10th Asian Regional Conference of IAEG (2015)

Analyzing the Differences of Initiation Strength Between Embedded Crack and
Penetrating Crack
Han BAO(1,2), Faquan WU(1), Jinyuan CHANG(3), Ning LIANG(1,2)(1) Key Laboratory of Shale Gas and Geoengineering, Institute of Geology and Geophysics,
Chinese Academy of Sciences, Beijing 100029, China
E-mail: baohangeo@163.com(2) University of Chinese Academy of Sciences, Beijing 100049, China(3) College of Civil Engineering, Shaoxing University, Shaoxing Zhejiang, 312000, China
Abstract
Based on the maximum energy release rate criterion of fracture mechanics, we deduced
and compared the initiation strength expressions about embedded crack and penetratingcrack under three-dimensional condition. Results showed that the initiation strength of both
two types of cracks could be described by equal-slope linear function with confiningpressure as independent variable, and the initiation strength of embedded crack was larger
than that of penetrating crack. The relative differences of initiation strength between the two
types of cracks usually should not be ignored, they were related with confining pressure and

w(  w is the
4 2friction angle of crack), the differences would be minimum. And the differences tended to
be stable when the radius exceeded a certain value. Generally speaking, the differences of
initiation strength would be large for good quality rock or stable crack.

the parameters of rock and crack. When the dip angle of crack was+

Keywords: embedded crack, penetrating crack, initiation strength1.

also be observed in some experiments (Wong et al.,2006). Many factors might have effects on the crack
initiation strength, including crack attitude (Gong etal., 2005), crack length, dip angle, sample size (Wonget al., 2002), seepage pressure (Li et al., 2012),
microstructure (Hatzor et al., 1997), rock type,mineral particle size, cementing quality andschistosity (Zhang et al., 2011). Many studies also
showed that crack initiation strength of rock mass
was much lower than peak strength undercompressive stress (Nicksiar and Martin, 2014; Cai etal., 2004). Thus, it is of great significance to takecrack initiation strength into account for the stabilityof engineering.
In fact, several types of cracks exist inengineering rock mass, in which embedded andpenetrating cracks are two common types. Embedded
cracks are widespread in rock mass, while penetrating

Introduction

Cracks (joints) have remarkable effects on themechanical properties of rock mass. Many studies
have been done on the mechanical behavior of cracks,
in which crack initiation strength is always a hot spot.
When studying crack propagation mechanism ofbrittle materials, three important breakage initiation
criteria are usually used, i.e., the maximum tangential
stress criterion (σ-criterion), the maximum energy
release rate criterion (G-criterion) and the minimum
energy density criterion (S-criterion). Cracks havemultiple propagation modes under compressive stress.
In general, only mode I and mode II cracks (i.e.,tensile cracks and shear cracks) would be observed
originating from the original tips of pre-existing
cracks (Lajtai, 1974; Bobet and Einstein, 1998).
However, mode III cracks (i.e., tear cracks) could110th Asian Regional Conference of IAEG (2015)

cracks are often found in bedded rock mass, highdegree jointed rock mass, and shallow free faces ofslopes, tunnels and underground caverns. Though alarge amount of studies have been done on the crackmechanical behaviors (Haeri et al., 2013; Wong et al.,2002), we have rarely discussed embedded crack andpenetrating crack together, their differences on
initiation strength have also been rarely studied.
Sometimes, we even didn’t make a distinction
between embedded crack and penetrating crack instudies or applications, such as when generalizing
three-dimensional situations into two-dimensionalmodels. Due to the significant influences of initiation
strength on engineering stability, it is worthy to study
the differences of initiation strength between
embedded crack and penetrating crack.
Therefore, on the basis of the maximum energy
release rate criterion (G-criterion), we deduced
expressions of crack initiation strength undercomposite breakage. Depending on the deduced
expressions, we then analyzed the differences of
initiation strength between embedded crack andpenetrating crack and their influencing factors as well.
Our study was supposed to make a positive
contribution to the cognition of mechanical behaviors
of different types of cracks.

a. Embedded crack modelb. Penetrating crack model
Fig.1

Rock mass models with single crack

Thus, the normal direction cosine of the cracks
could be obtained as follows:

n1   cos  sin 

2. Conceptual models of cracked rock massn2  sin  sin (1)n3  cos 

In three-dimensional space, we establishedconceptual models of rock mass with embedded crack(Fig. 1a) and penetrating crack (Fig. 1b) as follows:
In three-dimensional Cartesian coordinate system,
the negative direction of X axes points to 0° and thepositive direction of Y axes points to 90°. Maximum
principal stress ( 1 ), medium principal stress (  2 )We define the surplus shear stress on the cracks as
 . By assuming  as the angle between X axes and , and  as the angle between X axes and edge
point of cracks (Fig. 2 ), the value of  can becalculated according to Eq. (2) .

and minimum principal stress (  3 ) are in the
directions of Z axes, X axes and Y axes, respectively.
It is noted that we set compressive stress to bepositive.  ,  ,  w , cw are the dip direction
angle, dip angle, fraction angle and cohesion of asingle crack, respectively. 0 , c0 and  areinternal friction angle, cohesion and Poisson’s ratio ofrock, respectively. The radius of embedded
penny-shape crack is a . The length of penetratingcrack is 2a (we treated a as the radius ofpenetrating crack), the crack tends to the same
direction of crack length, B is the penetrating
thickness of crack in rock mass.

Fig.2

t  s0 2

Shear stress direction on crack

ts
ts

(2)

10th Asian Regional Conference of IAEG (2015)

where t is the shear stress on the cracks and s is
the shear strength on the cracks. t and s can becalculated as follows:t  112 n32   222 n12  332 n22  (11n32   22 n12  33n22 )2s  (1n32  2n12  3n22 )tanw

can get the strain energy release rate of crack under
three failure modes based on G-criterion. The threestrain energy release rate of embedded crack arelisted in Eq. (6) (Wu, 1993).

(3)

1  v2K e 2E(1  v 2 )(2  v)
Ge 
K e 22E
(1  v)(2  v)
Ge 
K e 2
2 E (1  v)
Ge 

3. Initiation strength criteria of the two types ofcracks
3.1 Deduction of initiation strength criteria
Three modes of stress intensity factor (SIF) ofembedded crack and penetrating crack can beobtained from reference (Li et al., 2010) under
three-dimensional stress condition showed in Fig.1.
Noting that the mode I crack is tensile cracks whichreflects the reaction of tensile stress, a parameter k
is introduced to reflect stress state on the normal
direction of crack. If the stress is tensile stress, k  1 ,or k  0 .
The SIFs of embedded crack are shown as
follows:

K e 
K e Thus, the strain energy release rate of composite
mode crack is
G  GIe  GIIe  GIIIe

1  v2 2 2  v 22v
2
K e K 
[ K e e]E
22(1  v)2

(7)

The three strain energy release rate of penetratingcrack are
1  2 2
K Ip
E
1  2 2
GIIp 
K IIp
E
1  2
GIIIp K IIIp
E
GIp 

2



k n a4 a(2  v) K e  (6)cos(   )4(1  v) a(2  v) 

(4)

(8)

sin(   )Then the strain energy release rate of composite mode
crack can be written as

where  n is the normal stress of crack. The
G  GIp  GIIp  GIIIp

conceptual model with penetrating crack can begeneralized to a two-dimension model, so the SIF of
penetrating crack are expressed as Eq. (5) under thecondition of plane strain.

(9)G-criterion states that if energy release rate isequal to consumed surface energy, the system will bein a critical state, i.e., G reaches the critical value ofK p  k n  a
K p   a  a1  2 2 1  2 2 1   2
K Ip 
K IIp K IIIp
E
E
E(5)

GJc , and the crack starts propagation. Assuming theK p   B  aconsumed surface energy of composite breakage isequal to mode I propagation, namely new equalsuperficial area, then we can get Eq. (10).

Where  a and  B are components of  along
length and penetrating direction of crack,
respectively.
Assuming the rock is linear elasticity material, we

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10th Asian Regional Conference of IAEG (2015)

G  Gc 

1  2 2K cE(k n2   a2 For embedded crack, Eq. (10) can be derived asK 2c  K 2e 

2v 2
2v
2
K e K e
2
2(1  v)22 2 4
( k 
 ) a  K 2c
2v


(13)

 B2
) a  K 2c
1 

(14)

Assuming rock mass is under compressive stress with(11)

a constant confining pressure, we can have  2   3
and k  0 . For penetrating crack,  a  

Substituting Eq. (4) into Eq. (11), we can have
2
n

1
2
K 
p
1 v

K 2c  K 2p  K 2p 

(10)

and

 B  0 . Substituting Eq. (1), Eq. (2) and Eq. (3) into
(12)

Eq. (12) and Eq. (14), we can get the crack initiation
strength criteria of embedded crack and penetratingcrack as follows.

For penetrating crack, Eq. (13) and Eq. (14) are
deduced with the same method mentioned above.
For embedded crack:sin(2    w )  sin  w
 1e 3 
sin(2    w )  sin  w

 (2  )) cos  w
8asin(2   w )  sin  w

2(c w  K c

(15)

For penetrating crack:1
2(c w  K c
) cos  w
sin(2    w )  sin  w
a
1 p 3 
sin(2    w )  sin  w
sin(2    w )  sin  w

Eq. (15) and Eq. (16) indicate that crack initiation
strength is not only related to dip angle, length,
friction angle and cohesion of crack, but also related
to mode I fracture toughness and Poisson’s ratio ofrock. Additionally, the crack initiation strength is also
positively related to confining pressure and varies
with dip angle. Through derivation we can get theminimum values of Eq. (15) and Eq. (16) both when




4

+

w
2

(16)

Taking the deep buried marble at Jinping IIhydropower station as original sample, Xiao et al.
(2012) made the physical model samples withstandard size of  50 mm  100 mm using highstrength silica powder mortar, in which two groups ofmodel samples had single pre-crack with 13mm
length and 19mm length, respectively, and the dip
angles of the pre-cracks were all 60°.
Table 1 shows the related indices about marble
and physical model. Two groups of samples werecompressed to failure under confining pressure of 7,
14 and 21 MPa, and the values of peak strength wererecorded. We used Eq. (16) to calculate the initiation
strength and then compared it to the peak strength. If the crack is continuous, there is no

fracture mechanical effect on the crack, then K c  0 .
Thus Eq. (15) and Eq. (16) can be transformed into
Mohr-Coulomb criterion.

(Table 2). The value of Kc was calculated following

3.2 Verification of initiation strength criteria
We verified the reasonability of initiation strength
criteria using a series of physical model tests withpenetrating crack conducted by Xiao et al. (2012).

Zhang (2002), i.e., K c   t / 6.88 .

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10th Asian Regional Conference of IAEG (2015)

Table 1 Physical-mechanical indices of silicon powder mortar model samples and marble original sample(Data source: Xiao et al. (2012) )Material

ρ/(g·cm-3)

E/GPa

ν

σc/MPaσt/MPac0/MPa

φ0 /°

φw /°

Physical model

2.207

14.19

0.12

70.12

2.16

20.73

35.0

30

Marble

2.700

25.20

0.14

199.20

5.57

61.13

35.8

--

*ρ is material density; E is elasticity modulus; ν is Poisson’s ratio; σc is uniaxial compressive strength; σt is uniaxial tensile strength;
c0 is cohesion of material; φ0 is internal friction angle of material; φw is friction angle of crack.

Table 2 Strength analysis of physical models
Samples

Group1

Group2Cracklength/mm

Peakstrength/MPa

Confiningpressure/MPa

Initiation
strength/MPa

Initiation strength/peakstrength

13

89.50

7

28.61

0.319

13

99.86

14

49.61

0.497

13

114.58

21

70.61

0.616

39

53.50

7

25.39

0.475

39

82.08

14

46.39

0.565

39

109.59

21

67.39

0.615

Results show that the initiation strength increases
with the increase of confining pressure but decreases
with the increase of crack length. And the ratio of
initiation strength to peak strength increases withboth confining pressure and crack length, with values
ranging from 0.32 to 0.62 (Table 2). The ratio values
obtained in Liu et al. (2012) and Martin and Chandler(1994) for marble at Jinping II hydropower station
and brittle rock, respectively, are around 0.4 to 0.5,
which indicate that our results are similar to theexisting studies. However, the ratio values of rockmass with macrocrack have larger range than that of
intact rock. The reason is that macrocrack is easier toproduce stress concentration than microcrack so thatshows smaller values in low confining pressure. But
when the confining pressure increasing, the structure
control become weak and the strength of cracked rockmass increases so rapidly that make the ratio values
R 

 1e
 1
1 p

K c (

increase too. Additionally, the physical model
material may convert from brittleness to ductility
under the condition of high confining pressure, while
the criteria derived in this paper cannot be applied toductile material, so we can get higher ratio values inhigh confining pressure.
4. Comparison of the initiation strength between
embedded crack and penetrating crack

We define R as the initiation strength ratio
between embedded crack and penetrating crack toexpress the relative differences. Thus, R can bederived form Eq. (15) and Eq. (16) as: (2  )
8a



1
)
a

(17)
1
(sin  cos   sin  tan  w ) 3  cw  K camechanics properties, geometry and mechanics of
crack and confining pressure. In order to figure out
Obviously, the initiation strength of embedded
crack is higher than that of penetrating crack, so
the influences of factors on R ，we adopt singleR  1 , where R is related with rock massfactor analysis method to study the change of25

10th Asian Regional Conference of IAEG (2015)

R

under

different

confiningpressure.

radius is over some certain limit which is not affectedThe

by confining pressure, R tends to be stable. The

parameters of rock are based on the parameters of
marble in Table 1 and the parameters of Jinping
marble joints (cracks) are based on Chen et al. (2008),

certain value in this case is about 0.5 m. As to K c ,in which radius a  0.382 m, dip angle   40 ,cohesioncw  0.07 MPa,andfrictionR is positively correlated with K c and increases

rapidly with the increase of K c ( Fig. 3e). In Fig. 3c,angle w  30.44 . The results are showed in Fig. 3.

R decreases firstly and then increases with the

The variation ranges of factors in Fig. 3 are consistent
with the principle of rock mass mechanics and canincrease of dip angle, with the minimum value atreflect the real varying patterns of R . Results show


4

+

w
2

. Fig. 3 indicates that confining pressure

has significant effects on R , and R decreases
that R is in negative but weak correlation with

gradually and tends to be 1 with the increase ofconfining pressure.
Based on the discussion above, we can conclude
that there are differences between the initiation
strength of embedded crack and penetrating crack, thedifferences are influenced by confining pressure andparameters of rock and crack. In some cases, thedifferences maybe reduce to elimination. Therefore,

cohesion, friction angle and Poisson’s ratio (Fig. 3b,
3d and 3f), i.e., R slowly decreases with theincrease of these factors. However, with the increaseof radius, R decreases rapidly at first and thenbecome stable gradually (Fig. 3a), which
demonstrates that the increase of radius could
efficiently reduce the differences of initiation strength
between the two types of cracks. Especially, when

Fig.3

Relationship between R and parameters of rock mass under different confining pressure

6

10th Asian Regional Conference of IAEG (2015)

whether to distinguish the differences between theembedded crack and penetrating crack in practicedepends on properties of rock and crack. In the case

Crack types influence the strength of rock mass.
In practical applications, we usually regard natural
crack as embedded penny-shape. In fact, penetratingcracks are often found in nature because of joints
intersection, especially in bedded rock mass. In
addition, human activities can lead to new cracks
generation, which may help cracks penetrate in one
dimension. For instance, slope excavation can
generate many tensile cracks paralleling to slope
surface, the phenomenon observed in Xiaowanhydropower station is a typical example (Wu et al.,2009). In deep rock mass engineering, zonal
disintegration is another phenomenon influenced byhuman activity which may also transform crack type
(Qian and Li, 2008). Therefore, we need to
distinguish rock types carefully and pay more
attention on the selection of expressions for initiation
strength calculation. Crack type is related to rockmass stability, we may get lager safety factors if we
use wrong expressions, because the stability ofembedded crack may 20% higher than penetratingcrack as Jinping marble shows.

of Jingping marble, the value of R we get is 1.2,
which indicates that the differences between
embedded crack and penetrating crack at Jingping IIhydropower station should be considered.
When excluding confining pressure, the ratio R
under the condition of uniaxial compression is:R 

K (  2 (2  ) / 8  1)
 u1e
 1  c
 u1 pK  c  cw  a

Where  u1e and(18)

 u1 p are initiation strength ofembedded and penetrating cracks under uniaxial
stress, respectively. Eq. (18) reveals that, under thecondition of uniaxial compression, the relative
differences between embedded crack and penetratingcrack are independent of dip angle and friction angleof crack. The values of R are usually large as Fig.

6. Conclusion

3 showing. Hence embedded crack and penetratingcrack should be treated specifically under thecondition of uniaxial compression.
We note that the cohesion of crack is usually

In this paper, based on the energy criterion of
fracture mechanics, we deduced the initiation strength
criteria of embedded crack and penetrating crack
under three-dimensional stress and obtained thefollowing results by comparing the two criteria.
1. The initiation strength of embedded crack islarger than that of penetrating crack. The initiation
strength of embedded crack and penetrating crack can
be both expressed by equal-slope linear function withconfining pressure as independent variable, and theysmall and even negligible. Thus, when cw  0 , theratio R 

 u1e
 2 (2  )
. Here R is only u1 p8

related with Poisson’s ratio and ranges from 1.36 to
1.57 depending on the value of  .

both have the minimum values when  =In general, R will be large with a large K c

 w

+
.
4 2
2. Under a condition of three-dimensional stress,
the relative differences of initiation strength between
embedded crack and penetrating crack decrease with
the increase of confining pressure, and are in positive
correlation with mode I fracture toughness of rock but
in negative correlation with radius, dip angle,
cohesion, friction angle of crack, and Poisson’s ratioof rock. Generally speaking, the differences will belarge for good quality rock or stable crack.
3. The relative differences of initiation strengthand a small Poisson’s ratio, which means goodquality rock. Also, when excluding effects of friction
angle, R will be large with a small radius, dipangle and cohesion of crack, which means stablestructure plane. Therefore, the differences will belarge when rock material is good and structure planeis relatively stable.
5. Application condition7

10th Asian Regional Conference of IAEG (2015)

initiation characteristics of opening-mode crack
embedded in rock-like material under seepage
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between embedded crack and penetrating crack will
be minimum when  = w+, and tends to be stable
4 2
when radius exceeds a certain value.

Acknowledgements

This work was financially supported by Key
program of National Natural Science Foundation ofChina (Grant NO. 41030749).
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