# Related Rule Study of Subcritical Crack Growth and Threshold Values in Transversely Isotropic Slates

Authors: Gang Ma; Jiangteng Li; Huiwen Wang
Journal: Geofluids
 (2020)

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## Abstract

Elastic parameters and the subcritical crack growth of different bedding angle slate specimens were studied using uniaxial compression testing and the double torsion constant displacement load relaxation method using SANS and MTS Insight machines. To study the relations of the mode-I stress intensity factorKI versus the subcritical crack growth velocity V, the fracture toughness KIC, the stagnation speed, and the threshold values, the double torsion constant displacement load relaxation method was carried out. The related rules between the bedding angles (β) and the uniaxial compressive strength, fracture toughness, and threshold values were investigated. Experimental results show that the uniaxial compression, the fracture toughness, and the threshold value curves move to the bottom then increase with the increase of the β angle. In addition, its fracture toughness is minimal when the β angle of the slates is 45°, and crack initiation and crack propagation are generated under load, which can lead to the failure of the slate. lgKI-lgV relations of transversely isotropic slates measured by this method are in accordance with linear rules, which is in good agreement with the Charles theory. The range of K0/KIC for these different bedding angle slates is from 0.511 to 0.789. The test results would provide the basis for studying seepage and time-dependency of rock engineering stability.


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## Body

## 1. Introduction

Rocks are mineral aggregates after a long geological process; rocks of layered structure can be simplified into a transversely isotropic body, whose physical and mechanical properties in parallel to the bedding plane are different from those in perpendicular to the bedding plane. The stability analysis of the layered rock is complicated by the isotropic characteristics of the bedding surface and the coupling effect of the seepage field and stress field [1]. Thus, to reveal the transversely isotropic permeability mechanism, it is necessary to investigate the transversely isotropic fracture characteristics.At present, the researches on anisotropic rock properties are mainly concentrated on the following areas: (1) the measurement method study of elastic deformation parameters [2], (2) the anisotropy strength and yield criterion study [3, 4], (3) anisotropic rock constitutive model study [5, 6], and (4) anisotropic rock deformation characteristics and mechanical property study [7, 8]. Transverse isotropy is a special case of anisotropy; these studies have a great significance on helping us have a better understanding of the permeability and mechanical properties of transversely isotropic rocks. The instability instances in situ and rock laboratory examinations show that the rock failure is due to crack growth and propagation in rock mass [9], and the crack is the main seepage channel of the rock mass, which controls the seepage characteristics of the rock mass. According to the fracture mechanics and stress corrosion theory [10, 11], when a crack tip stress intensity factor KI exceeds its fracture toughness KIC, the crack will quickly expand; when the stress intensity factor is less than its fracture toughness, the crack growth will process at the speed of stress corrosion subcritical crack growth. The study shows that the crack in certain circumstances propagates until the material fractures with a collective effect of tensile stress and corrosion medium, which is called stress corrosion [12]. Usually, the mode-I stress intensity factor KI at the crack tip is regarded as a parameter to control the crack growth. The stress corrosion lower limit is K0; when the stress intensity factor is less than K0, the crack will not extension; when the stress intensity factor is between KIC and KI, the crack grows at a certain velocity; the reasons can be listed as follows: (1) the stresses at the crack tip increases in the process of forming a microcrack and interspace and (2) the chemical bond fracture stresses decrease by the effects of the environmental medium, so the crack grows at a stable and static state in the process, which is called subcritical crack growth. The crack growth in the rock mass happens after the subcritical cracks propagate to a certain extent. Therefore, studies of the subcritical crack growth laws and fracture properties are important to rock engineering long-term stability. The elastic parameters and the principle of subcritical crack growth can be obtained in the paper. The related laws between the bedding angles (β) and compressive strength, fracture toughness, and threshold values had been analyzed. The relations of the mode-I stress intensity factor KI versus the subcritical crack growth velocity V and the fracture toughness KIC will be able to predict when the crack will be extended to the critical crack length under certain stress conditions, the result of which can cause structural damage, which gives the permeability and mechanics study a new approach.


## 2. Transverse Isotropy of Different Bedding Angle Slates

### 2.1. Constitutive Relations of Transversely Isotropic Elastic Body

Transversely isotropic elastic body is made of an elastic symmetry plane, whose physical and mechanical properties in parallel to the elastic symmetry plane are different from those in perpendicular to the elastic symmetry plane. For the layered rock, engineering generally regards the bedding plane as an elastic symmetry plane; it is believed that it has the same physical and mechanical properties in parallel to the bedding plane which are different from those that are vertical to the elastic symmetry plane. As shown in Figure1, the XOY plane is assumed as the elastic symmetry plane; the transversely isotropic elastic parameters should satisfy the following conditions [13]:
(1)Ex=Ey=E,Ez=E′,μxy=μyx=μ,μzx=μzy=μ′,Gxy=E21+μ,Gyz=Gxz=G2.Figure 1
Transversely isotropic slate schematic view (β=0°).The transversely isotropic elastic body constitutive relation can be simplified as follows [14]:
(2)εxεyεzγxyγyzγxz=Aσxσyσzτxyτyzτxz=a11a12a13000a11a23000a33000a4400a550a55σxσyσzτxyτyzτxz,where A is the compliance matrix, which can be can be calculated by
(3)a11=1E,a12=−υE,a13=−υ′E,a33=1E′,a44=21+υE,a55=1G′,where E and ν are Young’s modulus of rock and Poisson’s ratio in the direction parallel to the transversely isotropic plane, respectively; E′ and ν′ is Young’s modulus of rock and Poisson’s ratio in the direction perpendicular to the transversely isotropic plane, respectively; and G′ is the shear modulus for the plane normal to the transversely isotropic plane.Because the stiffness matrix and the compliance matrix are inverse matrix, so the stiffness matrixC can be calculated as follows:
(4)c11=λn1−nv′2,c12=λnv+nv′2,c13=λnv′1+v,c33=λ1−v2,c44=G′,c66=E21+υ,wheren=E/E′andλ=E′/1+υ1−υ−2nυ′2.When an orthotropic body degenerates into a transversely isotropic elastic body, the nine independent elastic parameters are reduced to the five independent elastic parameters, respectively;E, E′, G′, ν, ν′, and G are not independent elastic parameters, because of G=E/21+ν.


### 2.2. Coordinate Conversion of the Anisotropic Elastic Constants

Due to the different mechanical properties of the rock in all directions in anisotropic media, the coordinate system often needs to be converted to solve the elastic parameters, and coordinate transformation of anisotropic elastic parameters generally follows the tensor rule.The global coordinate system (X,Y,Z) and the local coordinate system (X′,Y′,Z′) of a transversely isotropic body are shown in Figure 2; the local coordinate system around the z-axis rotates the β angle to get the global coordinate system. Stiffness matrix C in the local coordinate system can be converted to the stiffness matrix C′ in the global coordinate system. It can be expressed as [13]
(5)C′=LTCL,where L is the transformation matrix, which can be expressed as follows:
(6)L=cos2βsin2βsinβcosβsin2βcos2β−sin2βcosβ−2sinβcosβ2sinβcosβcos2β−sin2β.Figure 2
Transverse isotropic material schematic view.Transversely isotropic elastic parameters in the local coordinate system can be transferred into the elastic parameters in the global coordinate system, which are shown as follows:(7)εxεyεzγxyγyzγxz=Aσxσyσzτxyτyzτxz=a11a12a1300a16a12a22a2300a26a13a23a3300a36a44a450a45a550a16a26a36a66σxσyσzτxyτyzτxz,where a11,⋯,a66 can be calculated by the following equations:
(8)a11=1Ecos4β+1E′sin4β+1G′−2υ′Esin2βcos2β,a22=1Esin4β+1E′cos4β+1G′−2υ′Esin2βcos2β,a12=1E+1E′+2υ′E−1G′sin2βcos2β−υ′E,a13=−υ′E′sin2β−υEcos2β,a23=−υ′E′cos2β−υEsin2β,a33=1E,a44=sin2βG+cos2βG′,a45=1G′−1Gsinβcosβ,a55=cos2βG+sin2βG′,a36=2υE−υ′E′sinβcosβ,a16=2sin2βE′−cos2βE+1G′−2υ′Ecos2β−sin2βsinβcosβ,a26=2cos2βE′−sin2βE+1G′−2υ′Ecos2β−sin2βsinβcosβ,a66=41E+1E′+2υ′E−1G′sin2βcos2β+1G′.


### 2.3. Thermodynamic Constraint of Transversely Isotropic Constant

To make the rock strain energy positive in this experiment, the 5 elastic parameters of the transversely isotropic rock should meet the related inequality of hot mechanical constraints, which are expressed as follows [15]:
(9)E,E′,G′>0,−1≤υ≤1,−E′E1−υ2≤υ′≤E′E1−υ2.


## 2.1. Constitutive Relations of Transversely Isotropic Elastic Body

Transversely isotropic elastic body is made of an elastic symmetry plane, whose physical and mechanical properties in parallel to the elastic symmetry plane are different from those in perpendicular to the elastic symmetry plane. For the layered rock, engineering generally regards the bedding plane as an elastic symmetry plane; it is believed that it has the same physical and mechanical properties in parallel to the bedding plane which are different from those that are vertical to the elastic symmetry plane. As shown in Figure1, the XOY plane is assumed as the elastic symmetry plane; the transversely isotropic elastic parameters should satisfy the following conditions [13]:
(1)Ex=Ey=E,Ez=E′,μxy=μyx=μ,μzx=μzy=μ′,Gxy=E21+μ,Gyz=Gxz=G2.Figure 1
Transversely isotropic slate schematic view (β=0°).The transversely isotropic elastic body constitutive relation can be simplified as follows [14]:
(2)εxεyεzγxyγyzγxz=Aσxσyσzτxyτyzτxz=a11a12a13000a11a23000a33000a4400a550a55σxσyσzτxyτyzτxz,where A is the compliance matrix, which can be can be calculated by
(3)a11=1E,a12=−υE,a13=−υ′E,a33=1E′,a44=21+υE,a55=1G′,where E and ν are Young’s modulus of rock and Poisson’s ratio in the direction parallel to the transversely isotropic plane, respectively; E′ and ν′ is Young’s modulus of rock and Poisson’s ratio in the direction perpendicular to the transversely isotropic plane, respectively; and G′ is the shear modulus for the plane normal to the transversely isotropic plane.Because the stiffness matrix and the compliance matrix are inverse matrix, so the stiffness matrixC can be calculated as follows:
(4)c11=λn1−nv′2,c12=λnv+nv′2,c13=λnv′1+v,c33=λ1−v2,c44=G′,c66=E21+υ,wheren=E/E′andλ=E′/1+υ1−υ−2nυ′2.When an orthotropic body degenerates into a transversely isotropic elastic body, the nine independent elastic parameters are reduced to the five independent elastic parameters, respectively;E, E′, G′, ν, ν′, and G are not independent elastic parameters, because of G=E/21+ν.


## 2.2. Coordinate Conversion of the Anisotropic Elastic Constants

Due to the different mechanical properties of the rock in all directions in anisotropic media, the coordinate system often needs to be converted to solve the elastic parameters, and coordinate transformation of anisotropic elastic parameters generally follows the tensor rule.The global coordinate system (X,Y,Z) and the local coordinate system (X′,Y′,Z′) of a transversely isotropic body are shown in Figure 2; the local coordinate system around the z-axis rotates the β angle to get the global coordinate system. Stiffness matrix C in the local coordinate system can be converted to the stiffness matrix C′ in the global coordinate system. It can be expressed as [13]
(5)C′=LTCL,where L is the transformation matrix, which can be expressed as follows:
(6)L=cos2βsin2βsinβcosβsin2βcos2β−sin2βcosβ−2sinβcosβ2sinβcosβcos2β−sin2β.Figure 2
Transverse isotropic material schematic view.Transversely isotropic elastic parameters in the local coordinate system can be transferred into the elastic parameters in the global coordinate system, which are shown as follows:(7)εxεyεzγxyγyzγxz=Aσxσyσzτxyτyzτxz=a11a12a1300a16a12a22a2300a26a13a23a3300a36a44a450a45a550a16a26a36a66σxσyσzτxyτyzτxz,where a11,⋯,a66 can be calculated by the following equations:
(8)a11=1Ecos4β+1E′sin4β+1G′−2υ′Esin2βcos2β,a22=1Esin4β+1E′cos4β+1G′−2υ′Esin2βcos2β,a12=1E+1E′+2υ′E−1G′sin2βcos2β−υ′E,a13=−υ′E′sin2β−υEcos2β,a23=−υ′E′cos2β−υEsin2β,a33=1E,a44=sin2βG+cos2βG′,a45=1G′−1Gsinβcosβ,a55=cos2βG+sin2βG′,a36=2υE−υ′E′sinβcosβ,a16=2sin2βE′−cos2βE+1G′−2υ′Ecos2β−sin2βsinβcosβ,a26=2cos2βE′−sin2βE+1G′−2υ′Ecos2β−sin2βsinβcosβ,a66=41E+1E′+2υ′E−1G′sin2βcos2β+1G′.


## 2.3. Thermodynamic Constraint of Transversely Isotropic Constant

To make the rock strain energy positive in this experiment, the 5 elastic parameters of the transversely isotropic rock should meet the related inequality of hot mechanical constraints, which are expressed as follows [15]:
(9)E,E′,G′>0,−1≤υ≤1,−E′E1−υ2≤υ′≤E′E1−υ2.


## 3. Elastic Parameters of Transversely Isotropic Slate Specimen

Elastic modulus, Poisson’s ratio, and uniaxial compressive strength values of the cylinder slates whose diameter is 50 mm and height is 100 mm with different bedding angles (0°, 30°, 45°, 70°, 80°, and 90°) can be obtained by uniaxial compression test using the SANS machine, which are listed in Table1.Table 1
Elastic modulus, Poisson’s ratio, and uniaxial compressive strength.
β (°)SpecimenCompressive strength (MPa)The average of compressive strength (MPa)Elastic modulus (MPa)The average of elastic modulus (MPa)The average of Poisson’s ratioμxzμyz0°0-170.0874.7936.6738.380.190.190-263.8641.020-390.4337.4530°1-135.5632.2125.0023.000.200.211-230.4523.451-330.6520.5545°2-132.4524.9625.5621.770.230.202-220.5619.002-321.8720.7570°3-130.2027.0020.3823.980.220.243-228.3724.383-322.4327.1880°4-156.3663.3930.5532.960.300.184-267.8532.894-365.9635.4490°5-171.1670.5042.2241.850.250.235-269.3942.745-370.9540.59According to the translation of rock mechanics by Milou and the relation between the constant in compliance matrix and elastic constants of rock, the following relationship can be obtained [16]:
(10)1Ez′=sin4βE1+cos4βE2+1G2−2ν1E2sin2βcos2β,νxz′=−εxεz=E1ν1E1sin2β+ν2E2cos2β,νyz′=−εyεz=ν2E1E2−1+E1E2+2ν2E1E2−E1G2sin2βcos2β,1Gxy′=sin2βG2+21+ν1cos2βE1,1Gxz′=cos2βG2+21+ν1sin2βE1,1Gyz′=1E141+E1E2+2ν2E1E2−E1G2sin2βcos2β+E1G2,where E1 and ν1 are the elastic modulus and Poisson’s ratio of the slate with a bedding angle of 90°, respectively; E2 and ν2 are the elastic modulus and Poisson’s ratio of the slate with a bedding angle of 0°, respectively; and Eβ is the elastic modulus of the slate with β bedding angle.For the slates with a certain angle between the transversely isotropic plane and horizontal plane,Gxy, Gxz, and Gyz of the slate specimen in three orthogonal directions can be obtained by Equation (10), which are listed in Table 2. Based on the transversely isotropic and loading direction of the slate, we select Gxz as the shear modulus of the slate specimen in the calculation of its fracture toughness.Table 2
Shear modulus of slate rock specimen at different angles.
β0°30°45°70°80°90°Gxy16.8913.208.386.055.695.57Gxz5.578.388.7013.6515.9616.89Gyz5.5713.0516.687.695.6816.89


## 4. The Relations between Bedding Angles of Slate Specimen and Uniaxial Compressive Strength and Elastic Modulus

For the layered rock with a single layer, its bedding angles have a great influence on its destructive strength. For the slates in this experiment, it can be considered as the transversely isotropic constitutive model which has only one set of parallel beddings with different angles loaded. We can obtain the related law between the bedding angles of the slate specimen and the uniaxial compressive strength and elastic modulus from Table1, which can be expressed as Figure 3, from which we can know that the bedding angles have a great effect on the uniaxial compressive strength and elastic modulus of the slates. What is more, the uniaxial compression and elastic modulus decreases then increases with the increase of β angle and the relation curves show a “U” type. The results show that when the β angle of the slate is 45°, its uniaxial compression and elastic modulus are minimal. Therefore, it is most likely to be damaged when the bedding angle of slates is 45°.Figure 3
The relations betweenβ angles and compressive strength and elastic modulus.


## 5. Double Torsion Method

### 5.1. Principle of Double Torsion Testing

The double torsion specimen was first adopted by Williams and Evans [17] to decide the fracture properties of glass and ceramics, which is shown in Figure 4. Since 1977, Ciccotti et al. [18], Saadaoui et al. [19], and Nara and Kaneko [20] applied it to rock research. The double torsion specimen can be assumed as two elastic torsion bars, of which each cross-section is a rectangle.Figure 4
Schematic diagram of double torsion specimen: (a) specimen plane; (b) “V”-style cut schematic; (c) crack growth.
(a)(b)(c)For the small deformation, when the width of the torsion bar is much larger than the thickness of the torsion bar, the torsion strain can be calculated by the following equation [21]:
(11)θ≅ywm≅6Tawt3G,where T is the torque Pwm/2, P is the applied load on the torsion bar, G is the shear modulus of the rock, a is the crack length, t is the specimen thickness, w/2 of the torsion bar’s width, and wm is the length of twisted arm, and Equation (11) can be rewritten as
(12)yp≅3wm2awt3G≅C,where C is the elastic compliance of the double torsion specimen. The relation among crack strain energy release rate g and the flexibility of the specimens satisfies the following relationship:
(13)g=p22dCdA=p22tndCda,where A is the area of the crack and tn is the thickness of plane where crack exists. For the double torsion specimen, strain energy release rate can be defined as
(14)g=3p2wm22wt3tnG.Based on the relations between stress intensity factorKI and the strain energy release rate g and the relationship between elastic modulus E and shear modulus G, a lot of research has been done, such as by Cao et al. [22], Andreas et al. [23], Ciccotti et al. [18, 24], and Zhao et al. [25]. The stress intensity factor is introduced in this study.
(15)KI=Pwm3E2wd3dnG1/2.When the loadP reaches a critical value of the PC on the specimen, the crack quickly grows, So KI reaches a critical value KIC, which is shown as follows:
(16)KIC=PCwm3E2wd3dnG1/2.Tests showed that the compliance change rate can be obtained by the displacement change rate with the given loads or by the load change rate with the given displacement. Under the constant displacement conditions, through the derivative of Equation (12), it can be obtained that
(17)dadt=−wt3Ey6wm2p21+υdpdt.Equation (17) shows that if the specimen’s size and displacement are known, the subcritical crack growth velocity is related to the load relaxation rate with the given displacement state.


### 5.2. Stress Intensity Factor and Crack Velocity of the Double Torsion Method

The stress intensity factor was independent of the crack length in this study, and the double torsion test is convenient when it uses opaque materials, such as rocks for which crack length measurements are inconvenient. There are three methods to obtain the crack velocity in the double torsion tests, each of which has a different loading process. The constant load method was used by Li et al. for the original double torsion test [26]. Two other methods are the constant displacement rate and the constant displacement load relaxation method which were introduced by Evans [27]. In the constant displacement load relaxation method, the displacement of the loading point is kept constant during the experiment and the load relaxation is measured with the crack growth. Because the stress intensity factor is the function of the loading and the crack growth rate is the function of the temporal load and decreasing rate of the loading, the relation between the stress intensity factor and the crack growth rate can be obtained over a wide range of loads using a single experimental run. For this reason, the constant displacement load relaxation method was adopted in this paper.


### 5.3. Experimental

#### 5.3.1. Specimen Preparation

Slates of different bedding angles (0°, 30°, 45°, 70°, 80°, and 90°) were machined as a rectangle plate double torsion specimen whose standard size was180mm×60mm×5mm and width-to-thickness ratio was 15 : 1 ~ 12 : 1. There was a longitudinal groove on the bottom surface, which was 1 mm in width and t/3 in depth, and a notch was cut at the end. The crack front proceeded with a straight line starting from a machined initial notch and guided by a side groove thus producing a relaxation of the load. The parallel error between the top surface and the bottom surface of specimens was less than 0.025 mm. The mechanical parameters are listed in Table 3.Table 3
The specimen mechanical parameters.
βNumber of sampleCompressive strength (MPa)Elastic modulus (GPa)μxzμyz0°374.7938.380.190.1930°332.2123.000.200.2145°324.9621.770.230.2070°327.0023.980.220.2480°363.3932.960.300.1890°370.5041.850.250.23


#### 5.3.2. Experimental Procedure

The constant displacement load relaxation method only needs to monitor the decrease of the load with time to measure both the stress intensity factorKI and the subcritical crack growth velocity V, regardless of the crack length measurement. The double torsion test was processed using an MTS Insight machine in the testing center of the Central South University. Firstly, the slate specimen should be precracked to obtain reasonable KIC and KI-V data [28, 29], and precrack velocity was 0.05 mm/min. The load-time curve of a typical specimen in precrack is shown in Figure 5. Secondly, the precracked specimen was loaded in 5 mm/min displacement velocity during constant displacement load relaxation testing [19]. Then, the displacement of the slate specimen was kept constant when the load comes up to 95% of the precrack load and the relaxation testing was done [30, 31]. The load-time data was recorded until the load relaxation was stabilized. The load-time curve and the displacement-time curve of a typical specimen are shown in Figures 6 and 7, respectively.Figure 5
Load-time curve of typical specimen in precrack.Figure 6
Load-time curve of typical specimen.Figure 7
Displacement-time curve of typical specimen.The precracked specimen was loaded in 20 mm/min displacement velocity during the fracture toughnessKIC testing, and the critical load PC was recorded. The fracture toughness KIC can be obtained by Equation (l7). The load-time curve of a typical specimen in testing KIC is shown in Figure 5.


#### 5.3.3. Results and Analysis

(1)KI-V Relations in Transversely Isotropic Slates. Based on the double torsion testing data, the relation between subcritical crack growth velocity V and the stress intensity factor KI can be obtained by Equations (15) and (17). According to the Charles theory [32], to analyze the relation between the subcritical crack growth velocity V and the stress intensity factor KI, a linear function of double logarithmic space can be used, and their lgKI-lgV coordinate points in logarithm coordinates are shown in Figures 8–13. lgKI-lgV data are regressed as follows:
(18)lgV=a+blgKI,where a and b are the fitting coefficients.Figure 8
lgKI-lgV curve for the slate specimen of 0° bedding angles.Figure 9
lgKI-lgV curve for the slate specimen of 30° bedding angles.Figure 10
lgKI-lgV curve for the slate specimen of 45° bedding angles.Figure 11
lgKI-lgV curve for the slate specimen of 70° bedding angles.Figure 12
lgKI-lgV curve for the slate specimen of 80° bedding angles.Figure 13
lgKI-lgV curve for the slate specimen of 90° bedding angles.The power function relation betweenV and KI is defined as
(19)V=AK1n,where A and n are the fitting coefficients.Table4 shows each specimen’s subcritical crack growth parameters. Researching rock seepage and stable time effect must be based on the KIC and the relation KI-V, from the perspective of fracture mechanics, and these obtained parameters provide basic information for further study of geotechnical engineering seepage and time-dependent stability.Table 4
The parameters of subcritical crack growth.
βSpecimen no.An0°0-11.213×10−3967.0100-22.333×10−65116.8910-34.775×10−3861.58830°1-12.009×10−3655.5341-21.780×10−3558.6441-31.982×10−3460.49545°2-11.984×10−3559.5522-21.992×10−3461.5042-31.935×10−3365.53270°3-11.387×10−555.7433-32.089×10−456.70080°4-11.986×10−1368.1854-21.849×10−1768.1854-34.764×10−2079.71090°5-13.972×10−1784.6155-23.656×10−1392.0645-33.622×10−1081.102(2)Fracture Toughness (KIC) and the Threshold Values of Different Bedding Angles of Slates. Based on the testing data, the fracture toughness (KIC) can be obtained by Equation (16), which is shown in Table 5.Table 5
Fracture toughnessKIC testing value (MN·m-3/2).
βSpecimen no.KICAverageKIC0°0-14.144.210-24.250-34.2430°1-13.153.021-23.001-32.9145°2-12.602.652-22.722-32.6370°3-12.922.793-32.6680°4-12.552.814-23.004-32.8890°5-13.152.955-22.905-32.80Typically, during the testing process of the rock subcritical crack growth rate, with the expansion of the crack, the crack tip’s stress intensity factor decreases and the crack extended speed is reduced. When approaching subcritical crack extension threshold value, the crack growth rate is difficult to measure [21]. For rock-like material, the previous researchers did not detect the threshold value, at present. When Wilkins did the research of the subcritical crack expansion of granite, he still did not find the threshold value (K0) with the expansion speed of 10-12 m/s [17]. In this experiment, the lowest subcritical crack extension speed is in the order of 10-5 to 10-6. From the point of view of engineering significance, it can be considered that crack growth has been stopped at this time. In this article, the lowest subcritical crack extension velocity is regarded as the stagnation speed of the crack propagation which is shown in Table 6. According to Equation (19), the stress intensity factor KI related to the crack propagation can be obtained as the threshold values K0, which are shown in Table 7.Table 6
The stagnation speed.
ParameterStagnation speed0°4.01671×10−730°5.87673×10−745°3.93884×10−670°3.04077×10−680°1.05808×10−690°1.36032×10−6Table 7
Threshold valuesK0.
βSpecimen no.K0K0KIC0°0-13.2680.7890-23.3310.7840-33.2390.764Average3.2790.77930°1-12.2180.7041-22.2180.7391-32.6210.901Average2.3520.78145°2-11.3480.5182-21.4400.5292-31.3450.511Average1.3780.52070°3-11.5480.5303-31.7130.618Average1.6310.57480°4-11.8380.6864-21.6380.6424-32.0160.672Average1.8380.66190°5-12.3910.7595-22.2400.7725-32.1660.774Average2.2660.768(3)The Relations between the KIC and the Bedding Angles of Slates. The relation between the bedding angles (β) and the fracture toughness (KIC) can be obtained, as shown in Figure 14. It is clear that the basic relation can be expressed as KIC=0.00043β2−0.05β+4.19, R2=0.97, from which we can see that fracture toughness decreases then increases with the increase of the β angle and the relation curves show a “U” type; the result can well reflect the change rule of the KIC with the increase of β angles, which is consistent with the theoretical result. Fracture toughness is an important parameter to characterize the rock to resist crack initiation and propagation. This experiment shows that when the bedding angle of the slates is 45°, its fracture toughness is minimal; it means that its capacity to resist crack propagation is minimal, and it is most likely to be damaged during loading.Figure 14
Relation between fracture toughness andβ angle.(4)The Relation between the Threshold Values and the Bedding Angles of Slates. Figure 15 shows the relation between K0 and β angles for different bedding angles’ slates. The basic relationship is K0=0.00055β2−0.063β+3.369, R2=0.96. According to Figure 15, it is found that threshold values (K0) decrease then increase with the increase of the β angle. Because of the different bedding angles corresponding to different thresholds, we can know that the threshold values of the slate specimen have obvious anisotropy. When the bedding angle of the slates is 45°, its threshold value is minimal; it means that when the slate specimen is under load, crack initiation and expansion are most likely to happen, which can cause slate specimen failure.Figure 15
Relation between threshold values andβ angle.


## 5.1. Principle of Double Torsion Testing

The double torsion specimen was first adopted by Williams and Evans [17] to decide the fracture properties of glass and ceramics, which is shown in Figure 4. Since 1977, Ciccotti et al. [18], Saadaoui et al. [19], and Nara and Kaneko [20] applied it to rock research. The double torsion specimen can be assumed as two elastic torsion bars, of which each cross-section is a rectangle.Figure 4
Schematic diagram of double torsion specimen: (a) specimen plane; (b) “V”-style cut schematic; (c) crack growth.
(a)(b)(c)For the small deformation, when the width of the torsion bar is much larger than the thickness of the torsion bar, the torsion strain can be calculated by the following equation [21]:
(11)θ≅ywm≅6Tawt3G,where T is the torque Pwm/2, P is the applied load on the torsion bar, G is the shear modulus of the rock, a is the crack length, t is the specimen thickness, w/2 of the torsion bar’s width, and wm is the length of twisted arm, and Equation (11) can be rewritten as
(12)yp≅3wm2awt3G≅C,where C is the elastic compliance of the double torsion specimen. The relation among crack strain energy release rate g and the flexibility of the specimens satisfies the following relationship:
(13)g=p22dCdA=p22tndCda,where A is the area of the crack and tn is the thickness of plane where crack exists. For the double torsion specimen, strain energy release rate can be defined as
(14)g=3p2wm22wt3tnG.Based on the relations between stress intensity factorKI and the strain energy release rate g and the relationship between elastic modulus E and shear modulus G, a lot of research has been done, such as by Cao et al. [22], Andreas et al. [23], Ciccotti et al. [18, 24], and Zhao et al. [25]. The stress intensity factor is introduced in this study.
(15)KI=Pwm3E2wd3dnG1/2.When the loadP reaches a critical value of the PC on the specimen, the crack quickly grows, So KI reaches a critical value KIC, which is shown as follows:
(16)KIC=PCwm3E2wd3dnG1/2.Tests showed that the compliance change rate can be obtained by the displacement change rate with the given loads or by the load change rate with the given displacement. Under the constant displacement conditions, through the derivative of Equation (12), it can be obtained that
(17)dadt=−wt3Ey6wm2p21+υdpdt.Equation (17) shows that if the specimen’s size and displacement are known, the subcritical crack growth velocity is related to the load relaxation rate with the given displacement state.


## 5.2. Stress Intensity Factor and Crack Velocity of the Double Torsion Method

The stress intensity factor was independent of the crack length in this study, and the double torsion test is convenient when it uses opaque materials, such as rocks for which crack length measurements are inconvenient. There are three methods to obtain the crack velocity in the double torsion tests, each of which has a different loading process. The constant load method was used by Li et al. for the original double torsion test [26]. Two other methods are the constant displacement rate and the constant displacement load relaxation method which were introduced by Evans [27]. In the constant displacement load relaxation method, the displacement of the loading point is kept constant during the experiment and the load relaxation is measured with the crack growth. Because the stress intensity factor is the function of the loading and the crack growth rate is the function of the temporal load and decreasing rate of the loading, the relation between the stress intensity factor and the crack growth rate can be obtained over a wide range of loads using a single experimental run. For this reason, the constant displacement load relaxation method was adopted in this paper.


## 5.3. Experimental

### 5.3.1. Specimen Preparation

Slates of different bedding angles (0°, 30°, 45°, 70°, 80°, and 90°) were machined as a rectangle plate double torsion specimen whose standard size was180mm×60mm×5mm and width-to-thickness ratio was 15 : 1 ~ 12 : 1. There was a longitudinal groove on the bottom surface, which was 1 mm in width and t/3 in depth, and a notch was cut at the end. The crack front proceeded with a straight line starting from a machined initial notch and guided by a side groove thus producing a relaxation of the load. The parallel error between the top surface and the bottom surface of specimens was less than 0.025 mm. The mechanical parameters are listed in Table 3.Table 3
The specimen mechanical parameters.
βNumber of sampleCompressive strength (MPa)Elastic modulus (GPa)μxzμyz0°374.7938.380.190.1930°332.2123.000.200.2145°324.9621.770.230.2070°327.0023.980.220.2480°363.3932.960.300.1890°370.5041.850.250.23


### 5.3.2. Experimental Procedure

The constant displacement load relaxation method only needs to monitor the decrease of the load with time to measure both the stress intensity factorKI and the subcritical crack growth velocity V, regardless of the crack length measurement. The double torsion test was processed using an MTS Insight machine in the testing center of the Central South University. Firstly, the slate specimen should be precracked to obtain reasonable KIC and KI-V data [28, 29], and precrack velocity was 0.05 mm/min. The load-time curve of a typical specimen in precrack is shown in Figure 5. Secondly, the precracked specimen was loaded in 5 mm/min displacement velocity during constant displacement load relaxation testing [19]. Then, the displacement of the slate specimen was kept constant when the load comes up to 95% of the precrack load and the relaxation testing was done [30, 31]. The load-time data was recorded until the load relaxation was stabilized. The load-time curve and the displacement-time curve of a typical specimen are shown in Figures 6 and 7, respectively.Figure 5
Load-time curve of typical specimen in precrack.Figure 6
Load-time curve of typical specimen.Figure 7
Displacement-time curve of typical specimen.The precracked specimen was loaded in 20 mm/min displacement velocity during the fracture toughnessKIC testing, and the critical load PC was recorded. The fracture toughness KIC can be obtained by Equation (l7). The load-time curve of a typical specimen in testing KIC is shown in Figure 5.


### 5.3.3. Results and Analysis

(1)KI-V Relations in Transversely Isotropic Slates. Based on the double torsion testing data, the relation between subcritical crack growth velocity V and the stress intensity factor KI can be obtained by Equations (15) and (17). According to the Charles theory [32], to analyze the relation between the subcritical crack growth velocity V and the stress intensity factor KI, a linear function of double logarithmic space can be used, and their lgKI-lgV coordinate points in logarithm coordinates are shown in Figures 8–13. lgKI-lgV data are regressed as follows:
(18)lgV=a+blgKI,where a and b are the fitting coefficients.Figure 8
lgKI-lgV curve for the slate specimen of 0° bedding angles.Figure 9
lgKI-lgV curve for the slate specimen of 30° bedding angles.Figure 10
lgKI-lgV curve for the slate specimen of 45° bedding angles.Figure 11
lgKI-lgV curve for the slate specimen of 70° bedding angles.Figure 12
lgKI-lgV curve for the slate specimen of 80° bedding angles.Figure 13
lgKI-lgV curve for the slate specimen of 90° bedding angles.The power function relation betweenV and KI is defined as
(19)V=AK1n,where A and n are the fitting coefficients.Table4 shows each specimen’s subcritical crack growth parameters. Researching rock seepage and stable time effect must be based on the KIC and the relation KI-V, from the perspective of fracture mechanics, and these obtained parameters provide basic information for further study of geotechnical engineering seepage and time-dependent stability.Table 4
The parameters of subcritical crack growth.
βSpecimen no.An0°0-11.213×10−3967.0100-22.333×10−65116.8910-34.775×10−3861.58830°1-12.009×10−3655.5341-21.780×10−3558.6441-31.982×10−3460.49545°2-11.984×10−3559.5522-21.992×10−3461.5042-31.935×10−3365.53270°3-11.387×10−555.7433-32.089×10−456.70080°4-11.986×10−1368.1854-21.849×10−1768.1854-34.764×10−2079.71090°5-13.972×10−1784.6155-23.656×10−1392.0645-33.622×10−1081.102(2)Fracture Toughness (KIC) and the Threshold Values of Different Bedding Angles of Slates. Based on the testing data, the fracture toughness (KIC) can be obtained by Equation (16), which is shown in Table 5.Table 5
Fracture toughnessKIC testing value (MN·m-3/2).
βSpecimen no.KICAverageKIC0°0-14.144.210-24.250-34.2430°1-13.153.021-23.001-32.9145°2-12.602.652-22.722-32.6370°3-12.922.793-32.6680°4-12.552.814-23.004-32.8890°5-13.152.955-22.905-32.80Typically, during the testing process of the rock subcritical crack growth rate, with the expansion of the crack, the crack tip’s stress intensity factor decreases and the crack extended speed is reduced. When approaching subcritical crack extension threshold value, the crack growth rate is difficult to measure [21]. For rock-like material, the previous researchers did not detect the threshold value, at present. When Wilkins did the research of the subcritical crack expansion of granite, he still did not find the threshold value (K0) with the expansion speed of 10-12 m/s [17]. In this experiment, the lowest subcritical crack extension speed is in the order of 10-5 to 10-6. From the point of view of engineering significance, it can be considered that crack growth has been stopped at this time. In this article, the lowest subcritical crack extension velocity is regarded as the stagnation speed of the crack propagation which is shown in Table 6. According to Equation (19), the stress intensity factor KI related to the crack propagation can be obtained as the threshold values K0, which are shown in Table 7.Table 6
The stagnation speed.
ParameterStagnation speed0°4.01671×10−730°5.87673×10−745°3.93884×10−670°3.04077×10−680°1.05808×10−690°1.36032×10−6Table 7
Threshold valuesK0.
βSpecimen no.K0K0KIC0°0-13.2680.7890-23.3310.7840-33.2390.764Average3.2790.77930°1-12.2180.7041-22.2180.7391-32.6210.901Average2.3520.78145°2-11.3480.5182-21.4400.5292-31.3450.511Average1.3780.52070°3-11.5480.5303-31.7130.618Average1.6310.57480°4-11.8380.6864-21.6380.6424-32.0160.672Average1.8380.66190°5-12.3910.7595-22.2400.7725-32.1660.774Average2.2660.768(3)The Relations between the KIC and the Bedding Angles of Slates. The relation between the bedding angles (β) and the fracture toughness (KIC) can be obtained, as shown in Figure 14. It is clear that the basic relation can be expressed as KIC=0.00043β2−0.05β+4.19, R2=0.97, from which we can see that fracture toughness decreases then increases with the increase of the β angle and the relation curves show a “U” type; the result can well reflect the change rule of the KIC with the increase of β angles, which is consistent with the theoretical result. Fracture toughness is an important parameter to characterize the rock to resist crack initiation and propagation. This experiment shows that when the bedding angle of the slates is 45°, its fracture toughness is minimal; it means that its capacity to resist crack propagation is minimal, and it is most likely to be damaged during loading.Figure 14
Relation between fracture toughness andβ angle.(4)The Relation between the Threshold Values and the Bedding Angles of Slates. Figure 15 shows the relation between K0 and β angles for different bedding angles’ slates. The basic relationship is K0=0.00055β2−0.063β+3.369, R2=0.96. According to Figure 15, it is found that threshold values (K0) decrease then increase with the increase of the β angle. Because of the different bedding angles corresponding to different thresholds, we can know that the threshold values of the slate specimen have obvious anisotropy. When the bedding angle of the slates is 45°, its threshold value is minimal; it means that when the slate specimen is under load, crack initiation and expansion are most likely to happen, which can cause slate specimen failure.Figure 15
Relation between threshold values andβ angle.


## 5.3.1. Specimen Preparation

Slates of different bedding angles (0°, 30°, 45°, 70°, 80°, and 90°) were machined as a rectangle plate double torsion specimen whose standard size was180mm×60mm×5mm and width-to-thickness ratio was 15 : 1 ~ 12 : 1. There was a longitudinal groove on the bottom surface, which was 1 mm in width and t/3 in depth, and a notch was cut at the end. The crack front proceeded with a straight line starting from a machined initial notch and guided by a side groove thus producing a relaxation of the load. The parallel error between the top surface and the bottom surface of specimens was less than 0.025 mm. The mechanical parameters are listed in Table 3.Table 3
The specimen mechanical parameters.
βNumber of sampleCompressive strength (MPa)Elastic modulus (GPa)μxzμyz0°374.7938.380.190.1930°332.2123.000.200.2145°324.9621.770.230.2070°327.0023.980.220.2480°363.3932.960.300.1890°370.5041.850.250.23


## 5.3.2. Experimental Procedure

The constant displacement load relaxation method only needs to monitor the decrease of the load with time to measure both the stress intensity factorKI and the subcritical crack growth velocity V, regardless of the crack length measurement. The double torsion test was processed using an MTS Insight machine in the testing center of the Central South University. Firstly, the slate specimen should be precracked to obtain reasonable KIC and KI-V data [28, 29], and precrack velocity was 0.05 mm/min. The load-time curve of a typical specimen in precrack is shown in Figure 5. Secondly, the precracked specimen was loaded in 5 mm/min displacement velocity during constant displacement load relaxation testing [19]. Then, the displacement of the slate specimen was kept constant when the load comes up to 95% of the precrack load and the relaxation testing was done [30, 31]. The load-time data was recorded until the load relaxation was stabilized. The load-time curve and the displacement-time curve of a typical specimen are shown in Figures 6 and 7, respectively.Figure 5
Load-time curve of typical specimen in precrack.Figure 6
Load-time curve of typical specimen.Figure 7
Displacement-time curve of typical specimen.The precracked specimen was loaded in 20 mm/min displacement velocity during the fracture toughnessKIC testing, and the critical load PC was recorded. The fracture toughness KIC can be obtained by Equation (l7). The load-time curve of a typical specimen in testing KIC is shown in Figure 5.


## 5.3.3. Results and Analysis

(1)KI-V Relations in Transversely Isotropic Slates. Based on the double torsion testing data, the relation between subcritical crack growth velocity V and the stress intensity factor KI can be obtained by Equations (15) and (17). According to the Charles theory [32], to analyze the relation between the subcritical crack growth velocity V and the stress intensity factor KI, a linear function of double logarithmic space can be used, and their lgKI-lgV coordinate points in logarithm coordinates are shown in Figures 8–13. lgKI-lgV data are regressed as follows:
(18)lgV=a+blgKI,where a and b are the fitting coefficients.Figure 8
lgKI-lgV curve for the slate specimen of 0° bedding angles.Figure 9
lgKI-lgV curve for the slate specimen of 30° bedding angles.Figure 10
lgKI-lgV curve for the slate specimen of 45° bedding angles.Figure 11
lgKI-lgV curve for the slate specimen of 70° bedding angles.Figure 12
lgKI-lgV curve for the slate specimen of 80° bedding angles.Figure 13
lgKI-lgV curve for the slate specimen of 90° bedding angles.The power function relation betweenV and KI is defined as
(19)V=AK1n,where A and n are the fitting coefficients.Table4 shows each specimen’s subcritical crack growth parameters. Researching rock seepage and stable time effect must be based on the KIC and the relation KI-V, from the perspective of fracture mechanics, and these obtained parameters provide basic information for further study of geotechnical engineering seepage and time-dependent stability.Table 4
The parameters of subcritical crack growth.
βSpecimen no.An0°0-11.213×10−3967.0100-22.333×10−65116.8910-34.775×10−3861.58830°1-12.009×10−3655.5341-21.780×10−3558.6441-31.982×10−3460.49545°2-11.984×10−3559.5522-21.992×10−3461.5042-31.935×10−3365.53270°3-11.387×10−555.7433-32.089×10−456.70080°4-11.986×10−1368.1854-21.849×10−1768.1854-34.764×10−2079.71090°5-13.972×10−1784.6155-23.656×10−1392.0645-33.622×10−1081.102(2)Fracture Toughness (KIC) and the Threshold Values of Different Bedding Angles of Slates. Based on the testing data, the fracture toughness (KIC) can be obtained by Equation (16), which is shown in Table 5.Table 5
Fracture toughnessKIC testing value (MN·m-3/2).
βSpecimen no.KICAverageKIC0°0-14.144.210-24.250-34.2430°1-13.153.021-23.001-32.9145°2-12.602.652-22.722-32.6370°3-12.922.793-32.6680°4-12.552.814-23.004-32.8890°5-13.152.955-22.905-32.80Typically, during the testing process of the rock subcritical crack growth rate, with the expansion of the crack, the crack tip’s stress intensity factor decreases and the crack extended speed is reduced. When approaching subcritical crack extension threshold value, the crack growth rate is difficult to measure [21]. For rock-like material, the previous researchers did not detect the threshold value, at present. When Wilkins did the research of the subcritical crack expansion of granite, he still did not find the threshold value (K0) with the expansion speed of 10-12 m/s [17]. In this experiment, the lowest subcritical crack extension speed is in the order of 10-5 to 10-6. From the point of view of engineering significance, it can be considered that crack growth has been stopped at this time. In this article, the lowest subcritical crack extension velocity is regarded as the stagnation speed of the crack propagation which is shown in Table 6. According to Equation (19), the stress intensity factor KI related to the crack propagation can be obtained as the threshold values K0, which are shown in Table 7.Table 6
The stagnation speed.
ParameterStagnation speed0°4.01671×10−730°5.87673×10−745°3.93884×10−670°3.04077×10−680°1.05808×10−690°1.36032×10−6Table 7
Threshold valuesK0.
βSpecimen no.K0K0KIC0°0-13.2680.7890-23.3310.7840-33.2390.764Average3.2790.77930°1-12.2180.7041-22.2180.7391-32.6210.901Average2.3520.78145°2-11.3480.5182-21.4400.5292-31.3450.511Average1.3780.52070°3-11.5480.5303-31.7130.618Average1.6310.57480°4-11.8380.6864-21.6380.6424-32.0160.672Average1.8380.66190°5-12.3910.7595-22.2400.7725-32.1660.774Average2.2660.768(3)The Relations between the KIC and the Bedding Angles of Slates. The relation between the bedding angles (β) and the fracture toughness (KIC) can be obtained, as shown in Figure 14. It is clear that the basic relation can be expressed as KIC=0.00043β2−0.05β+4.19, R2=0.97, from which we can see that fracture toughness decreases then increases with the increase of the β angle and the relation curves show a “U” type; the result can well reflect the change rule of the KIC with the increase of β angles, which is consistent with the theoretical result. Fracture toughness is an important parameter to characterize the rock to resist crack initiation and propagation. This experiment shows that when the bedding angle of the slates is 45°, its fracture toughness is minimal; it means that its capacity to resist crack propagation is minimal, and it is most likely to be damaged during loading.Figure 14
Relation between fracture toughness andβ angle.(4)The Relation between the Threshold Values and the Bedding Angles of Slates. Figure 15 shows the relation between K0 and β angles for different bedding angles’ slates. The basic relationship is K0=0.00055β2−0.063β+3.369, R2=0.96. According to Figure 15, it is found that threshold values (K0) decrease then increase with the increase of the β angle. Because of the different bedding angles corresponding to different thresholds, we can know that the threshold values of the slate specimen have obvious anisotropy. When the bedding angle of the slates is 45°, its threshold value is minimal; it means that when the slate specimen is under load, crack initiation and expansion are most likely to happen, which can cause slate specimen failure.Figure 15
Relation between threshold values andβ angle.


## 6. Conclusion

Transversely isotropic elastic parameters and the subcritical crack growth of different bedding angle slate specimens were studied, and a series of uniaxial compression tests and the double torsion constant displacement load relaxation methods were carried out. The following conclusions can be drawn based on this study:(1)
Based on the slate specimen elastic parameters obtained from the uniaxial compression test and the double torsion test, it is clear that the slate is significantly transversely isotropic(2)
The relations of the mode-I stress intensity factorKI versus the subcritical crack growth velocity V and the fracture toughness KIC were obtained by the double torsion constant displacement load relaxation method; the threshold values were calculated. The calculation results indicate that lgKI-lgV relations of different bedding angle slate specimens measured are in accordance with the linear rule, and they are in great agreement with the Charles theory(3)
The threshold valueK0 of subcritical crack propagation was obtained from the perspective of engineering applications. In this experiment, the range of K0/KIC for these different bedding angle slates is from 0.511 to 0.789, and the lowest subcritical crack extension speed of the different bedding angle slates is in the order of 10-5 to 10-6(4)
The slate’s bedding angle has an inevitable impact on its mechanical properties. The uniaxial compression, fracture toughness, and threshold values decrease then increases with the increase of theβ angle and the relation curve is U-shaped. It is also observed that when the β angle of the slate is 45°, its fracture toughness is minimal. Therefore, crack initiation and crack propagation are generated under loading, resulting in the failure of the slate



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