A model for the wing crack initiation and propagation of the inclined crack under uniaxial compression

International Journal of Rock Mechanics & Mining Sciences 123 (2019) 104121

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A model for the wing crack initiation and propagation of the inclined crack under uniaxial compression Hongyan Liu a, b, *, Shuran Lv c a

School of Engineering and Technology, China University of Geosciences (Beijing), Beijing, 100083, PR China Key Laboratory of Deep Geodrilling Technology of Ministry of Natural Resources, Beijing, 100083, PR China c School of Management Engineering, Capital University of Economics and Business, Beijing, 100026, PR China b

A R T I C L E I N F O

A B S T R A C T

Keywords: T-stress SIF Fracture criterion Wing crack initiation angle Wing crack propagation path

The wing crack initiation and propagation of the inclined crack under uniaxial compression is the key factor affecting and controlling the cracked rock mass mechanical behavior. Therefore, in order to deeply investigate this issue, the stress intensity factor (SIF) at the crack tip, the wing crack initiation criterion and propagation path are studied with the theoretical method. First of all, the calculation method of the SIF at the crack tip is proposed on basis of analyzing the mechanical behavior of the crack under the compressive-shear stress. And it is found that KII calculated with the proposed method is larger than that obtained with other methods. And then ac­ cording to the analysis of the stress field near the crack tip under the compressive-shear stress, it is assumed that the nonsingular stress (T stress) should be taken into account in the wing crack initiation criterion, and then the revised maximum tangential stress (MTS) criterion by considering three T stress components, e.g. Tx, Ty, Txy is proposed. Thereafter the wing crack initiation angle is calculated with the revised MTS criterion. Finally, on basis of the above research results and the calculation model for the wing crack propagation path proposed by other researchers, a new calculation model for the wing crack propagation path of the crack under the compressiveshear stress is proposed. By comparison between the theoretical results and test ones, it is found that they agree with each other very well.

1. Introduction Through long and complicated geological processes, the rock mass has lots of natural defects such as joints and cracks. Many researches1,2 indicate that the rock mass failure is the outcome of the initiation, propagation and coalescence of these defects under loading. Therefore, it is very necessary to understand the rock mass failure mode and mechanism by studying the mechanical behavior, initiation and propa­ gation mechanism of the crack under loading. Meanwhile lots of researches3–5 show that the failure type of material (i.e. brittle and ductile failure), the type of loading (i.e. tension and compression), and the crack geometry and mechanical parameters all have much effect on the SIF at the crack tip, the wing crack initiation angle and its propa­ gation path. As is known to all, the traditional fracture mechanics mainly focuses on the crack under the tensile-shear stress. However, the rock mass in the practical engineering are often prone to the compressive-shear stress, and therefore how to perfectly study its me­ chanical behavior attracts enormous concern of the researchers.6–8 Up to

now, the study on the mechanical behavior of the crack under the compressive-shear stress mainly focuses on the following three aspects, e.g. the SIF at the crack tip, the wing crack initiation criterion and its propagation path. First of all, the calculation of the SIF KI at the crack tip under the compressive-shear stress should be studied first because there is still much divergence in it. Above all, some researchers9,10 assume that the calculation method of KI at the crack tip under the compressive-shear stress is the same as that under the tensile-shear stress, and then KI < 0 will be obtained by assuming the compressive stress to be nega­ tive with the traditional calculation method of KI. However Li et al.11 assumed that the crack face would close under compression, and therefore KI should be 0 because of the non-penetration of the substance. Meanwhile Zhu12 obtained that the analytical solution of KI was 0 with the complex stress functions. So, up to now the viewpoint of KI ¼ 0 under compression is widely accepted in the rock fracture mechanics circle. Secondly, there is still much divergence in the study on the SIF KII at the crack tip under the compressive-shear stress. Many researchers13,14

* Corresponding author. School of Engineering and Technology, China University of Geosciences (Beijing), Beijing, 100083, PR China. E-mail address: [email protected] (H. Liu). https://doi.org/10.1016/j.ijrmms.2019.104121 Received 29 April 2019; Received in revised form 18 September 2019; Accepted 25 September 2019 Available online 3 October 2019 1365-1609/© 2019 Elsevier Ltd. All rights reserved.

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International Journal of Rock Mechanics and Mining Sciences 123 (2019) 104121

assume that it is proportional to the effective stress on the crack face, which equals to the shear stress component induced by the external stress subtracts the friction stress on the crack face. However Lu15 assumed that KII obtained by other researchers was rather less, and then proposed a new calculation method. So how to accurately calculate KII is still an import issue in the rock fracture mechanics. Secondly, on the wing crack initiation criterion, many conventional criteria such as the MTS criterion,16 the maximum energy release rate criterion,17 and the minmum energy density criterion18 have been pro­ posed in the traditional fracture mechanics. However these traditional fracture criteria only consider the singular stress term of r1/2 in Wil­ liams’ series expansion and often neglect the higher order terms of O (r1/2) and non-singular term (T-stress) in the stress field at the crack tip. Lots of researchers19,20 assumed that it would lead to the disagreement between the theoretical predictions and the test results of the wing crack initiation angle, fracture toughness and so on. So, many researchers begin to consider the effect of T-stress and its corresponding strain on the rock fracture behavior. Li et al.21 studied the mechanical behavior of the crack under compression, and assumed that both Tx stress component parallel with the crack face and Ty stress component perpendicular to the crack face existed near the crack tip. Meanwhile the existence of Tx and Ty will decrease and increase the wing crack initiation angle respec­ tively, and finally increases the rock fracture toughness. In view of the MTS criterion, Tang et al.22 studied the effect of T-stress on the initiation and propagation of the crack under the compressive-shear stress, and assumed that Tx and Ty had much effect on the wing crack initiation angle and rock fracture toughness. Liu23 established the revised MTS criterion by considering T stress, and discussed the effect of the crack geometry, strength and deformation parameters on the wing crack initiation angle. However, Woo and Ling24 assumed that three T-stress components namely Tx, Ty and Txy existed near the crack tip at the same time, and accordingly established the corresponding MTS, minmum energy density and the maximum energy release rate criteria. The wing crack initiation angle obtained with the revised fracture criteria fitted with that from the test. Meanwhile by taking into account the effect of T-strain as well as the singular strain components, Mirsayar et al.25,26 developed an extended version of the maximum tangential strain cri­ terion to predict the crack initiation angle and the onset of fracture in the rock materials subjected to the mixed mode I/II loading. The average strain energy density criterion is also proposed by Aliha et al.27–29 to predict the fracture loads of the tested rock samples under mixed mode I/II conditions. In sum, the previous researches indicate that the wing crack initiation criterion by considering T stress can perfectly predict the wing crack initiation critical stress and initiation angle. Finally, the study on the wing crack propagation path is also an important issue in the fracture mechanics. It is known that the rock will not fail immediately after the initiation of the wing crack because af­ terwards there will be a period of wing crack propagation. The longer the wing crack propagation path is, the more the surface energy required for the newly formed wing crack is, and accordingly the rock strength is larger. Therefore, the wing crack propagation path has much effect on the rock mechanical behavior. So, many researches have been done on it, and however many of them mainly focus on the test and numerical ones.8,30 Then how to establish the universal theoretical prediction models appears to be more important. Sumi31,32 adopted a polynomial function to describe the curved propagation path of the wing crack under uniaxial compression, and utilized the first order perturbation method to determine its coefficients. However, its calculation process is rather complicated and difficult to generalize. Based on the character­ istics that the wing crack will firstly propagate along a curved path and then gradually be parallel to the direction of the loading under uniaxial compression, Li et al.33 set up a hyperbolic equation to describe the curved path of the wing crack. However they assumed that the wing crack initiation angle of the crack under the compressive-shear stress was always 70.5� , which was obviously unreasonable from other re­ searchers’ result.23,24 Afterwards, the difference of the wing crack

initiation angle will finally lead to much error in the wing crack prop­ agation path. Therefore, on basis of analyzing the mechanical behavior of the crack under the compressive-shear stress, the problems in the previous calculation method of the SIF KII at the crack tip is firstly investigated, and then the modification to it is proposed. Secondly, the effect of Tstress on the initiation angle of the wing crack under uniaxial compression is studied on basis of the revised MTS criterion. Finally the wing crack propagation path is calculated, and a comparison between the calculation result and the test one is made. 2. Overview of the previous researches on the wing crack propagation path Many researches34–39 have been conducted on the effect of the specimen geometry and shape and the loading condition on the wing crack propagation path in the cracked rock. Therefore, in order to more clearly compare the difference of the wing crack propagation path among the different specimen geometry and shape, Table 1 illustrates the wing crack propagation path reported in some samples such as Brazilian disc (BD), semi-circular disc (SCB), edge cracked triangular specimen (ECT), four-point bend (FPB), and diagonally loaded square plat (DLSP). It can be found from the sample 3 in Table 1 that with increasing the crack dip angle from 0� to 27� , the fracture mode changes from pure mode I fracture to pure mode II fracture, and accordingly the wing crack propagation path also varies greatly. Meanwhile, in order to avoid the difficulty in manufacturing the inclined pre-crack in the specimen, Ayatollahi et al.34 suggested an asymmetric three-point bend loading method to investigate the mixed mode fracture in brittle mate­ rials, for example the samples 1, 7, 8, 9 and 10. It can be seen that the specimen shape and loading condition (e.g. compression, tension, and asymmetric three-point bend loading) both play an important role in the wing crack propagation path. Meanwhile, they will also affect the stress field near the crack tip such as the stress intensity factors and T-stress, and for simplicity we do not state it here. 3. SIF at the crack tip under compressive-shear stress For the inclined crack in an infinite plate in Fig. 1(a) with length of 2a and dip angle of α, the crack closure and shear slippage will occur under uniaxial compression. The previous studies show that the SIF KI at the crack tip under uniaxial compression is 0, and the formula of the SIF KII at the crack tip is12,22,23 � 0 pffiffiffiffiffi tan α � f KІІ ¼ (1) σ cos αðsin α f cos αÞ πa tan α > f where, f is the friction coefficient on the crack face. It can be seen from Fig. 1 that the rock is in the critical state of sliding along the crack face when the shear stress on the crack face equals the friction on the crack face. For this case, the SIF KII at the crack tip is 0. Because the rock fracture toughness KIIC of type II crack must be a constant larger than 0, from the viewpoint of fracture mechanics, there is KII ≧ KIIC>0 if the crack is in the critical state of propagation. Obvi­ ously, this contradicts Eq. (1), therefore the calculation method of the SIF KII in Eq. (1) is questionable. In order to solve the above problem in Eq. (1), a new calculation method of the SIF KII at the crack tip is proposed and here the crack model in Fig. 1(a) is still taken for an example. Under uniaxial compression, the crack face will firstly close and then slide, and there­ fore the compressive stress and its friction effect on the crack face must be taken into consideration at the same time. So, this model is equivalent to that under the far field compressive and shear stress, and the compressive stress and its friction effect on the crack face, shown in Fig. 1②. The model in Fig. 1① is translated into that in Fig. 1② in x-y coordinate system. According to the transformation of coordinates, the 2

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International Journal of Rock Mechanics and Mining Sciences 123 (2019) 104121

Table 1 The previous studies on the wing crack propagation path for different specimen geometry and shape under loading.

3

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International Journal of Rock Mechanics and Mining Sciences 123 (2019) 104121

Fig. 1. The rock sample with one central inclined crack under uniaxial compression. α is the crack dip angle and 2a is the crack length. ① the model with an inclined crack under uniaxial compression. ② the equivalence one of the model with an inclined crack under uniaxial compression.

far field stress of the model in Fig. 1② can be written as

σ x ¼ σ sin2 α; σy ¼ σα ¼ σ cos2 α; τxy ¼ τα ¼ σ sin α cos α

model in Fig. 1② can be seen as the superposition of that in these three stress conditions shown in Fig. 2, namely KІІ ¼ KІІ1 þ KІІ2 þ KІІ3 . Because the direction of the shear stress is opposite to that of the friction stress, the SIF at the crack tip of the model Fig. 1① can finally be expressed as

(2)

where, σα and τα are the normal and shear stresses on the crack face of the model in Fig. 1① respectively.

pffiffiffiffiffi KІІ ¼ KІІ1 þ KІІ2 þ KІІ3 ¼ 0 þ τxy πa

1 pffiffiffiffiffi 2 πa

Z

rffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffi aþx dx ¼ τxy πa τf a x a

a

pffiffiffiffiffi 0:5τf πa ¼ τxy

As shown in Fig. 1②, the crack will close under the compressiveshear stress, and then the friction stress τf on the crack face can be expressed as τf ¼ f σ n , where σ n ð¼ σy ¼ σα Þ is the normal stress on the crack face. When f � tan α, the rock will not slide along the crack face. The stress field at the crack tip is not singular, therefore KI ¼ KII ¼ 0 and the crack does not initiate and propagate. However, when f < tan α, the rock will have the trend of sliding along the crack face. The singularity of the stress field at the crack tip is controlled by the shear stress, therefore KI ¼ 0, KII>0 and the crack is a special type II one. It is noted that although KII is larger than 0 when σ is small, it is still less than the rock type II fracture toughness KIIC. Therefore, although the rock has the trend of sliding along the crack face, it does not slide and make the wing crack initiate yet. After that, with increasing σ , KII will become larger than KIIC at some moment, where the wing crack will initiate and propagate. According to the linear elastic fracture mechanics, KII of the

0:5τf

�pffiffiffiffiffi πa

(3)

pffiffiffiffiffi It can be seen from Eq. (3) that KII will increase from 0 to 0:5τf πa when τxy on the crack face gradually increases from 0:5τf to τf . Although it is larger than 0 in this process, KII does not reach the rock type II fracture toughness KIIC. Therefore, although the rock has the trend of sliding along the crack face, the slide does not occur yet. However, with increasing σ, the wing crack will begin to initiate when τxy ¼ τf and at pffiffiffiffiffi this moment KII is equal to 0:5τf πa. According to the fracture me­ chanics, at this moment KII exactly equals KIIC. Substituting Eq. (2) into Eq. (1), we obtain KII with the traditional method �pffiffiffiffiffi KІІ ¼ τxy τf π a (4) By comparing Eq. (3) with Eq. (4), the following two differences can

Fig. 2. Solution of the SIF at the crack tip. For the model ① under the far and near field compressive stress, the crack faces will close and KІІ1 ¼ 0 is obtained because pffiffiffiffiffi of the nonpenetration of the substance. For the model ② under the far field shear stress, KІІ2 ¼ τxy πa is obtained. For the model ③ under the symmetrical shear qffiffiffiffiffiffi R a 40 stress, KІІ3 ¼ 2p1ffiπffiaffiffi a τf aþx a xdx is obtained according to the handbook of the SIF. 4

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International Journal of Rock Mechanics and Mining Sciences 123 (2019) 104121

increasing r. Meanwhile many researchers have verified the existence of T stress and its effect on rock fracture by experiments.19,20 Through the study on the crack under compression, Li et al.21 and Tang22 assumed that both Tx stress component parallel to the crack face and Ty stress component perpendicular to the crack face existed near the crack tip. In view of the previous researches, Liu23 proposed the revised MTS crite­ rion by considering Tx and Ty, and studied the effect of crack dip angle, friction coefficient, normal and shear stiffness on the wing crack initi­ ation angle. However, it can be seen from Eq. (5) that the wing crack initiation angle is also affected by KII at the same time. Meanwhile KII obtained with Eq. (1) is less than that with Eq. (3), however KII in the study by Liu23 is still calculated with Eq. (1), which will lead to the decrease in the wing crack initiation angle obtianed the method pro­ posed by Liu.23 However, Woo and Ling24 assumed that T stress near the crack tip included not only Tx and Ty stress components, but also Txy stress component. Therefore, the revised MTS criterion of the wing crack under the compressive-shear stress by considering T stress and KII is proposed here, and then the wing crack initiation angle is calculated. In the polar coordinate system in Fig. 4, the stress field near the crack tip by considering three T stress components namely Tx, Ty and Txy can be expressed as (KI ¼ 0 for the case)43 9 K θ > σrr ¼ pffiІІffiffiffiffiffiffi ð3 cos θ 1Þ⋅sin þ Tx cos2 θ þ Ty sin2 θ þ Txy sinð2θÞ > > > 2 > 2 2πr > > > = 3KІІ θ 2 2 σθθ ¼ pffiffiffiffiffiffiffi sin θ cos þ Tx sin θ þ Ty cos θ Txy sinð2θÞ (6) > 2 2 2πr > > > > > � K θ 1 > τrθ ¼ pffiІІffiffiffiffiffiffi cos ð3 cos θ 1Þ þ Ty Tx sin 2 θ þ Txy cosð2θÞ > ; 2 2 2 2π r

Fig. 3. Variation of the ratio k with the crack dip angle α and friction coeffi­ cient f.

be obtained. Firstly, it can be seen from Eq. (4) that KII is larger than 0 when τxy > τf namely tan α > f satisfies. However, we find from Eq. (3) that KII is larger than 0 when τxy > τf =2 namely tan α > f=2 satisfies. That is to say, KII is already larger than 0 before the wing crack initiates, but it is still less than KIIC, and therefore the wing crack does not initiate and propagate. So the calculation method in Eq. (3) is more reasonable. Second, KII obtained with Eq. (4) is always less than that obtained with Eq. (3), and if let the ratio of the former to the latter is k, the variation of k with the friction coefficient f and crack dip angle α is shown in Fig. 3. It can be seen that k increases with the increase in f when α is the same, and k decreases with the increase in α when f is the same. Therefore, KII obtained with the proposed method is larger especially when the crack dip angle and friction coefficient are little and large respectively, which will finally lead to much difference in the wing crack initiation angle and propagation path.

where, Tx, Ty and Txy are three T stress components, and they can be expressed as Tx ¼ σ sin2 α, Ty ¼ σ cos2 α, Txy ¼ f σ cos2 α for the inclined crack in Fig. 1(a).24 According to Eq. (6), T stress always exists no matter which fracture criterion is adopted, and it will undoubtedly have effect on the wing crack initiation angle, initiation stress and propagation path. Ayatollahi and Aliha44 also pointed out that the singular stress term was much larger than T stress near the crack tip, and T stress could be ignored. However, with increasing the distance from the crack tip, the singular stress term dramatically decreases and accordingly the proportion of T stress in the whole stress field becomes much larger which cannot be ignored any more. Under uniaxial compression in Fig. 1(a), the normal stress σ α and shear stress τα on the crack face with dip angle α are respectively

4. The wing crack initiation criterion 4.1. The revised MTS initiation criterion by considering T stress In order to study the mechanical behavior of the rock sample with a central inclined crack in Fig. 1, the wing crack initiation characteristics such as the initiation criterion and initiation angle of the wing crack under compressive-shear stress should be studied first. As stated above, many criteria have been proposed in the traditional fracture mechanics. For instance, the MTS criterion which is often adopted in the traditional fracture mechanics to calculate the wing crack initiation angle is:6 KІ sin θ þ KІІ ð3 cos θ

1Þ ¼ 0

σ α ¼ σ cos2 α; τα ¼ σ sin α cos α

(7)

According to the statement above, for the crack under uniaxial compression in Fig. 1(a), KI is 0, and KII can be expressed as � 0 pffiffiffiffiffi f � 2 tan α (8) KІІ ¼ ½τα 0:5f σα � πa f < 2 tan α

(5)

According to the viewpoint stated above, KI at the crack tip under the compressive-shear stress should be 0, therefore the wing crack initiation angle θ is always 70.5� from Eq. (5), which is irrelevant of the crack dip angle, friction coefficient on the crack face and so on. It is obviously unreasonable, which is proved by lots of test results.24,41 Through enormous efforts of many researchers, it is found that the primary cause leading to it is no consideration of T stress in the stress field near the crack tip.20,21 In the analyzing the stress field near the crack tip, the previous study only considers the r1/2 singular term in Williams’ se­ ries,42 and the non-singular term namely T stress and higher order terms O(r1/2) are often ignored because they are assumed to have little effect on the stress field near the crack tip. However, up to now it is found that T stress has much effect on the rock fracture with the ongoing research. This is because the higher order terms O(r1/2) can be ignored when r→0, but T stress is constant and does not vary with r. Therefore, T stress has more and more effect on the stress field near the crack tip with

Fig. 4. Stress at the crack tip. σr,σθ, τrθ are the stress components at the point A near the crack tip in the r-θ local polar coordinate system.

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International Journal of Rock Mechanics and Mining Sciences 123 (2019) 104121

With the MTS criterion, substituting Eqs.(7) and (8) into the second term of Eq. (6) yields

ðσ θθ Þmax ¼

MethAcrylate) sheets with cracks of different dip angles tested by an Instron universal testing machine, where rc/a ¼ 0.01, namely η ¼ 0.14 is

8 > <

σ sin2 α sin2 θ þ σ cos2 α cos2 θ

3σ � > : pffiffiffiffiffiffiffiffiffi sin α cos α 2 2r=a

� θ 0:5f cos2 α cos sin θ þ σ sin2 α sin2 θ þ σ cos2 α cos2 θ 2

f σ cos2 α sin 2 θ

(10)

pffiffiffiffiffiffiffiffiffiffiffi Assume η ¼ 2rc =a, and substituting Eq. (9) into Eq. (10) yields. For f � 2 tan α � sin 2 θ cos 2 α þ 2f cos2 α cos 2 θ ¼ 0 (11) cos 2 θ cos 2 α 2f sin 2 θ cos2 α < 0 For f < 2 tan α

3 � sin α cos α 2η 3 � sin α cos α 2η

f σ cos2 α sin 2 θ

f < 2 tan α

(9)

adopted. The variation of the wing crack initiation angle with the crack dip angle is shown in Fig. 5. It can be seen that the wing crack initiation angle θ obtained with the proposed method is not a constant of 70.5� , but varies with the crack dip angle, which fits with the test results by Woo and Chow.45 When the crack dip angle is 0� , the direction of the crack is perpendicular to the loading direction, and then the crack will close under compression. The wing crack perpendicular to the direction of the crack will occur from the crack tip, and finally the split failure happens in the sample. For this case, the wing crack initiation angle is 90� . When the crack dip angle is 90� , the direction of the crack is parallel to the loading direction, and the wing crack parallel to the direction of the crack will occur from the crack tip under compression. For this case, the wing crack initiation angle is 0� , which is logically reasonable. Meanwhile the comparison of the wing crack initiation angle between

The MTS criterion assumes that the wing crack initiation angle θ should satisfy

∂σ θθ ∂2 σθθ <0 ¼ 0; ∂θ ∂θ2

f � 2 tan α

� � 9 � θ 1 θ > sin sin θ þ sin 2 θ cos 2 α þ 2f cos2 α cos 2 θ ¼ 0 > 0:5f cos2 α cos cos θ > = 2 2 2 � � > � 5 θ θ > ; cos sin θ þ sin cos θ 2 cos 2 θ cos 2 α 4f cos2 α sin 2 θ < 0 > 0:5f cos2 α 4 2 2

(12)

the calculation result with the proposed method and one with the method by Woo & Ling24 is made, in which the latter is always less than the former. By comparison with the test result, it can be seen that the error between the calculation result by Woo & Ling24 and the test one is larger when the crack dip angle α is little, and it gradually becomes less

where, rc is the critical radius from the crack tip, which is the material mechanical property.42 It can be seen from Eq. (12) that the wing crack initiation angle θ is related to not only the crack length 2a and crack dip angle α but also the critical radius rc from the crack tip. 4.2. Analysis of the calculation example The test results by Woo and Chow45 are adopted to validate the proposed method. The model of the inclined cracks under compression was carried out using 1/8 inches thick PMMA (PolyMethyl

Fig. 5. Relationship between the wing crack initiation angle θ and the crack dip angle α under uniaxial compression.

Fig. 6. The model for the wing crack propagation path of the closed crack. α is the crack dip angle and θ is the wing crack initiation angle. 6

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International Journal of Rock Mechanics and Mining Sciences 123 (2019) 104121

and less with increasing α. This is because KII is calculated with Eq. (1) in the method proposed by Woo & Ling,24 and however in the proposed method, KII is calculated with Eq. (3). So, it is assumed that the proposed method is more reasonable.

x1 ¼ a cos α; y1 ¼ a sin α

Second, the tangential equation at the wing crack initiation point is y ¼ a sin α þ ðx

5. The wing crack propagation path

y¼

6. Verification of the proposed model Here the tests reported by Lee and Jeon46 are taken as an example to validate the proposed model for the initiation and propagation of the wing crack under uniaxial compression. In the experiment, they pre­ pared the rectangular prismatic specimens of PMMA, 60 mm � 120 mm � 25 mm in size, and one central crack of 1 mm in aperture thickness and 20 mm in length was produced with a water-jet system. The physical and mechanical parameters of PMMA in the test are as follows. Its mass density, elastic modulus, uniaxial compressive strength and Poisson’s ratio are 1.18 g/cm3, 2.9 GPa, 139 MPa and 0.44 respectively, and η ¼ 0.14, namely rc ¼ 0.1 mm is adopted here. The test results of the initiation and propagation of the wing crack with dip angle for α ¼ 0� , 30� , 45� , 60� and 75� are shown in Fig. 7 (a1)~ (a5). It can be seen that the tensile wing cracks firstly initiate from the crack tips, then propagate curvilinear for a short distance, and finally grow parallel to the loading direction. The similar test results are also obtained by many other researchers.8,47 It can be explained from the following two aspects. On one hand, it can be explained from the viewpoint of the rock strength property. As is known to all, rock is a special material with its tensile strength much lower than its compres­ sive strength, which often leads to the rock tensile failure. Therefore, the MTS criterion is taken here as the wing crack initiation criterion. Namely, at the very time of the wing crack initiation, a certain angle exists between the directions of the MTS and original crack. Therefore, curved propagation of the wing crack occurs at the initial stage, and then after a short curve, the wing crack propagates nearly parallel to the compressive loading direction. It indicates that the MTS along the compressive loading direction is the maximum. If the wing crack at the previous moment is assumed to be the original crack, the direction of the original crack is parallel to the compressive loading direction. Because the newly formed wing crack at the next moment still propagates along the direction of the original crack, it is found that the initial angle of the wing crack is 0� , which also indicates the conclusion that the initial angle of the wing crack is always 70.5� is false. On the other hand, from the viewpoint of the least energy consumption principle, the wing crack initiation and propagation will proceed along the direction of the least energy consumption. So, at the moment of the wing crack initiation, the initiation resistance is the minimum along the direction deviating from the strike direction of the crack for a certain degree. Afterwards, the wing crack also propagates along the direction of the least energy con­ sumption too. After a short curvilinear propagation, the wing crack will propagate parallel to the loading direction because of the propagation

(14)

(15)

Therefore the friction coefficient f on the crack face must lie between these two extreme cases. According to the linear interpolation method, when the effective shear stress on the crack face is τ, the asymptotic line equation of the wing crack propagation path is

τmax τ ðxmax τmax τmin

xmin Þ

(16)

When the friction coefficient on the crack face is f, the effective shear stress on the crack face is

τ ¼ σ sin α cos α

f σ cos2 α

(17)

Substituting Eqs.(13)~(15) and (17) into Eq. (16), the asymptotic line equation of the wing crack propagation path yields � x ¼ fa cos2 α sin α (18) Therefore, the wing crack propagation path in Fig. 6 can be described with the following hyperbolic equation ðx

x0 Þðy

y0 Þ ¼ c

af cos α cot αÞ2 tanðθ0 þ αÞ þa cos αð1 f cot αÞtanðθ0 þ αÞ x af cos α cot α (23)

Accordingly, the effective shear stress on the crack face for these two cases is

x¼

ða cos α

þ a sin α

Second, the friction coefficient on the crack face is assumed to be maximum, namely fmax ¼ tan α. For this case, the wing crack will firstly occur from the crack tip and then propagate along the loading direction after a short curvilinear propagation. Its asymptotic line equation is

τmin ¼ 0 ðfmax ¼ tan αÞ or τmax ¼ σ cos α sin αðfmax ¼ 0Þ

(21)

By substituting Eq. (22) into Eq. (19), the equation of the upper wing crack propagation path is obtained

(13)

xmax ¼ acosα

a cos αÞtanðθ þ αÞ

where, θ is the wing crack initiation angle, which can be obtained with the method proposed in this paper. Finally, the asymptotic line equation of the wing crack propagation path is shown in Eq. (18). Combining Eqs. (18), (20)~(21) yields 9 � x0 ¼ fa cos2 α sin α = (22) y0 ¼ y1 þ ðx1 x0 Þtanðθ0 þ αÞ ; 2 c ¼ ðx1 x0 Þ tanðθ0 þ αÞ

The wing crack propagation path under uniaxial compression is shown in Fig. 6, from which it can be seen that the wing crack gradually tends to an asymptotic line parallel with the loading direction after a short curvilinear propagation. Here in view of the idea proposed by Li et al.,33 the wing crack propagation path is re-studied by revising some unreasonable assumptions by them. For example, Li et al.33 assumed that the wing crack initiation angle under uniaxial compression was always 70.5� , which is not reasonable. The difference of the wing crack initiation angle will inevitably lead to much discrepancy in the wing crack propagation path. Therefore, on basis of the research results ob­ tained in this study, a new calculation method of the wing crack prop­ agation path is proposed. Because these two wing cracks up and down the original crack is symmetrical, here only the upper wing crack propagation path is taken for an example. Because the asymptotic line characteristics of the wing crack propagation path under uniaxial compression is related to the crack face coefficient, the following two extreme conditions are firstly studied according to the friction coeffi­ cient on the crack face. First, the friction coefficient on the crack face is assumed to be minimum, namely fmin ¼ 0. For this case, the asymptotic line of the wing crack propagation path under uniaxial compression is one which is parallel to the loading direction and passes through the center of the crack. Its equation is: xmin ¼ 0

(20)

(19)

where, x0, y0, c are the constants to determine, and they can be solved with the following known conditions. First of all, the wing crack initiation point is the end point (x1, y1) of the crack 7

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International Journal of Rock Mechanics and Mining Sciences 123 (2019) 104121

Fig. 7. Comparison of the wing crack propagation path between the calculation results and test ones. (a1)~ (a5) are the test results of the wing crack propagation path in PMMA. Here T denotes the tensile wing crack, and its subscript number “1” and “2” denotes the order of initiation. Two T1s in (a4) and (a5) mean that these two wing cracks initiate simultaneously. (b1)~ (b5) are the comparison of the wing crack propagation path between the theoretical results and test ones. (c1)~ (c5) are the comparison of the local magnification of the wing crack propagation path between the theoretical results and test ones.

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International Journal of Rock Mechanics and Mining Sciences 123 (2019) 104121

also should be noted that there is much error between the theoretical result and test one for the specimen in Fig. 7(b1). Meanwhile the similar test in Fig. 8 by Sun et al.48 shows that the occurrence of the wing crack in the specimen with crack dip angle 0� is at random for some degree. However in most cases, the vertical wing crack will often initiate from the crack tip and propagate. Therefore, the theoretical result in Fig. 7 (b1) is reasonable. As is known to all, rc has some effect on the calculation results, therefore the above specimen with crack dip angle α ¼ 45� is taken as an example to study its effect on the initiation angle and propagation path of the wing crack. Here rc ¼ 0.05 mm, 0.1 mm and 0.2 mm namely η ¼ 0.1, 0.14 and 0.2 are also adopted respectively, and the corre­ sponding calculation results are shown in Fig. 9. It can be seen from that the wing crack initiation angle will change from 71.9� to 72.5� and 73.4� with rc increasing from 0.05 mm to 0.1 mm and 0.2 mm respectively. And accordingly the wing crack path will be also different, but their difference is not much. It indicates that the effect of the critical radius rc from the crack tip on the initiation angle and propagation path of the wing crack is little for the specimen with crack dip angle α ¼ 45� . 7. Discussions and conclusions

Fig. 8. Model fracture of physical test sample.48 The rock-like material is made of cement type C425, washed sand and water; the volume blending ratio is cement/sand/water ¼ 2:4:1. The rectangular specimens with a size of 100 mm � 50 mm � 50 mm (height � width � thickness) and the crack dimen­ sion of 25 mm � 1 mm � 50 mm (height � width � thickness). The crack is created manually by inserting a predesigned acrylic sheet into the mold before pouring concrete. The uniaxial compression test is conducted by a WEP-600 hydraulic screen display universal testing machine.

One of the works of this paper is to take into account the effect of the nonsingular stress (T stress) in the Williams’ series solution for the stress near the crack tip on the wing crack initiation and propagation, and accordingly the revised MTS criterion by considering three T stress components, e.g. Tx, Ty, Txy, is proposed. Afterwards, the wing crack initiation angle and propagation path is then calculated on basis of the revised MTS criterion. However, many researchers49–54 make much deeper studies on this issue and assume that it is not enough to consider only the singular and nonsingular stress terms in the Williams’ series expansion. Akbardoost et al.49 assumed that the second and third terms in the Williams’ series expansion for the tangential stress near the crack tip should be considered in order to predict the size-dependent fracture resistance of Guiting limestone under mixed-mode loading. Saghafi et al.50 adopted a semi-circular specimen containing a vertical edge crack subjected to three-point bending to study its mixed mode fracture toughness of the marble rock, and found that the first three terms of tangential stress infinite series near the crack tip should be used to satisfy this condition. Meanwhile Nasiraldin Mirlohi and Aliha51 took into account the first six terms of the Williams’ series expansion, and observed very good agreements between the experimental and theo­ retical initial initiation angle and propagation path of the wing crack in the angled crack problem. For the estimation of crack propagation di­ � and Veselý,53 and Wei et al.54 assumed that at rection, Lucie,52 Malíkova least three or four higher-order terms of the Williams’ expansion should be used for the description of the stress field near the crack tip, especially for small-size specimens and when the criterion is applied at a larger distance from the crack tip. Therefore it can be seen that in some certain cases, the higher terms of the Williams’ series expansion have some effect on the initial initiation angle of the wing crack and its propagation path. However, from Figs. 5 and 7, it can be found that good agreement of the wing crack angle and its propagation path between the theoretical predictions and experimental results is achieved, and then the effect of the higher terms of the Williams’ series expansion on the theoretical predictions is not discussed here. But it is a significant issue to study in the fracture mechanics, and we will study it in the future. In all, this study proposes a calculation method of the SIF at the crack tip by considering the crack mechanical behavior under the compressive-shear stress. By comparision with the result obtained with the traditional method, the SIF obtianed with the proposed method is more reasonable. Meanwhile, in view of the deficiency of the traditional MTS criterion for the wing crack initiation, the revised MTS criterion is proposed by taking into consideration three T stress components near the crack tip on basis of analyzing the mechanical behavior of the crack under the

Fig. 9. Effect of rc on the wing crack initiation angle and propagation path for the specimen with crack dip angle α ¼ 45� .

resistance along this direction is the minimum. It can also be seen from Fig. 7(b2)~(b5) that the theoretical results of the wing crack propagation path obtained with the proposed method fits with the test ones very well. It indicates that the proposed theoretical model for the wing crack initiation angle and propagation path is reasonable. However, from Fig. 7(c2)~(c5), we can also find that there is still some error between the theoretical results and test ones. It indicates that although the hyperbola model can simulate the wing crack propa­ gation path on the whole, and the improvements should also be made in the selection of the hyperbola function and its parameters. However, it 9

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International Journal of Rock Mechanics and Mining Sciences 123 (2019) 104121

compression. The calculation result indicates that the wing crack initi­ ation angle obtained with the revised MTS criterion agrees with the test result much better. Finally, a new model for the wing crack propagation path of the inclined crack under uniaxial compression is proposed by incorporating the calculation methods of the SIF at the crack tip and the wing crack initiation angle. The calculation example shows that the wing crack propagation path obtained with the proposed method fits with the test one very well. Overall, the proposed model provides a new way to simulate the wing crack initiation and propagation of the inclined crack under uniaxial compression.

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