angle calculated according to Eq. (15) is 180°, which is an extreme case, and this does not actually exist. According to previous experimental results

Bond

10

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Fig. 14. Schematic diagram of the wing crack initiation in the PFC simulation and CAD drawing.

direction of crack is usually perpendicular to the flaw surface. Therefore, strictly speaking, in the case of a small inclination angle, the crack initiation angle is 90°, but the initiation position is not at the tip. However, the theoretical formulas obtained in Chapter 2 and 3 only consider the stress field near the tip, so there are indeed problems with the Eq. (15) for the part of small inclination angle, which need to be revised. But even when the crack angle is relatively small, sometimes cracks extension still start from the tip, as shown in Fig. 16. It can be seen that at initiation moment, the crack still is perpendicular to the flaw surface, and then, after a small extension, it rapidly deflects and continues to extend. Therefore, the initiation angle data in the range of 0–15 degrees in Fig. 16 can be considered as the angle at which the extremely tiny crack perpendicular to the crack surface continues to be

Table 4 Crack initiation angle at the flaw tip for different flaw inclination angles in PFC simulation. Flaw inclination angle, α /° Crack initiation angle, θ0 /°

0 125

15 117

30 88

45 86

60 71

75 45

90 –

[31], for the large flaw inclination angle, crack usually initiates at the tip of open flaw. Whereas when the flaw inclination angle is relatively small (such as 0–15°), the cracks often tend to initiate closed to the middle of the flaw surface. Moreover, there is no external boundary condition on the open flaw surface where the maximum principal stress is parallel to the flaw surface. And at the moment of crack initiation, the 11

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Table 5 Direction of maximum principal stress at the tip of microcrack. The flaw inclination angle/° Direction of maximum principal stress/ °

0 130

5 129

10 129

15 127

20 124

25 120

30 116

theory of Tirosh and Catz [32] also applies to the initiation of rock materials. Hence, the experimental and simulation results are close to the theoretical results. The trend is basically the same, that is, the initiation angles decrease with the increase in the inclination angle of the pre-existing flaws. Accordingly, the correctness of the crack initiation angle formula of the open flaw established in this paper is verified. 5. Conclusions (1) The open and the closed crack significantly differ. Thus, the two cracks should be treated respectively. Under uniaxial compression, the stress field form of the open flaw tip is consistent with that of uniaxial tension, in which σ is substituted by a negative value, − σ . The effect of a negative KI cannot be ignored when the open flaw is not closed under uniaxial compression. (2) Under uniaxial compression, the wing crack first appears at the open flaw tip, and the crack initiation angle is related to the ratio of the flaw tip KII / KI . Theoretical analysis shows that, as the flaw inclination angle increases, the crack initiation angle decreases monotonically and approaches 70.5° gradually. (3) The crack initiation angle at the open flaw tip under compression is obtained statistically using the laboratory test and PFC numerical simulation of the specimen with a single open flaw. The results are in good agreement with the theoretical results as a whole, and yet the theoretical results in the range of small angles need to be revised. Thus, the correctness of the theoretical formula of the crack initiation angle obtained in this paper is verified.

Fig. 15. Comparison of the results of the initiation angle from experiment and simulation and theoretical curve.

deflected by the stress field in Eq. (7). To this end, a simulation was carried out in ANSYS to simulate the uniaxial compression test of the specimen with an opening flaw at a small inclination angle, and a tiny crack perpendicular to the flaw surface was prefabricated at the flaw tip. The maximum principal stress direction of the tiny crack tip (angle with the flaw surface) was recorded and the results are shown in Table 5. It is almost the same as the test result of Fig. 15. Thus, for the error of the Eq. (15) at a small angle, the correction can be made by utilizing the extension direction of the tiny crack perpendicular to the pre-existed flaw. At α=75°, similar to the results of the laboratory experiments, the wing cracks at the crack tip also barely appeared in the numerical simulation. Only a few breaks are found between the particles near the tip without forming a long crack path. The crack angle is approximately 45°, which is consistent with the results of Tirosh and Catz [32]. The curve in Fig. 8 is obtained by not considering the crack thickness, the thickness is 0. However, in the simulation test, the prefabricated flaw has a certain extent of openness. Thus, the obtained initiation angle results are quite different. This finding shows that the

Acknowledgment This paper gets its funding from project (51774322;51774107;51774131) supported by National Natural Science Foundation of China; Project (2018JJ2500) supported by

Fig. 16. Details of crack initiation at the tip of small inclination angle flaw. 12

Theoretical and Applied Fracture Mechanics 104 (2019) 102358

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Hunan Provincial Natural Science Foundation of China. The authors wish to acknowledge these supports. The anonymous reviewer are gratefully acknowledged for his valuable comments on the manuscript.

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Declaration of Competing Interest The authors declare no conflicts of interest.

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Appendix A. Supplementary material

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Supplementary data to this article can be found online at https:// doi.org/10.1016/j.tafmec.2019.102358.

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