2

Running title: A constitutive model for Shale

A Strain Rate Dependent Constitutive Model for Longmaxi Formation

Shale

HOU Zhenkun1,2*

, LIANG Huqing3, MA Shuqi

5*, GAO Ruchao

4, WEI Xiang

6, GUO Yintong

7, and YANG

Chunhe7

1 Guangzhou Institute of Building Science Co., Ltd., Guangzhou 510440, China; 2 School of Materials Science and Engineering, South China University of Technology, Guangzhou 510640, China; 3 Guangzhou Municipal Construction Group Co., Ltd., Guangzhou 510030, China; 4 CCCC Second Harbour Engineering Co., Ltd., Wuhan，Hubei 430040，China; 5 Department of Civil and Environment Engineering, Colorado School of Mines, Golden, CO 80401, USA; 6 Chongqing Technology and Business University, Chongqing 400067, China; 7 State Key Laboratory of Geomechanics and Geotechnical Engineering, Institute of Rock and Soil Mechanics, Chinese

Academy of Sciences, Wuhan, Hubei 430071, China.

Abstract: Shale gas, as a relative clean source, is a realistic option for low-carbon energy consumption. Shale, as a kind of

brittle rock, often exhibit different nonlinear stress-strain behavior, failure and time-dependent behavior under different strain

rates. To capture these features, triaxial compression tests under axial strain rates 1 ranging from 510-6s-1 to 110-3s-1 are

conducted and the following conclusions are drawn: Both elastic modulus and peak strength have a positive correlation

relationship with strain rates 1 , with both following exponential relationships. These strain rate-dependent mechanical

behaviors of shale are originated from damage growth, which is described by a damage parameter D . When axial strain

1 is the same, D is positively correlated with 1 . When 1 is the same, with an increase in 1 , D decreases

firstly from an initial value (about 0.1 to 0.2), soon reaches its minimum (about 0.1), and then increases to an asymptotic

value of 0.8. Based on the experimental results, taking yield stress as the cut-off point and considering damage variable

evolution, a new measure of micro-mechanical strength F is proposed. Based on the Lemaitre’s equivalent strain

assumption and the new variable F , a statistical strain-rate dependent damage constitutive model for shale that couples

physically meaningful model parameters is established. Numerical back-calculations of these triaxial compression tests

results demonstrate the ability of the model to reproduce the primary features of the strain rate dependent mechanical

behavior of shale.

Key words: Longmaxi Shale; different strain rates; triaxial compression test; damage variable; constitutive model.

E-mail: zhenkunhoucqu@163.com; shuqima.ma@gmail.com.

1 Introduction

Shale as a kind of rock material often appears in many major geotechnical projects, such as tunnel engineering, petroleum engineering, road engineering, civil engineering, mining, slopes and foundations (Heng et al. 2015). In petroleum engineering and geotechnical engineering, the development of modern technologies in the field of drilling has transformed shale formation from an otherwise uneconomic rock material to a profitable resource of natural gas (Mahanta et al. 2016; Middleton et al. 2015). Thus, as a rich carrier material of shale gas, shale has earned increasing importance in the recent years (Zhang et al. 2018).

The mechanical properties of shale have been studied by many scholars. The anisotropy characteristics, compression strength, failure modes and microstructure of shale have been widely studied based on uniaxial and triaxial compression tests (Abousleiman et al. 2010; Sedman et al. 2012; Hou et al. 2015). Many researchers have carried out shear tests with different bedding orientations and found that the bedding planes were significantly weaker than those of the shale matrix (Abousleiman et al. 2010; Sedman et al. 2012). Tensile strength is a critical parameter that determines the resistance of shale to deformation and failure (Zhang et al. 2018) and fracturing in shale is produced mainly by tensile failure, so many researchers have performed a large number of Brazilian tests, which is a simple and effective way to measure the tensile strength of shale (Vervoort et al. 2014; Zhang et al. 2018). It is found that there is a considerable influence from different layer orientations on shale failure modes, and significant effects on the Brazilian tensile strength are also observed.

As described above, almost all the tests were conducted on shale with a certain static loading rate, there are limited investigations addressing the strain rate-dependent mechanical behaviors of shale. In fact, there is a clear

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time effect on many non-shale rock materials, for example high strain rates can enhance the peak strength and elastic modulus of many non-shale rocks, such as sandstone and asphalt concrete (Katsuki and Gutierrez 2011; Gong and Zhao 2014). But we do not know if it is the same for shale.

In petroleum engineering, during site hydraulic fracturing operation, the state of the stress applied to shale is varying including loading rates, which can be proved by the stress environment around shale. In 1957, an analytical model was developed by Hubbert and Willis (Hubbert and Willis 1957) to relate hydraulic fracture initiation pressures, according to the principle of this model, the tensile stress 0T applied to shale can be obtained as follow: 0 03i h HT P P (Frash 2007). In this equation, iP is the injection pressure. Pump pressure curves are jagged (Peng et al. 2017; Zhang et al. 2009), that is to say, both the loading rates and flowback rates of the injection pressure iP of lab and field hydraulic fracturing tests are always varying. All these above can cause the stress environment around shale to change, especially the magnitude and rate of the loads acting on shale. Thus, to be economically attractive, it is important to characterize the strain rate-dependent mechanical behavior of shale.

Given this, the triaxial compression tests of shale at different strain rates were carried out in this paper. The mechanical characteristics, damage characteristics and constitutive model of shale at different strain rates were carefully analyzed. It is envisioned that the presented results can provide some relevant parameters for hydraulic fracturing and brittleness evaluation of shale.

2 Test materials and procedures 2.1 Test materials

The test samples were taken from a fresh shale outcrop (a natural extension of the shale gas reservoir in the Fuling District of Chongqing) in Dayou Town, Chongqing City, China. As shown in Fig. 1, the shale cores were drilled parallel to the bedding plane and then were trimmed into standard rock cores in the laboratory with diameter and height of 50 and 100 mm, respectively (GT /T23561, 2009; ISRM 2007).

(Location of Fig. 1)

Fig. 1 Test samples used in the experiments 2.2 Test procedures

Triaxial compression tests were conducted with a US-made MTS815 loading frame using axial displacement control. In order to simulate realistic in situ conditions, the confining pressure 3 was set at 50 MPa corresponding to the burial depth of the shale gas in Fuling of about 3 km (Zhou 2006). According to the definition of Hudson et al. and Zhou, the strain rate range of quasi-static loading is 10

-4 to10

-6s

-1 and the loading

rate during production is mostly within this range (Hudson et al. 1997; Zhou 1990). Therefore, the strain rates

1 are set as 510-6

s-1

, 110-5

s-1

, 110-4

s-1

, 510-4

s-1

and 110-3

s-1

, respectively. The experimental results and other basic test parameters are described in Table 1. Table 1 Experimental program and testing parameters.

Test

number

3

/MPa

1

/s-1

E

/GPa

Mean value of

E 1 3 max- （ ）

/MPa

Mean value of

1 3 max- （ ）

(MPa/s)%

Mean value of

YS-1 50 5*10-6 24.22

23.25

207.75

210.24

0.12

0.10 YS-2 50 5*10-6 22.54 202 0.09

YS-3 50 5*10-6 23 220.96 0.1

YJ-1 50 1*10-5 24.67 23.88

207.35 211.79

0.21 0.20

YJ-3-1 50 1*10-5 23.09 216.23 0.19

YS-10 50 1*10-4 24.92

25.05

226.6

222.75

2.04

2.06 YS-11 50 1*10-4 25.45 232.55 2.05

YS-12 50 1*10-4 24.79 209.11 2.1

YS-13-1 50 5*10-4 25.02 25.55

225.13 227.44

10.61 10.71

YS-14 50 5*10-4 26.07 229.75 10.8

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YS-16-1 50 1*10-3 26.78 26.83

236.74 233.23

21.89 22.05

YS-18-1 50 1*10-3 26.88 229.71 22.2

3 Analysis of test results 3.1 Analysis of stress-strain curves

As shown in Fig. 2, the initial stage is obvious from the initial concave up shape of the stress-stain curve, indicating the shale has initial damage from the micro-cracks. The lower the strain rate 1 , the larger the axial strain at the initial compression stage. At elastic stage, a constant slope of the stress-strain curve occurs, and then the slope decreases at which stage the level enters the yield stage. At this time, the micro-cracks in the sample begin to grow and propagate resulting in lower stiffness. Following the yield stage, each stress-strain curve reaches its peak point and then drops rapidly. At peak point, failure is a non-stable resulting from uncontrolled fracture propagation. This process is uncontrollable, indicating that shale has a high brittleness.

As shown in Fig. 2, in spite of the change in the strain rates of more than 103 times, the axial strain values at

the peak points are observed in a narrow range from 1% to 1.15%. But for peak strength, elastic modulus and residual strength, they have a positive correlation with 1 . When the strain rate 1 is low, the micro-stress at the shale grains can be fully accommodated and the cracks have sufficient time to evolve, resulting in lower shale strength and lower stiffness. When

1 is low, the asperity damage of fracture surface will be sheared more completely, resulting in lower residual strength. Thus, with the increase of 1 , the peak strength, elastic modulus and residual strength all show an increasing trend. There is a slight rebound of the stress-strain curve before reaching the residual strength at a high strain rate. The reason is as follows: at high 1 , the confining pressure is too late to compress the damaged shale tightly because the shale samples rupture too quickly, which leads to an extreme stress drop. Subsequently, the crushed shale is gradually compacted by the confining pressure and the stress is gradually increased to its residual strength.

(Location of Fig. 2)

Fig. 2 Deviatoric stress vs. axial strain curves for different axial strain rates.

3.2 Effect of strain rate on elastic modulus and peak strength of shale

Table 1 and Fig. 3 show elastic modulus E and peak strength 1 3 max- （ ） for tests at different strain rates. It

is noted that both parameters are important indexes in the design of hydraulic fracturing and evaluation of the brittleness of shale. Besides, the determination of “sweet spot” is the primary task to achieve efficient and stable production, and “sweet spot” need to first possess “excellent friability”, which are mainly related to elastic modulus and peak strength (Hou et al. 2018).

(Location of Fig. 3)

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Fig. 3 Strain rate sensitivity of elastic modulus E and peak deviatoric strength 1 3 max- （ ） of shale.

The strain rate dependencies of the mechanical properties of shale are shown in Fig. 3 in terms of the elastic modulus E (or the peak strength

1 3 max- （ ） ) as function of the strain rate 1 in a E –log( 1 ) plot (or

1 3 max- （ ） –log( 1 ) plot). As shown in Fig. 3, as 1 increases, there is a definite increase in E . Although E is somewhat scattered, the mean values increase as 1 increases from 510

-6s

-1to 110

-3s

-1. This

is because at high strain rates, the deformation of the shale lags behind the stress, resulting in partial deformation that has not yet taken place when the stress reaches a large value, so the shale exhibits a higher elastic modulus. In Fig. 3, the mean values of E have an exponential function relationship with 1 , as represented by the correlation given in Eq. (1). Fig. 3 and Table 1 show the peak strength 1 3 max- （ ） as function of strain rate 1 . Higher strain rates lead to higher peak strength, which indicates that 1 has a strengthening effect on the shale. Similar to the elastic modulus, an exponential function relationship exists between the mean values of 1 3 max- （ ） and 1 , which can be fitted as Eq. (2).

0.0235 2

1=31.117( ) ; 0.937E R (1)

0.0191 2

1 3 max 1- =264.91( ) ; 0.984R （ ） (2)

4 Constitutive model for strain rate-dependent behavior of shale

Studies relating macroscopic mechanical response to microstructural properties, evolution and behavior are necessary to develop reliable constitutive models for shale subjected to varying loading rates. In order to be fully functional and relevant, constitutive models should be described by intrinsic mechanical properties and behavior of the constituent materials. This paper presents a relatively simple rate-dependent constitutive model for shale that requires only three easy-to-determine model parameters apart from 7 other parameters that can be acquired from experiments directly. The model accounts for the rate-dependent response of shale, using a viscoelastic damage formulation that accounts for nonlinear elastic behavior, failure and strain softening. The model is based on the results of the triaxial tests at different strain rates described above. A procedure is proposed to statistically quantify the damage evolution of shale during shearing. A damage evolution function in the proposed model is then based on a statistical distribution model originally proposed to describe the probability of failure of engineering materials. The proposed model is validated against experimental data.

4.1 Statistical damage model for shale at different strain rates

The random distribution of micro-fissures during loading is the source of damage in materials. Based on the Lemaitre “strain equivalence” assumption (Lemaitre 1984), the relationship between the nominal stress i and the effective stress i

as function of the damage variable D can be obtained

(1 )i i D (3)

According to the generalized Hooke's law

'( )i i j kE (4)

where , , 1,2,3i j k , i and i

are the Poisson's ratio and micro-strain of the undamaged part of the rock, respectively. Substituting Eq. (4) into Eq. (3), yields

1 1 2 3(1 ) ( )(1 )E D D (5)

According to the Lemaitre strain equivalence assumption(Lemaitre 1984)

i i (6)

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where i is the nominal strain and

i is the effective micro-strain.

According to Eq. (6) and the definition of Poisson's ratio

3 1 3 1/ = / ' (7)

Using Eq. (3) the following equivalence is obtained for 3 2 in conventional triaxial compression test

3 2 3 2= (1 )= (1 )D D (8)

Substituting Eqs. (7) and (8) into Eq. (5) yields

1 1 3(1 ) 2E D (9)

Eq. (1) can be written as:

0 1( )rE k (10)

where 0k and r are the fitting parameters at different strain rates. Combining Eqs. (9) and (10) yields the following model for the rate-dependent stress-strain behavior of shale

under triaxial compression:

1 0 1 1 3( ) (1 ) 2rk D (11)

4.2 Damage variable D

According to the definition of damage variable D (Ma and Chen 2014):

/dD N N (12)

Where N is the total number of micro-elements and dN is the number of damaged micro-elements. Assuming that the micro-element strength F obeys the Weibull random probability distribution, then D

can be written as follows (Weibull 1951):

0

1 exp

m

FD

F

(13)

where 0F and m are Weibull distribution parameters. In order to find the appropriate formulation for F , it is necessary to carry out a preliminary study on the

variation of D . Only in this way can Eq. (13) describe the damage variable D more accurately. Eq. (9) can be inverted to yield D :

1 3

1

-21D

E

(14)

Where, 1 is the axial stress and 1 is the total axial strain. For triaxial compression experiment, as shown in Fig. 2, the axial stress and axial strain measured by linear

variable differential transformers are deviatoric stress (denoted as 1s ) and deviatoric axial strain (denoted as

1s ), respectively, that is to say:

1 1 3= -s (15)

1 1 0= -s (16)

Where, 1s and 1s are deviatoric stress and deviatoric axial strain collected by linear variable differential transformers, respectively, which are experimental data in Fig. 2. 1 and 3 are axial load and confining pressure, respectively, 1 is the total axial strain, 0 is the axial strain generated in the axial direction of the specimen under confining pressure 3 .

If shale is completely elastic material (in fact it is not), let 1 3= in Eq. (9), and where the initial damage variable is assumed to be very small (approximately zero) then the 0 can be approximately obtained as follows

3

0

1- 2

E

(17)

Substituting Eq. (15) to (17) into Eq. (14) yields:

1 1

1 3

-

+ 1- 2

s s

s

ED

E

(18)

According to Eq. (18), the D - 1 curves are obtained. As shown in Fig. 4a, shale specimen has an

initial damage value 0D (about 0.1) at 1 01 . As the axial strain goes above 01 and increases,

D decreases slowly (implying with the increase of axial stress, microcracks inside shale specimen

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are gradually compressed), goes through a minimum value (approaching 0.05) at 1 02 , then

increases gradually at 1 below 03 , implying new microcracks in shale specimen initiated and

begin to gradually expand with the original microcracks. When 1 is greater than 03

(corresponding to the peak point of the stress-strain curve), D increases linearly with a significant

slope, which demonstrates that the speed of the microcracks formation is accelerated and the density

of cracks increases sharply. When 1 is increased to 04 and beyond, D tends to level off and

approach 0.80. (Location of Fig. 4)

Axial strain, ε (%)1

5·10 s- 6 - 1

1·10 s- 5 - 1

1·10 s- 4 - 1

5·10 s- 4 - 1

1·10 s- 3 - 1

0.5 1.0 1.5 2.0 2.5 3.0

0.0

0.2

0.4

0.6

0.8

1.0D

am

ag

e p

ara

mete

rD

(1)

,

ε04ε03

ε02ε01

(a)

Axial strain, ε (%)1

0.5 1.0 1.5 2.0 2.5 3.0

0.0

0.2

0.4

0.6

0.8

1.0

5·10 s- 6 - 1

1·10 s- 5 - 1

1·10 s- 4 - 1

5·10 s- 4 - 1

1·10 s- 3 - 1

Da

ma

ge

para

me

ter

D(1

),

(b)

Fig. 4 Damage parameter D vs. axial strain 1 curves at different strain rates (a) (calculated from

experimental data); (b) calculated from Eqs. (13), (27) and (28). 4.3 Micro element strength F

From Fig. 4a it is observed that for the same D , the smaller the strain rate 1 , the greater the axial strain

1 . Similarly, for the same 1 , the damage variable D has a positive correlation with the strain rate 1 . These characteristics should be reflected when micro element strength F is used to characterize the damage variable D in Eq. (13).

Currently, there are a variety of expressions for the micro element strength F , it is considered that F can be represented by axial strain 1 because damage increases with the increasing of 1 . However, this view is limited for F is closely related to stress state (Katsuki and Gutierrez 2011). Some new methods for determining the micro-element strength were proposed based on the rock failure and yield criterion (Cao et al. 2007; Cao et al. 2013). The problem is how to select the appropriate failure criterion. When the Drucker-Prager criterion is used to characterize F , the results will be conservative (Alejano et al. 2012). The conventional criterion

fails to describe the magnitude of failure in high-stress zones and tensile stress zones (Bai and

Wierzbicki 2010), which is not applicable in this paper because of high confining pressure. The Hoek-Brown empirical criterion can characterize non-linear damage characteristics of rock, and is more suitable for rock failure under various stress conditions. However, this criterion contains a large number of parameters and some parameters need to be empirically determined (Benz et al. 2008).

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The conventional criterion approximates the Mohr envelope as a straight line, which has good application in low stress conditions. However, under high confining conditions, the envelope will gradually deviate from the straight line (as shown in Fig. 5) (Hou 2018). In fact, the envelope under high stress can be expressed in parabolic form, which is an assumption based on an extension of Griffith (Cai et al. 2002; Hou 2018).

2 ck (19)

where k and c are parabolic fitting parameters. Suppose that the center of the Mohr’s circle for stress is at 1 3 / 2 along the normal stress axis and

the radius is 1 3( ) / 2r . The distance from any point on the parabola to the center of the circle is as follows

2

2 1 3 c2

d k

(20)

Since the parabolic failure criterion (Eq. 19) is tangent to the Mohr’s circle for stress, d should take the minimum value, which is also equal to r , then the following can be obtained

2

1 32( ) 02

dk

（ ） (21)

From above, 0 can be obtained

1 30

2 2

k

(22)

Substituting Eq. (22) and Eq. (20) into 2 2d r , yields the principal stress expression of the criterion at high

stress level

2 2

1 3 1 3( ) ( )c 0

4 2 4

k k (24)

From Lemaitre’s equivalent strain assumption, Eq. (24) can be modified and the effective stress expression of the criterion at high stress level can be derived

2 2

1 3 1 3( ) ( ) 4c

4 2 4

k k

(25)

Substituting Eqs. (2) and (14) into Eq. (25), yields the nominal stress expression of the criterion at high stress levels

2 2 2 2

1 3 1 1 3 1

2

1 3 1 3

( ) ( ) 4c -=

4 2 2 2 4

E k E k

（ ） （ ） (26)

Based on traditional M-C criterion ( 1 3(1 sin ) (1 sin ) =2 cosc ), micro-element strength F can be defined as (Cao et al. 2007)

1 3(1 sin ) (1 sin ) -2 cosF c (27)

The damage variables D can be established based on Eqs. (13) and (27). It can be calculated that D is generated from the peak strength, which clearly does not match Fig. 4a. In Fig. 4a, D can be divided into two parts taking the yield stress point(

s ，s ) as the cut-off point. Referring to the principle of Eqs. (27) and

according to Eqs. (26), the micro element strength F can be modified as follows: For

1 s , F can be constructed as:

2 2 2

1 3 s 1 1 3 s 11 2

1 3 1 3

( ) - ( ) ( - )

4 2 2 2

E k EF

（ ）

（ ） （ ） (28)

For 1 s , F can be constructed as:

2 2 2 2

1 3 1 1 3 12 2

1 3 1 3

( ) ( ) 4c -

4 2 2 2 4

E k E kF

（ ） （ ） (29)

According to Eqs. (28) and (29), when 1= s , we can get

1 2 0F F , that is, the damage variable at the yield point is 0. Combining Eq. (13) and Fig. 4a, we can verify that Eq. (28) and Eq. (29) are reasonable.

Assuming that the yield point at the deviator stress-strain curve is( 1 ss ， 1 ss ), then it can be shown that

1 s 3= -s s (30)

1 s s 0= -s (31)

Where, 1 ss and 1 ss are deviatoric stress and deviatoric axial strain of the yield point in Fig. 2,

which are measured by linear variable differential transformers, respectively; s and s are the

axial stress and axial strain of the yield point, respectively.

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4.4 Determination of model parameters

The statistical damage constitutive model of shale under different strain rate and high confining pressure can be obtained by substituting Eqs. (28), (29) and (13) into Eq. (9):

10 1 1 3 1

0

1

20 1 1 3 1

0

( ) exp 2 ,

( ) exp 2 ,

a

b

m

r

s

a

m

r

s

b

Fk

F

Fk

F

(32)

In addition to the Weibull distribution parameters 0aF ,am , 0bF and

bm in Eq. (32), the other parameters can be obtained from experiments. Therefore, the method of calculation of the Weibull distribution parameters will be given in the following section.

The following boundary conditions can be obtained from the different stages of the stress-strain curves of shale (Cao et al. 2013).

① 00, 0 D D ， ； ② 0, / 0;d d ③ 1 1 1,c c s c （ ） ； ④ 1 , / 0;c d d

1c and 1c are the peak stress and strain, respectively. When

1 s , the following equation can be obtained by differentiating Eq. (32) with respect to the axial strain:

1

1 2 1 2 20 1

1 0 0 1

( ) exp 1-

bb

b

mm

r b

m

b b

d F m F Fk

d F F

(33)

where

2 2

1 3 1 1 32

2

1 1 3 1 3

( ) ( )=

2 2 2 2

E k EF

（ ） （ ） (34)

Substituting boundary condition ③ into Eq. (32), yields

21 0 1 1 3

0

( ) exp 2

bm

r cc c

b

Fk

F

(35)

where

2 2 2 2

1 3 1 1 3 12c 2

1 3 1 3

( ) ( ) 4c -

4 2 2 2 4

c c c c

c c

E k E kF

（ ） （ ） (36)

Substituting boundary condition ④ into Eq. (33), yields

1 1

1

2 1 2 20 1

0 0 1

( ) exp 1- 0

bb

c

mm

r c b c c

m

b b

F m F Fk

F F

(37)

where

1 1

2 2

1 3 1 1 32

2

1 1 3 1 3

( ) ( )

2 2 2 2c

c c c

c c

E k EF

（ ） （ ） (38)

From Eq. (37), the following can be obtained

1 1

1

1 2 2

0 1

1- 0b

b

c

m

b c c

m

b

m F F

F

(39)

Eq. (39) can be inverted to yield the following equation by multiplying both sides by 2cF simultaneously

1 1

2 2

0 1 2 1

1

/

b

c

m

c cb

b c

F Fm

F F

(40)

Eq. (35) can be inverted to yield Eq. (41)

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2 1 3

0 0 1 1

- 2ln

( )

bm

c c

r

b c

F

F k

(41)

Combining Eqs. (40) and (41) yields bm

1 1

2

1 3 21

0 1 1 1

=-2

ln( )

c

cb

ccr

c

Fm

F

k

(42)

Substituting Eq. (42) into Eq. (41), yields

20 1

1 3

0 1 1

- 2ln

( )

b

cb

mc

r

c

FF

k

(43)

Parameters 2cF and 1 12 1/

cF in Eqs. (42) and (43) can be obtained by using Eqs. (36) and (38),

respectively. When1 s , substituting boundary condition ① into Eqs.(13) and (32), yields Eqs. (44) and

(45)

100

0

1 exp

am

a

FD

F

(44)

2 2 2

1 3 1 s 1 3 1 s10 2

1 3 1 3

( ) ( )

4 2 2 2

s sE k EF

（ ） （ ） (45)

The yield stress point of the deviator stress-strain curve and the stress-strain curve are ( 1 ss ， 1 ss ) and ( s ，

s ), respectively. Eq. (44) can be inverted to yield 0aF :

10

0a 1/

0ln 1am

FF

D

(46)

In fact, the Weibull distribution parameter m has little effect on the elastic response (Cao et al. 2013). Therefore, we can assume a bm m . Thus, all the parameters of the Weibull distribution can be calculated from Eqs. (42) (43) and (46).

5 Discussions and modification 5.1 Determination of laboratory parameters

Combining triaxial compression tests conducted by Hou 2018 and Eq.(30) yields the s of shale under

different confining pressures. Then the yield stress-confining pressure curve can be shown as in Fig. 5. As can be seen from Fig. 4a, the damage variable is 0 at the yield stress point. Based on this, combining Eq. (3) and Eq. (25) yields the following equation:

2 2

3 3( ) ( ) 4

4 2 4

s sk c k (47)

(Location of Fig. 5)

20 40 60 80 100 120

Confining pressure σ (MPa), 3

150

200

250

300

350

Yie

ld s

tre

ss σ

(MP

a)

,S Experimental curve

Best-fit curve

The traditional M-C criterion

Fig. 5 The yield stress vs. confining pressure curves (Hou 2018).

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Eq. (47) can be inverted to yield Eq. (48):

3 32s k c k (48)

According to Eq.(48), the yield stress-confining pressure curve in Fig. 5 can be fitted as follows:

3 352.34 2 52.34 - 443s ; R2=0.996 (49)

The traditional Mohr-Coulomb criterion assumes the relationship between the yield stress and confining pressure is a straight line. It can be seen from Fig. 5 that the yield stress gradually deviates from the straight line at high confining pressure, which demonstrates that Eq. (48) is more accurate.

5.2 Analysis of reasonableness of damage variables

According to Eqs. (13), (28) and (29), the damage variable D can be determined and shown in Fig. 4b. The following observations can be made: (1) The initial damage 0D is not 0; (2) D in the elastic phase is almost zero; (3) D begins to increase gradually from the yield stage; and (4) When the stress nears its peak, D increases sharply. Due to local damage, stress reduction, secondary compaction and other effects, the damage variables in this region fluctuate slightly. Eventually, D tends to a certain value in the residual phase. The variation of D in Fig. 4b is almost the same as in Fig. 4a. Both curves will have the same law, which demonstrates that D is correct.

5.3 Weibull distribution parameters

Weibull distribution parameters have a large influence on the shape of stress-strain curves. In this paper, the influence of 0bF and bm is analyzed based on the test results on sample 12. As shown in Fig. 6, the initial stress-strain curves are almost unaffected by Weibull distribution parameters. Both 0bF and bm have a great influence on the yield stage and the subsequent stages. As shown in Fig. 6a, the greater the value of 0bF , the greater the peak strength. After the peak, the decreasing sections of the curve parallel each other, which show that 0bF does not affect the post-peak stress drop rate, that is to say, 0bF has little effect on the brittleness of shale. Via calculation, when 0bF is increased by 100%, the average values of stress 1 , strain 1 and post-peak stress drop rate will increase by 10.38%, 14.11% and 0%, respectively. As shown in Fig. 6b, when the value of bm is greater, the peak strength will be greater and the rate of stress drop after the peak will be significantly accelerated, which means the brittleness is significantly increased. When bm is increased by 100%, the average values of stress 1 , strain 1 and post-peak stress drop rate will increase by 5.86%、3.80%、75.88%, respectively.

(Location of Fig. 6)

Axial strain, ε (%)1

Ax

ial

stre

ss,

σ(M

Pa)

1

0.0 0.5 1.0 1.5 2.00

100

200

300

400

From left to right

F =5000 8000 12221,0 b , , 18000, 25000, 30000

(a)

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Axial strain, ε (%)1

Ax

ial

str

ess

,σ(M

Pa

)1

0.0 0.5 1.0 1.5 2.00

100

200

300

350

50

150

250

From right to left

m=1, 2, 2.9, 5, 10

(b)

Fig. 6 Predicted stress-strain curves of shale for different values of Weibull parameters. (a) Effect of different

values 0bF ; (b) Effect of different values m ( a bm m m )

In summary, 0bF has a greater effect on peak strength and can be integrated to reflect the strength

characteristics of shale. Although bm has a certain effect on the peak strength, it has more influence on the post-peak stress drop rate.

6 Model verification

Verification of constitutive model at different strain rates-As shown in Fig. 7, the predicted model curves agree well with the original stress-strain curves at different strain rates, especially the peak-front curves before the peak strain, but the model can not reflect the post-peak characteristics for the following reasons: Shale will collapse and destroy suddenly after reaching peak strength because of its high brittleness, coupled with the limited accuracy due to the model’s reliance on empirical data, resulting in a larger post-peak increase. The model is based on the Lemaitre strain equivalence assumption, which assumes that rock is no longer capable of bearing after damage, so the residual strength can't be described. The model needs to be further improved to improve the results of the post-peak curve.

(Location of Fig. 7)

Axial strain, ε (%)1

Ax

ial

stre

ss,

σ(M

Pa)

1

(a)S train rate:5·10 s- 6 - 1

0.0 0.5 1.0 1.5 2.0 2.5

0

50

100

150

200

250

300

350

Test result

F itting curve

Axial strain, ε (%)1

Ax

ial

stre

ss,

σ(M

Pa)

1

(b)Strain rate:1·10 s- 5 - 1

0.0 0.5 1.0 1.5 2.0 2.5

0

50

100

150

200

250

300

350

Test result

F itting curve

Axial strain, ε (%)1

Ax

ial str

ess

,σ(M

Pa)

1

(c)S train rate:1·10 s- 4 - 1

0.0 0.5 1.0 1.5 2.0 2.50

50

100

150

200

250

300

350

Test result

F itting curve

Axial strain, ε (%)1

Ax

ial str

ess

,σ(M

Pa)

1

(d)Strain rate:5·10 s- 4 - 1

0.0 0.5 1.0 1.5 2.0 2.50

50

100

150

200

250

300

350

Test result

F itting curve

Axial strain, ε (%)1

Ax

ial str

ess

,σ(M

Pa)

1

(f)S train rate:1·10 s- 3 - 1

0.0 0.5 1.0 1.5 2.0 2.50

50

100

150

200

250

300

350

Test result

F itting curve

Fig. 7 Comparisons of predicted experimental stress-strain curves with experimental results for different strain

rates.

7 Conclusions

The following conclusions can be obtained from the experimental and constitutive modeling results: (1)Higher strain rates lead to higher elastic modulus and peak strength. Both elastic modulus and peak

strength have exponential function relationships with the strain rate.

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(2)The damage parameter D is slightly bigger than zero in the compaction stage, and is almost zero in the elastic phase. Then D begins to increase gradually during the yield stage. When the stress nears its peak, D increases sharply and fluctuates slightly. Eventually, the damage variable tends to a certain value in the residual phase. For the same axial strain 1 , when the strain rate 1 is higher, D is also higher.

(3)Taking the yield stress as the cut-off point and fully considering the changes in the damage variable, a new method of micro-element strength measurement is proposed. Based on the Lemaitre strain equivalence assumption and the new micro-element strength measurement, a statistical damage constitutive model of shale at different strain rates is established and the physical meaning of the relevant parameters is given: 0bF has a greater effect on peak strength and can be integrated to reflect the strength characteristics of shale. Although

bm has an effect on peak strength, it has more influence on the rate of decline of post-peak stress and can comprehensively reflect the brittle characteristics of shale. Acknowledgements

The research was supported by the China Scholarship Council project, the National Natural Science Foundation of China (grant No. 51574218, 51678171, 51608139, U1704243 and 51709113), National Science and Technology Major Project of China (2016ZX05060-004 and 2017ZX05036-003), Guangdong Science and Technology Department (Grant No.2015B020238014), Guangzhou Science Technology and Innovation Commission (Grant No.201604016021), and High-level Talent Research Launch Project (No. 950318066). References Abousleiman, Y.N., Hoang, S.K., and Tran, M.H., 2010. Mechanical characterization of small shale samples subjected to

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About the first author HOU Zhenkun Male; born in 1988 in Dancheng City, Henan Province; doctor; post doctorate at Guangzhou Institute of Building Science Co., Ltd. and South China University of Technology; He is now interested in the study on hydraulic fracturing, rock mechanics and pipe pile. Email:zhenkunhoucqu@163.com; phone: 17875316866.

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