Underlying fracture trends and triggering on Mode-II crack branching and kinking for quasi-brittle solids

Accepted Manuscript Underlying fracture trends and triggering on Mode-II crack branching and kinking for quasi-brittle solids J. Li, Y.J. Xie, X.Y. Zheng, Y.M. Cai PII: DOI: Reference:

S0013-7944(18)31361-4 https://doi.org/10.1016/j.engfracmech.2019.02.036 EFM 6377

To appear in:

Engineering Fracture Mechanics

Received Date: Revised Date: Accepted Date:

6 December 2018 11 February 2019 26 February 2019

Please cite this article as: Li, J., Xie, Y.J., Zheng, X.Y., Cai, Y.M., Underlying fracture trends and triggering on Mode-II crack branching and kinking for quasi-brittle solids, Engineering Fracture Mechanics (2019), doi: https:// doi.org/10.1016/j.engfracmech.2019.02.036

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Underlying fracture trends and triggering on Mode-II crack branching and kinking for quasi-brittle solids J. Li a, b, Y. J. Xie a 1, X. Y. Zheng a, Y. M. Cai a a b

Department of Mechanical Engineering, Liaoning Shihua University, Fushun, 113001, LN, P. R. China

College of Pipeline and Civil Engineering, China University of Petroleum (East China), Qingdao, 266000, SD, P. R. China Abstract: A geometrical model on multiple cracks initiation from boundary with singular stress fields has been

proposed for the fracture analysis of Mode-II crack. The conservation law has been explored utilizing a partial integral path, from which an analytical solution of the energy release rate for multiple cracks initiation from a crack tip has been established. Some underlying fracture trends of cracks initiation from a crack tip, including crack side-branching and crack kinking, for quasi-static Mode-II deformation have been theoretically investigated, which implies actually some degree of instability for Mode-II fracture. The K-based criterions and fracture toughness for Mode-II crack side-branching and kinking have been defined. The fracture configurations of crack side-branching and kinking under Mode-II loading and corresponding fracture toughness predicted by the present modelling agree well with the experimental observations on the rock fracture reported in the literatures. Keywords: Multiple crack initiation; Mode-II crack; Crack-branching; Crack kinking. Nomenclature 

angle of the boundary translation

c

critical cracking angle for crack kinking and branching for Mode-II crack

i

intersections of the θ-axis and integrand curve of normalized J  -integral

n

normalized phase angle

 ij

stress components



Poisson’s ratio

ei

boundary shifting tensor

E

elastic modus

F, f, 

interim parameters

energy release rate for boundary cracking

G

Gsideb max

(G ) k

GC

energy-based driving force for crack side-branching

energy-based driving force for crack kinking

max

energy-based fracture toughness

Gside-b , Gk energy release rates for crack branching and kinking

G



wing side-b max



ext , Gside -b



max

energy-based driving forces for wing crack and crack extension respectively when Mode-II

crack side-branching 0 0Gside -b , Gside-b energy release rates for Mode-II crack side-branching with a weak Mode-I loading disturbance

Gk

1

energy release rates for Mode-II crack kinking with a weak Mode-I loading disturbance

Corresponding authors. Y.J. Xie, E-mail address: [email protected]. 1

G 

 k max

G

energy-based driving force for Mode-II crack kinking with a weak Mode-I loading disturbance



0side-b max

0 , Gside -b max energy-based driving force for Mode-II crack side-branching with a weak Mode-I loading

disturbance conservation integrals

Ji

K I , K II

stress intensity factors for the Mode-I and Mode-II loading

K IC , K IIC K-based Mode-I and Mode-II fracture toughness

K IIC k , K IIC sideb

K 

wing

IIC side-b

 

, K IIC

fracture toughness for Mode-II crack kinking and side-branching respectively

ext side-b

K-based Mode-II fracture toughness for wing crack and crack extension respectively when

Mode-II crack side-branching P

load

s, si boundaries or integration paths around the crack tip

sin l

integral path within solids

Ti

stress vector acting on the integration path

ui

displacement components

vi

directions of local boundary si shifting

w

strain energy density

1. Introduction Crack branching from a single well-defined crack tip is interesting, and it occurs not only from a Mode-I crack tip but also from a mixed Mode-I/II crack tips. A good understanding of the crack branching conditions is important for the fundamental study on complex failure mechanisms of brittle materials, such as rocks and rock-like materials, which is also essential for prevention of geotechnical and mining failures. A pure Mode-I crack and Mode-II crack can be considered as two extreme cases for a mixed Mode-I/II crack. Although tensile fractures are frequently observed in the geomechanical applications, shearing is the most common mode of failure in rocks under the compressive stress states, and the failure of rock bridges between two adjacent discontinuities in rock masses mostly develops as shear fracture [1]. Therefore, shearing is another important failure mechanism in rock engineering, especially in the macroscopic sense. The crack branching under Mode-II loading is a typical and important failure behaviour in rock engineering, such as the earthquake analysis [2,3], which shows the impotence to insight into the branching and failure mechanism of Mode-II crack in engineering. However, with few exceptions, most of the previous theoretical studies on crack branching are focused on the mechanisms of dynamic Mode-I crack branching based on the experimental observations. In contrast to Mode-I crack branching, Mode-II crack branching has not been investigated to any depth in the literatures. The reason for that may be a well spread opinion that Mode-II crack branching is seldom seen or no fracture criterion could be used to interpret the Mode-II crack branching. Because of this common understanding, the possibility of the other peaks of the strain energy release for Mode-II crack branching at a certain angle has hardly attracted any attention. Consequently, some experimental phenomenon on multiple cracks originated from Mode-II crack tip, as reported in the literatures [4-6], have not been fully understood and even handled as an abnormal data. One cannot predict the mixed Mode-I/II crack branching, kinking and propagation path without having the knowledge of the fracture criterion. The study of crack initiation angle is also as much important in dealing with 2

arresting the crack [7]. There are some theories so far for evaluating the direction of crack initiation and fracture toughness for mixed Mode-I/II crack problems [8]. The maximum tangential stress (MTS) criterion [9] (Erdogan and Sih), the minimum strain energy density (SED) criterion [10] (Sih) and the maximum energy release rate (G) criterion [11] (Hussain et al) are some of the well-known mixed mode fracture criteria. Additionally, there are some other methods such as [12] can be used also to predict the crack kinking angle. In order to get better theoretical predictions on the fracture behaviors of quasi-brittle materials, such as the rocks and ceramics, some modified forms of above criterions, such as MTS criterion [13,14], the maximum tangential strain (MTSN) [15] based on KI, KII and T-stress and the average strain energy density (ASED) criterion over a control volume [16,17], had been suggested and used to predict the fracture behavior of rock materials concerning the fracture initiation angles, mixed Mode fracture toughness and critical fracture load. It is worth noting that a good agreement had been found between the experimentally obtained fracture behavior of the rock-liker materials and the theoretical predictions in the entire range of Mode-I/II mixities [18-22]. Additionally, some researches indicated that geometry and size of the specimens effect substantially on the fracture trajectory in a limestone rock under mixed Mode loading [23]. The positive/negative T-stress that exists in different types of specimens may be responsible for different fracture initiation angles under the same mixed Mode-I/II loading [19-23]. It was shown experimentally that the fracture toughness values of the tested rock material obtained from different test specimens are not consistent. Depending on the geometry and loading type of the specimen, noticeable discrepancies can be observed for the fracture toughness of a same rock material. The difference between the experimental mode I fracture resistance results is related to the magnitude and sign of T-stress that is dependent on the geometry and loading configuration of the specimen [22, 24]. The instability of fracture behaviors has already received much attention. All the methods mentioned above work well only for predicting one crack initiation from a crack tip, which is not applicable to analyze the multiple cracks initiation, such as the case of crack-branching under quasi-static loads and dynamic loads. An energy-based modelling was suggested by Xie et al.[25-27] to capture the essential physical processes of multiple cracks initiation from a crack tip under special Mode loading, i.e., pure Mode-I loading and mixed Mode-I/II loading with iso-stress intensity factors KI and KII . Some possible fracture trends and the inherent instability of fracture had been revealed. In present article, based on this fracture modelling, crack branching, kinking and the instability on cracks initiation from a Mode-II crack tip are investigated in detail. The possible critical crack initiation angles and associated critical energy release rates are found. The concept and definition of the Mode-II crack-branching toughness and energy-based crack-branching driving force are proposed. During the course of the experiment, it is very difficult to maintain a pure Mode-II loading. The Mode-II fracture in vast majority of experiments is actually a special mixed case of dominant Mode-II and weak Mode-I loading, which means a slight perturbation to Mode-II loading. In this case, by introducing a mixity parameter  n , the fracture mechanism of the perturbation to the fracture configuration can be functionally captured. The concerned underlying fracture behaviors on crack-branching and kinking for dominantly Mode-II case are theoretically predicted based on the energy-based modelling. 2. Modelling of multiple cracks initiation and energy release rate From the geometrical point of view, multiple cracks initiation from a boundary with singular stress can be understood and regarded as the multiple sub boundaries shift in different directions. Then a geometrical model for multiple fractures initiation in n directions from a boundary s with s  s1  s2      sn  0 is given in Fig.1 in 3

the case of n  3 and s  s1  s2  s3  0 . The energy release rate for multiple cracks initiation can be given by [25-35] n   G  lim  wei ni ds    lim  wei ni ds  , s 0 s  0   l 1 s s  

(1)

l

l

where e1  cos  l and e2  sin  l for the two-dimensional solids, the  l is an angle between shifting direction vl of boundary sl and x1 as shown in Fig.1. From the conservation law [28, 30], Eq.(1) can be written as n





G   J1 sl  |sl 0 cos  l  J 2 sl  |sl 0 sin  l ,

(2)

l 1

where J j sl  |s 0  lim  wn j ds  l

sl 0

sl

 wn j  Tiui, j ds , j=1, 2,

(3)

sin l

here, sin l , (l=1, 2…n), is an integration path within the area closed by the solids boundary, and sl  sin l , (l=1, 2…n), forms a closed path. The J j sl  |s 0 in Eq.(3) denotes the energy release rate of the boundary sl shifting l

in x j -direction or the driving force for crack initiation at the boundary sl in x j-direction when the limits taken exist.

S v1

S

s 0

v2 v3

s1 s

s2

s3

(a) Multiple-notch modelling for multiple boundaries si shifting.

(b) Modelling for multiple cracks initiation.

Fig.1. Modelling of multiple cracks initiation from a boundary with singular stress (in case of n=3) when s  0 for two-dimensional solids. The asymptotic singular stress fields exist not only in crack tip region, but also in the edge of the Mode-I indentation [36-38]. Above analysis defines actually a universal modelling used to analyse such as crack tip fracture or the boundary fracture of Mode-I indentation. It is more interested and powerful to capture the physical processes on the multiple cracks initiation. It should be pointed out that the boundary shift or fracture must be inwards to the solids and J j sl  |s 0 must be positive, which means the energy release otherwise means the energy l

absorbs [25-27, 38]. The boundary fracture inducing energy absorbs is impossible for an actual state of the solids under critical loading. Eq.(2) must include all the contributions of the positive energy release for each sl , (l=1, 2…n), shifting. 3. J i -integral for Mode-II crack 3.1 Basic characteristics on the J i -integral around mixed-Mode-I/II crack tip

4

An enlarged and U-turn-shaped crack tip with the limit ro  0 and a circular arc path s12 are shown in Fig.2. Let sl  s34 in Eq.(3), a portion of the crack tip boundary, and sin l  s12  s23  s41 within the K-dominant region. From Eq.(3), the energy-based driving forces in x j -directions can be given by J j sl  | s 0  lim  wn j ds  l

sl  0

sl

s12

 wn j  Ti ui, j ds s s 23

r  ro



41

{

 wn j  Ti ui, j ds

(4)

s12

}

For any integral path sl = s ro , θ ∈[θ0 , θ ] as shown in Fig. 2, substituting the stress and displacement within the K-dominant region into Eq.(4), the J i -integral over the path sl can be found as J1 sl  

1 2 2E

 K I 1  cos 2   K II 1  cos 2   2K I K II sin 2 d

(5)

  K I sin 2  K II sin 2  2K I K II 1  cos 2 d .

(6)

KI , K II

(7)



2

2

0

and

1 2 2E By introducing J 2 sl  

 2

 n  arctan



2

2

0

where  n is a normalized phase angle and also a mixity parameter varying from 0.0 (for pure Mode-II) to 1.0 (for pure Mode-I), which depicts the relative strengths of K I and K II similar to the parameter M e proposed by Shih in [39], then the Eqs. (5) and (6) can be rewritten as J1 sl  

and J 2 sl  

1  μ K 2

2 I

E

1   K 2

 K II2





 cos 0

2 I

E

 K II2



2

( 

 2

 n )d

(8)



 cos  sin(  n )d .

(9)

0

x2 2

r n

θ 1

sl

3

ro

4

θo

x1

o o

a

Crack tip

Fig.2. The local boundary of the crack tip and ro→0+. The integrand of the normalized J 1 -integral, i.e., cos 2  - n / 2  0 for -     , indicates that J1 -integral is always positive as shown in Fig.3, which means the energy release when any part of boundary sl around crack tip shift in x1 -direction. The J 1 -integral has a substantial contribution to the energy release rate for multiple cracks initiation from a crack tip. Unlike the J 1 -integral over any part of the crack tip boundary, the integrand of the normalized J 2 -integral shows a different and significant behaviour as shown Fig.4. 5

1.0

Inte gra

Mode-II φn=0.0 0.1 0.2 0.3

0.8

nd of J1π

0.6

0.4 0.5 0.6

E/( 1-μ 2

0.4

0.7 0.8 0.9 φn=1.0 Mode-I

)(K 2

I

+

2 0.2

KII )

0.0

-180.

-150.

-120.

-90.

-30.0

-60.

0.0

60.0

30.

90.0

120.

150.

180.0

Polar coordinate θ deg. Fig.3. Distribution of the integrand of normalized J 1 -integral along the circle path. 0.

Inte gra nd of J2π E/( 1-μ 2 )(K 2 + I

Mode-I φn=1.0

0.

Mode-II φn=0.0

0.9

0.1

0.8 0.

0.2 0.3 0.4 0.5

0.7 0.6 0.5

-0.

2

KII )

-0.

-0.

-1. -180.0

-150.0

-120.0

-90.0

-60.0

-30.0

0.0

30.0

60.0

90.0

120.0

150.0

Polar coordinate θ deg.

Fig.4. Distribution of the integrand of normalized J 2 -integral along the circle path. 3.2 J 2 -integral around Mode-II crack tip When n  0.0 , which defines actually a special case of pure Mode-II deformation as shown in Fig.4, the integrand of the normalized J 2 -integral cos  sin  0 for 0     2 and -    -  2 ; cos  sin   0 for  2     and -  2    0 . These characters determine that the whole boundary of a crack tip should be divided into four parts, i.e.,

s1  s ro , ∈0,  2 ,

s2 = s ro , θ ∈[π 2 , π ] , s3  s ro , ∈  ,  2 and

{

}

s4  s ro , ∈  2 ,0 as shown in Fig.5. The normalized J i -integrals over the paths s1，s2，s3 and s4 are given

in Table 1, which imply different physical meanings. For the upper half crack tip boundaries s1 and s2 ,

6

180.0

J 2 (s1 ) > 0 and J 2 (s2 ) < 0 indicate theoretically that if the boundary s1 moves in x2-direction, the cracked

solids will release the strain energy; if the s2 moves in the x2-direction, the cracked solids will absorb or increase energy, which is actually impossible to occur naturally. So, the contribution of J 2 s2  for Mode-II crack should be excluded in the analysis of the crack tip fracture. For the lower half crack tip boundaries s3 and s4 , when s3 moves in the opposite x2-direction, i.e., fracture is inward to the solids, the energy release rate - J 2 s3   0 , which means that the fracture will lead to energy-absorbing and is not going to happen; when s4 moves in the opposite x2-direction, - J 2 s4   0 implies that the fracture of boundary s4 inward to the solids is possible. Then the underlying moving trends of boundaries s1 , s2 , s3 and s4 can be schematically shown by red arrows in Fig.5. Table 1. Normalized Ji -integrals over the paths s1 , s2 , s3 and s4 . s1  s ro , ∈0,  2

{

s2  s ro , ∈ 2 ,  

}

s3 = s ro , θ ∈[- π, - π 2]

s4  s ro , ∈-  2 ,0

2J1 si E /(1   2 ) K II2

π/2

π/2

2J 2 si E /(1   2 ) K II2

1

-1

P s2 s1 P

P s4 s3 P

Fig.5. The interval partitioning around crack tip and the potential moving trend. 3.3 On the quasi-Mode-II crack It is nearly impossible to maintain an accurate experimental condition of Mode-II loading. The relationship of

n  0.0 , which may stand for a case of Mode-II ( n  0.0 ) loading with a slight disturbance of Mode-I loading during the experiment process as shown in Fig.6, depicts actually a Mode-mixity of weak Mode-I and Mode-II. In this case, Eq.(9) indicates that the four θ-axis intersection points can be found from integrand cos  sin( - n )  0 for a given small value of  n , i.e.,

1，2  

 2

，  3，4  k  n ，k=0, -1.

(10)

These intersection points should divide the interval [-π, 0] for lower crack surface or [0, π] for upper crack surface into three subintervals as shown in Fig6. Generally, over the three subintervals, the J 2 -integrals are either 7

positive or negative as indicated in Fig.4, which can be used to identify how many cracks should initiate from a weak mixed-Mode crack tip and to analyse also the instability and underlying fracture trends. The J i -integrals along the three subintervals of upper or lower crack surface are given in the Tables 2 and 3. The underlying moving trends for all of the local boundaries around a crack tip are shown by red arrows in Fig.6 similar to case of pure Mode-II loading.

s2

s2

s6

s1 πφn

s1

s5

πφn s6

s4

s3

s4

s3

(a) πφn→0.0+

s5

(b) πφn→0.0-

Fig.6. The interval partitioning around crack tip and the potential moving trend.

Table 2. Normalized Ji -integrals over the paths around crack tip with n  0.0 . s1  s ro , ∈n ,  2 ;

s3  s ro , ∈   n ,   2



2EJ1 si  (1   2 ) K I2  K II2



  n 2   cos n    n  sin n

2πEJ 2 (si ) (1 - μ 2 )(K I2 + K II2 )

2



s2  s ro , ∈ 2 ,  

s4  s ro , ∈-  2 ,0

  sin n 2  cos n 

 2

sin n

s5  s ro , ∈0, n ;

s6  s ro , ∈  ,  n 

n  sin n n sin n

Table 3. Normalized Ji -integrals over the paths around crack tip with n  0.0- . s2  s ro , ∈ 2 ,  - n 

{ };  s r , ∈  ,   2

s1 = s ro , θ ∈[0, π 2]

s3



2EJ1 si  (1   2 ) K I2  K II2



2πEJ 2 (si ) (1 - μ 2 )(K I2 + K II2 )

o

 2

 sin n

cos n 

 2

sin n

s5  s ro , ∈n， 0;

s4  s ro , ∈-  2 , n 

s6  s ro , ∈ - n， 

  n 2

-n - sin n

   cos n   n   sin n 2 

n sin n

4. Underlying fracture trends for pure Mode-II crack As discussed in section 3, the diversity of potential moving or fracture trends for all parts of the crack tip boundary constitutes the complicated fracture configurations, which is interesting and quite helpful for understanding the fracture behaviours in engineering practice. Actually, the crack extension, crack kinking and branching can be considered as that some fractures initiation from a Mode-II crack tip boundary are triggered. 8

For a pure Mode-II crack, there are three underlying fracture configurations, i.e., the crack side-branching ( -shaped fracture shown in Fig.7) and the crack kinking ( -shaped fracture shown in Fig.9) and crack extension as shown in Fig.11. 4.1. Energy-based driving force and fracture angle for Mode-II crack side-branching Crack side-branching ( -shaped fracture) is schematically shown in Fig.7. Although J 2 s1   0 means the energy release when the s1 moving in x2-direction, the fracture of boundary s1 is, however, hardly to be triggered because the compressive and shear stress state and geometrical restraint. The possible fractures initiation from a Mode-II crack tip may be form when s1 , s2 moving in x1-direction and the s4 moving inward to solids as a wing crack as shown in Fig.8. In this case, the energy release rate responsible for Mode-II crack side-branching can be found as following from Eq. (2) and above discussions. Gside-b = J1 (s1 )cos α1 + J1 (s2 )cos α2 + J1 (s4 )cos α4 + J 2 (s4 )sin α4 .

(11)

Substitute the results in Table 1 into Eq. (11), the following result can be found.

(1 - μ )K 2

Gside-b =

4E

2 II

{cos α + cos α 1

2

}

+ 1 + 4 / π 2 cos (α4 + β ) ,

(12)

where β = tan -1 (2 / π ) = 32.48 deg.

(13)

The energy release rate is now maximised with respect to i by setting

Gsideb   , i=1, 2 and 4, i.e.,  i

sin1c  0 sin  c   and sin 4c      , which give the critical crack initiation angles

c   deg.,  c   deg. and  4c  -  -. deg.,

(14)

where 1c ,  2 c and α4 c are the theoretical expected crack initiation angles at which Gsideb exhibits a maximum value, and explicitly this is given by

Gsideb max  1  

2

K

2 II

Fside b ,

(15)

2  1  4 / 2  0.7964 , 4

(16)

E

where Fsideb 

which is the energy-based driving force for Mode-II crack side-branching as shown in Figs.7 and 8. It indicates that when Gsideb max  Gc the Mode-II crack will fracture theoretically in the form of crack side-branching.

9

P x2 Mode-II crack P

P

x1 a

a

P Fig.7. Illustration for enlarged crack tip side-branching.

Original crack tip

x2 s1+s2→0

ro

α4c

x1

s4→0 Fig.8. Enlarged geometrical modelling for crack side-branching. P x2 Mode-II crack P

P

x1 a

a

P Fig.9. Illustration for enlarged Mode-II crack kinking.

Original crack tip

x2

ro

α4c

x1

s4→0 Fig.10. Enlarged geometrical modelling for Mode-II crack kinking. 10

4.2. Energy-based driving force and fracture angle for Mode-II crack kinking From the geometrical point of view, crack kinking from a Mode-II crack tip means that the fracture of the local boundary s4 is triggered as shown in Figs.9 and 10. Then the energy release rate responsible for the crack kinking can be found as Gk = J 1 (s4 )cos α4 + J 2 (s4 )sin α4

(1 - μ )K 2

=

. 1+ 4/ π2 cos (α4 + β ) 4

2 II

E

(17)

From above equation, the maximum energy release rate, i.e., the critical energy release rate or energy-based driving force and critical kinking angle become

(1 - μ )K 2

2 II

(G )

=

Fk =

1+ 4 / π2 = 0.2964 4

k max

E

Fk ,

(18)

where (19)

and α4 c = - β = -32.48 deg.

(20)

4.3. On Mode-II crack extension The Mode-II Crack extension is schematically shown in Fig.11. This kind of fracture configuration should be formed if the moving of the boundaries s1 , s2 , s3 and s4 in x1-direction is triggered. Then from Eq.(2), the energy release rate responsible for this case can be found as Gext = J1 (s1 + s2 )cos α1-2 + J1 (s3 + s4 )cos α3-4 .

(21)

Substitute the results in Table 1 into Eq.(21), the following result can be found.

(1 - μ )K 2

Gext =

2 II

{cos α

1- 2

2E

+ cos α3- 4 },

(22)

It is clearly that the maximum energy release rate for crack extension can be given by

(G )

ext max

(1 - μ )K 2

=

E

2 II

,

(23)

where the critical crack initiation angles α1-2c = 0 and α3-4 c = 0 , which means actually that the crack initiating along the axis x1-direction from a Mode-II crack tip will remain the Mode-II and the energy release rate is identical with the classical Rice’s J-integral.

11

P x2 Mode-II crack P

P

x1 a

a

P Fig.11. Illustration for crack extension. 5. Mode-II crack branching and kinking with n  0.0 When the condition of pure Mode-II loading is slightly disturbed, the normalized phase angle  n will deviate from 0.0, i.e., n  0.0 , which define actually a mixed-Mode fracture problem with weak Mode-I and Mode-II. As discussed in section 3.3, the underlying fracture configurations include also the crack side-branching and kinking. 5.1. Mode-II crack side-branching with n  0.0 When n  0.0 as shown in Fig.6(a), the energy-based driving force for boundary s6 fracture is a small quantity as shown in Table 2, therefore it is difficult to trigger boundary s6 to fracture as a wing crack although boundary s6 demonstrates a fracture trend inward to solids. The fracture behaviours induced by boundary s6 cracking is not an essential issue for Mode-II crack fracture. As shown in Figs. 6(a) and 7，if the cracks initiation from the boundaries si (i=1, 2, 4, 5) are triggered, the -shaped crack branching should be formed under weak mixed-Mode-I/II loading. From Eq. (2) and Table 2 the energy release rate can be written as 0   Gside - b  J  ( s 4 ) cos    J 2 ( s 4 ) sin   



   K 

 I

E

 K II

F

0 4

∑J (s ) cos 

i ,  ,5



cos(     40  ) 

i

 i

∑J (si ) cos  i

, 0.0   n  0.5

(24)

i , ,5

where

     F40    sin n    cos n  sin n  , 0.0  n  0.5 , 2 2    

(25)

2 cos n   sin n ， 0.0   n  0.5 .   2 sin n

(26)

2

2

and

 40  arctan

By setting

0 Gside b  0 , i=1, 2, 4, 5, the critical crack initiation angles can be found as  i

0  4c  - 40 ,  ic    ( i=1, 2, 5 ), 0.0   n  0.5 .

12

(27)

The maximum energy release rate becomes

G 

0 side-b max



1 -  K 2

F40   , 0.0   n  0.5 .  1  cos n 

2 II

E

(28)

When n  0.0- , as shown in Fig.6(b), which implies a case of mixed Mode-I/II (compression-shear) loading and is the most common mode of failure in rock engineering, from Eq. (2) and Table 3 the energy release rate for -shaped crack branching can be written as 00Gside - b  J  ( s 4 ) cos    J 2 ( s 4 ) sin  4 



   K 

 I

E

 K II

F

04

∑J (s ) cos 

 i , ,5， 6

cos( -   40- ) 

i

 i

∑J (si ) cos  i

, -0.5   n  0.0

(29)

i , ,5,6

where 2

      F40-    n   cos n   n   sin n  , - 0.5  n  0.0 , 2 2     2

(30)

and

 40-

By setting

  cos n   n   sin n 2   arctan ， - 0.5   n  0.0 .   n 2

(31)

0Gside b  0 , i=1, 2, 4, 5, 6 the critical cracking angles can be found as  i

 c  -40- ,  ic   ( i=1, 2, 5, 6 ), -0.5   n  0.0 .

(32)

The maximum energy release rate becomes

G



0side -b max



1 -  K 

 II

E

F40-    n  sin n , -0.5   n  0.0 .    cos n 

(33)

The normalized energy-based driving force and the wing crack initiation angle with a slight disturbance of Mode-I loading to Mode-II crack side-branching are shown respectively in Figs.12 and 13. 2.5

2πE (Gsi de-b) / max (12 μ) K

2.0

1.5

1.0

2 II

0.5

0.0 -0.4

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

Normalized phase angle φn Fig. 12. The normalized energy-based driving force for Mode-II crack side-branching.

13

90.0

Crit ical fra ctu re an gle – 

0 4c

80.0 70.0 60.0 50.0 40.0

β=32.48deg.

30.0 20.0 10.0

de g.

0.0 -0.4

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

Normalized phase angle φn Fig.13. Critical fracture angle. 5.2. Mode-II crack kinking with n  0.0 As the fracture of boundary s4 is triggered as a form of crack kinking as shown in Fig.9, i.e.,

-shaped fracture,

from Eq.(2) the energy release rate under n  0.0 can be written as Gk  J (s4 ) cos 40  J 2 (s4 ) sin40 .

(34)

Substituting the J i s j  -integrals in tables 2 and 3 into above equation, we have Gk 

( -   ) K II F cos  0    , E 1  cos n





(35)

where 0  F4 , 0.0   n  0.5 F40   0   F4 , - 0.5   n  0.0

(36)

and     ,    ,

   

0.0   n  . - 0.5   n  .

.

(37)

From Eq.(35), it is not difficult to get the critical crack kinking angles

 40c  - 4 ,

(38)

0 where  4c is the theoretical predicted crack kinking angle under Mode-II loading with a weak Mode-I loading

disturbance and can be schematically shown in Fig.13, at which Gk0 exhibits a maximum value, and explicitly this is given by

G 

 k max

1 -  K 2



E

2 II

F4 .  1  cos n 

(39)

5.3. On the disturbance of Mode-III loading The effect of Mode-III loading is an important issue that may affect the fracture behavior of quasi-brittle solids as discussed in [40-46], which may naturally interfere the experiment process of pure Mode-II loading in a way. Similar to the case of above discussions, the fracture behaviors under mixed Mode-II/III (dominant Mode-II and

14

weak Mode-III) can be formulized by using the same method. However, it may be a secondary issue after all next to the mixed Mode-I/II particularly in the experiment process. 6. K-based fracture criterions for crack kinking and side-branching 6.1. K-based fracture criterion for crack kinking For a homogenous and isotropic brittle solid Griffith’s criterion [47] states that a crack will propagate when the critical value Gc of the solid is equal to the driving force or applied Gmax [47, 48] Gmax  Gc .

(40)

For Mode-I crack extension, when Gmax  Gc , we have

1   K 2

Gc 

2 IC

E

.

(41)

When the energy-based driving force for Model-II crack kinking reaches the critical condition, i.e., Gk max  Gc , from Eq.(18) we get

(1 - μ )(K )

2

2

IIC k

E

Fk = Gc ,

(42)

where K IIC k is K-based crack kinking toughness. Then from Eqs.(19), (41) and (42), the relationship between the Mode-I crack extension toughness K IC and the kinking toughness (K IIC )k can be found as

K 

IIC k

 f k K IC ,

(43)

where the f k is a Mode-II crack kinking factor, and is given by 1

fk =

FK

=

2 = 1.84 . (1 + 4 / π 2 )1 / 4

(44)

A K-based critical condition for Mode-II crack kinking, which defines a relationship between the applied stress intensity factor and fracture toughness, can be found as K II = (K IIC )k = f k K IC .

(45)

6.2. K-based fracture criterion for crack side-branching When fracture configuration of crack side-branching is triggered, i.e., two cracks initiate simultaneously from boundaries s1  s2 and s4 respectively as discussed in section 4.1, the energy-based driving forces for a wink crack initiation and crack extension can be found as =

(G )

=

wing side-b max

(1 - μ )K 2

(G )

2 II

E

1+ 4 / π2 4

(46)

and ext side-b max

(1 - μ )K 2

E

2 II

1 . 2

(47)

From Eqs. (40), (41), (46) and (47), it is not difficult to find the K-based fracture toughness

(K )

= 1.84 K IC

(K )

= 1.41K IC

wing

IIC side- b

(48)

and ext

IIC side- b

(49)

Then from Eq.(15), the critical energy release rate can be found as 15

1 -  K 

G  side-b

where

2 IIC side-b

2

 GC side-b  max

E

Fside-b ,

(50)

ext K IIC side-b  maxK IIC wing , K IIC side-b  1.84K IC . side -b

(51)

wing ) (Gsideext -b )max and Gside-b max can be schematically shown in Fig.14. The relationship between (Gside -b max ,

The K-based criterion for Model-II crack side-branching can be written as





side b

Normalized energy-based drive force

K II  K IIC

.

(52)

(K ) GmaxE/(1-μ2)KIC2

IIC

E / (1 - μ 2 )K IC2 side-b

(G )

side-b max

E / (1 - μ 2 )K IC2

Energy release rate for

(G )

E / (1 - μ 2 )K IC2

(G )

E / (1 - μ 2 )K IC2

ext side-b max

creating a new surface GC E / (1 - μ 2 )K IC2

wing side-b max

(K )

ext

IIC side- b

K IC = 1.41

(K )

IIC side-b

K IC = 1.84

Stress intensity factor K II K IC wing ) (Gsideext -b )max and (Gside-b )max . Fig.14. The relationship between (Gside - b max ,

7. Effects of small parameter  n on fracture toughness When the normalized phase angle  n is a nonzero value and n  0.0 , the fracture configuration would be disturbed as shown in Fig.6, which has actually an impact on fracture toughness and is helpful for better understanding the fracture behaviours in the experiments and engineering. In this case, according to the Grifith’s criterion the critical fracture condition for crack kinking can be given by

1 -  K 

2

2

IIC k

E

F40  Gc  1  cos n 

(53)

from Eqs. (39) and (41). Then the normalized K-based fracture toughness for mode-II crack kinking expressed as

K 

IIC k

K IC

 1  cos n   0  F4 

12

  ,  

(54)

which can be schematically shown in Fig.15.

16

3.0 2.5

(KIIC 2.0 )k/K

1.84

1.5

IC

1.0 0.5 0.0 -0.4

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

Normalized phase angle φn Fig.15. The effects of disturbance on fracture toughness.

Nor mal ize d str ess inte nsit y fact or KII/ KIC

2.5

2.0

Theoretical (KIIC)side-b/ KIC =1.84

1.5

1.41

1.

0.5

0.0 0.0

t0 Time t Fig. 16. Impulse Mode-II loading.

8. Triggering for Model-II crack branching As discussed in above sections, the Mode-II crack side-branching exists theoretically based on the energy-based fracture analysis, which is one of the underlying fracture configurations. Because of the quantitative difference wing ) (Gsideext -b )max , only one crack will initiate from crack tip as a form of crack kinking or main between (Gside -b max and

crack extension under quasi-static mode-II loading. Therefore, the Mode-II crack side-branching is rare in the quasi-static fracture tests. However, if increasing loading rate or even impulse Mode-II loading, as shown in Fig. ext ) (Gsidewing-b )max driving the boundaries s1 + s2 and s4 to fracture respectively may reach their 16, the (Gside - b max and

critical values simultaneously or in a enough short time period. The mode-II crack side-branching should be triggered. 9. Typical experimental examples for Model-II crack branching and kinking 9.1 Mode-II crack side-branching As pointed out in section 8, impulse loading may be an available method to trigger the Mode-II crack side-branching. By using the experimental method of dynamic punch-through shear, Yao et al. [49] had proposed a quasi-impulse loading technique to quantify the dynamic Mode-II fracture toughness of brittle solids. A 50 mm 17

diameter and 30 mm length core specimen with 10 mm depth circular notches at both ends, experiment principle and a specimen holder are shown in Fig.17. A split Hopkinson pressure bar (SHPB) system was utilized to exert the impulse load to the specimen, which was detailed in the literature [49]. Fangshan marble from Fangshan region of Beijing, China is a fine-grained homogeneous marble and had been used to conduct the experiments. The microscopic studies have been performed to measure its mineralogical composition and grain size [49-51]. The Fangshan marble consists of dolomite (~98%) and quartz (~2%) and the size of minerals is from 10 to 200 μm with the average dolomite size of 100 μm and the average quartz size of 200 μm. The physical and mechanical properties are listed in Table 4. (a)

(c)

P1

P2/2 (b)

P2/2

(d)

Fig. 17. The (a) side and (b) top of a typical virgin specimen, (c) the dimension of the punch-through shear specimen, (d) schematics of the specimen holder in dynamic tests with SHPB [49]. The experimental results indicate that the fracture configuration triggered in specimen as shown in Fig.18 is actually the so-called crack side-branching. The wing crack initiation angle is very close to the value α4c = -32.48 degree predicted by present analysis. Therefore, the dynamic Mode-II fracture toughness tested by Yao’s method is obviously related to the (K IIC )side-b . From Eq.(51) and Table 4, the Mode-II crack side-branching toughness

(K )

IIC side-b

= 1.84 K IC = 2.76 MPa·m1/2, which is very closes to the experimental value as shown in Fig.19 under the

loading rate K II = 71.7 GPa·m1/2/s. Fig.20 [49] shows that the fracture configuration on the Mode-II crack side-branching isn't very sensitive to loading rate, which shows a good experimental repeatability and is important both for theory and applications. Some similar fracture behaviours can be found also in the literature [6].

18

α4c

α4c

Fig. 18. The CT image of a typical tested rock specimen under Mode-II loading [49], i.e. the so called the punch-through shear test.

Str es s int en sit y fac tor (M Pa ·m 1/2 )

Theoretical (KIIC)side-b=2.76 MPa·m 1/2 Experimental KIIC

Impulse loading

Time·(μs) Fig. 19. Typical SIF-time curve [49] with pules loading and quasi-pules loading.

Table 4. Summary of physical and mechanical properties of the Fangshan marble (average value) [49] Material Property Density Young’s modulus Poisson’s ratio

Symbol ρ E μ 19

Unit

Value 3

Kg/m GPa

2850 85 0.3

Uniaxial compressive strength Tensile strength Mode-I fracture toughness

σc σt KIC

MPa MPa MPa·m1/2

155 9.5 1.5

Fig. 20. The CT images of specimens under different loading rates [49]. (a) 25 GPa·m1/2/s; (b) 43 GPa·m1/2/s; (c) 76 GPa·m1/2/s; (d) 96 GPa·m1/2/s. It is worth noting that crack branching is a highly localized event and the relative size ratio of notch-width/grain-size will substantially influence the strength and fracture toughness of ceramics and rock [52, 53]. Although present work is limited to continuum mechanics similar to the classical theories [9-11], i.e., the model material is homogeneous and elastic, the proposed theory will, however, be approximately effective if the materials, such as some kinds of rocks and ceramics [27], have finite anisotropy and heterogeneity. 9.2 On Model-II crack kinking A series of fracture toughness tests have been conducted by Aliha et al [20, 54] on a type of coarse grain marble obtained from Harsin (in Kermanshah province, west of Iran). The tests were carried out either in pure Mode-I or in pure Mode-II conditions by using the cracked chevron notched Brazilian disc (CCNBD) specimen as shown in Fig.21. The CCNBD specimen is a circular disc of radius R and thickness B. Two chevron notches are cut at the center of disc from each side by using a rotary diamond saw. Pure Mode-I and pure Mode-II can be easily achieved in the CCNBD specimen by changing the direction of crack line relative to the applied diametric compressive load P. A total number of 44 CCNBD specimens were tested, half in pure Mode-I and the rest in pure Mode-II, to obtain reliable values for Mode-I and Mode-II fracture toughness K IC and K IIC of the tested marble. Crack growth will start from the tip of the chevron notch and stably continue along the crack line until a critical crack length a, from which unstable brittle fracture initiates as a form of crack kinking. The length a can be estimated by averaging a0 and a1 shown in Fig. 21. The values of Mode-I and Mode-II fracture toughness obtained from the experiments are shown in Fig.22. Similar to other brittle materials, there is a natural scatter in both Mode-I and Mode-II test results. However, it is seen in Fig.22 that the scatter in the test results obtained for Mode-II fracture toughness is more than that for Mode-I fracture toughness [20, 54]. 20

The average results for Mode-I and Mode-II fracture toughness of coarse grain marble were found to be 1.12 MPa·m1/2 and 2.25 MPa·m1/2, respectively. The average value of K IIC is about twice the average value of K IC ( K IIC/K IC =2.01). The Mode-II fracture toughness K IIC in [20, 54] is actually the (K IIC )k in present article as shown in Figs.9, 10 and 23(b). Now the fracture toughness ratio K IIC/K IC can be determined from present method, which gives K IIC/K IC =1.84 from Eq.(43). It is seen that the ratio K IIC/K IC predicted in this work agree well with the experimental value K IIC/K IC =2.01. Because of the impact of the Mode-II loading disturbance as shown in Fig.15, this experimental result is reasonable and effective based on present theory.

Additionally, by using the generalized MTS criterion, which takes into account the effects of a non-singular stress term called T-stress in addition to the conventional singular stresses around the crack tip, Aliha suggested K IIC/K IC =1.89 [20, 54], which is close to present value K IIC/K IC =1.84 obviously. As shown in Fig. 23, the fracture path is along the initial crack for Mode-I, whereas its path deviates from the direction of the initial crack for Mode-II specimens. The fracture initiation angle along the mid-thickness plane was measured for all specimen. The average value of measured fracture angles was about -42 deg. This is in much better agreement with the fracture angle α4c = -32.48 deg. given by Eq.(20) than the angle -70.5 deg. suggested by the conventional MTS criterion [9, 54]. Similarly, Fig.13 shows also certain reasonableness for this experimental result if the effect of Mode-II loading disturbance is considered.

P

A

a1 a a0

α

φ 2R

B 2a R

Rsaw O1 Section A-A

A P Fig.21. Cracked chevron notched Brazilian disc (CCNBD) specimen subjected to diametral compressive force P.

21

KIC , KII C

(M Pa ·m 1/2 )

Specimen number Fig.22. Mode-I and Mode-II fracture toughness of tested CCNBD specimens made of coarse grain marble [20].

(a) Mode-I

(b) Mode-II

Fig. 23. Fractured samples of CCNBD specimen made of marble in (a) pure Mode-I; (b) pure Mode-II [20, 54]. 10. Conclusions By further exploring the J i -integral over the local boundary of a Mode-II crack tip, an energy-based fracture modelling for multiple fractures initiation from a crack tip has been investigated and formulized. Unlike the traditional applications of the J-integral around a crack-tip, a new partial integral path and the associated calculation method on the energy release rate for multiple fractures initiation had been defined. The present analysis validates theoretically the underlying fracture trends and possibility for the Mode-II crack side-branching and kinking based on the energy-based method. The fracture mechanism for a Mode-II crack side-branching can be better explained from the present modelling, which is of great significance to understand the complex fracture phenomena of brittle materials and engineering applications such as hydraulic fracturing with multiple cracks. The crack side-branching and kinking toughness have been defined. The fracture initiation angles, the corresponding critical energy release rates and fracture criterions for Mode-II crack side-branching and kinking have been found. The fracture toughness predicted by the present modelling for Mode-II crack branching and kinking agree well with the experimental data available in the literatures. 22

The fracture nature of the experimental results can be easily recognized by using present analysis. By comparing the experimental fracture configurations and the theoretical critical one, which form of the fracture, such as crack side-branching and kinking, and fracture mechanism behind experimental phenomena can be identified. Then the corresponding energy-based driving force, fracture toughness and K-based criterion can be clearly known. Additionally, the present analysis shows a relativity between the dynamic fracture behaviours and underlying fracture trends for Mode-II crack. Moreover, if considering the contribution of mode III loading to the energy release rate, the present method can be extended to predict multiple-fractures initiation from the mixed Mode II/III and I/II/III crack tips. Additionally, the present method can be approximately used also to predict fractures initiation from a narrow U-shaped or V-shaped notch tip similar to some investigations [55-58], which is actually an approximate model in this case. The modelling as shown in Fig.6(b) can be used to analyse the crack branching and kinking for cracked rocks under compressive and shear loading. The concerned investigations will be given in the upcoming articles. Acknowledgements This work was supported by National Natural Science Foundation of China (Grant Nos.: 50771052, 50971068 and 11272141) and Natural Science Foundation of Liaoning (Grant Nos.: LS2010100 and 20102129). References [1]. Rao Q, Sun Z, Stephansson O, Li C. Stillborg B. Shear fracture (Mode-II) of brittle rock. Int J Rock Mech Min Sci. 2003; 40:355–75. [2]. Meng L, Ampuero J-P, Stock J, Duputel Z, Luo Y, and Tsai V C. Earthquake in a Maze: Compressional Rupture Branching During the 2012 Mw 8.6 Sumatra Earthquake. Science. 2012; 337:724-726. [3]. Solveig M. When does a crack grow under Mode-II conditions? Int J Fract. 1986; 30:103-114. [4]. Muhammad Aslam Md Yusof, Nur Adilla Mahadzi. Development of mathematical model for hydraulic fracturing design. Journal of Petroleum Exploration and Production Technology. 2015; 5: 269-276. [5]. Meng C F, Maerten F and Pollard DD. Modelling mixed-mode fracture propagation in isotropic elastic three dimensional solid. Int J Fract. 2013; 179:45–57. [6]. Lukic B. and Forquin P. Experimental characterization of the punch through shear strength of an ultra-high performance concrete. International Journal of Impact Engineering. 2016; 91: 34–45. [7]. Khan S M A, Khraisheh M K. Analysis of mixed mode crack initiation angles under various loading conditions, Engng Fract Mech. 2000; 67: 397-419. [8]. Aliha M R M, Ayatollahi M R. Analysis of fracture initiation angle in some cracked ceramics using the generalized maximum tangential stress criterion. Int. J. Solids Struct. 2012; 49: 1877–1883. [9]. Erdogan F, Sih G C. On the crack extension in plates under plane loading and transverse shear. J. Basic Engng. Trans. ASME. 1963; 85: 519–525. [10]. Sih G C. Strain-energy-density factor applied to mixed mode crack problems. Int. J. Fract. 1974; 10: 305–321. [11]. Hussain M A, Pu SL, Underwood J. Strain energy release rate for a crack under combined mode I and Mode II. Fracture Analysis, 1974. ASTM STP 560. American Society for Testing and Materials, Philadelphia, pp. 2–28. [12]. Cotterell B, Rice J R. Slightly curved or kinked cracks, Int. J. Fract. 1980; 16: 155-169. [13]. Akbardoost J, Ayatollahi M R, Aliha M R M, Pavier M J, Smith D J. Size-dependent fracture behavior of Guiting limestone under mixed mode loading, Int. J. Rock. Mech. Min. Sci. 2014; 71: 369–380. 23

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Highlights An energy-based modelling for multiple cracks initiation from a crack tip is suggested. Underlying fracture trends of cracks initiation from a Mode-II crack tip is revealed. Mode-II crack side-branching for rock can be triggered by using impulse loading.

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