# Determination of Rock Tensile Strength Based on Arc Loading Splitting Test considering Modulus Effect

Authors: Guirong Teng; Xiaodong Wang; Zenghui Zhao; Zhuoxiu Sun
Journal: Advances in Civil Engineering
 (2023)

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## Abstract

Tensile strength is an important control parameter for the design and stability analysis in rock engineering. In order to reveal the influence of elastic modulus on tensile strength test results, Brazilian splitting tests were carried out on specimens with different strengths, such as mudstone, white sandstone, green sandstone, red sandstone, and marble. Longitudinal and transverse strains were monitored to analyze the influence of elastic modulus on the maximum contact angle and peak contact load between head and disc. A numerical model of Brazilian splitting tests was built, and the contact load between the head and rock disc were detailed revealed. Based on the experimental and numerical results, a modified formula to calculate tensile strength considering the modulus effect was proposed theoretically. Results show that the maximum pressure between the head and disc increases and the contact angle decreases with the increasing of elastic modulus, which show great influence on the splitting failure; the distribution of contact load can be divided into three types: parabolic I, sinusoidal, and parabolic II, which are suitable for soft rock, medium-hard rock, and hard rock, respectively; the new models can improve the calculation accuracy of rock tensile strength.


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## Body

## 1. Introduction

The tensile strength of rock is far less than the compressive strength, so tension failure is frequently occurred in underground project, deep mining, and energy storage engineering. Rock tension failure is one of the main forms of surrounding rock instability [1]. Thus, for the design and stability analysis of various rock engineering, the tensile strength of rock is an important control parameter. Accurately obtaining the rock tensile strength is of great significance to ensure the safety of engineering construction and disaster prevention.There are two methods for testing the tensile strength, that is direct measurement and indirect measurement. Although the direct measurement method has high accuracy, it is difficult to realize, such as it is difficult to ensure that the axial load is completely along the axis of the specimen, and the fracture of the specimen is not necessarily in the middle, these difficulties will seriously affect the test results, so the indirect measurement is generally used. The Brazilian splitting method is the most widely used in the indirect measurement. There are three loading methods: diameter loading, platform loading, and arc loading. Among them, the arc loading mode is relatively reliable which can reduce the stress concentration near the contact area [2–5]. According to platform loading, You and Su [6] and Yu et al. [7] find that the measured tensile strength increases with the platform center angle 2α which should be 20°∼30°. Kaklis et al. [8] establish two kinds of platform splitting loading model and propose that the center angle of the disc center should be at least 2α ≥ 20°. It is suggested that the loading angle should be controlled within 2α ≤ 30° based on the platform splitting test conducted by Wang and Cao [9], Huang et al. [10] find that when the contact angle is too small, the pressure of head-disc contact area is significantly higher. With the increase of contact angle, the tensile stress at the center point of the disc end face has a decreasing trend. When the loading angle is ≥15°, the center crack of the end face can be guaranteed. Junhonget al. [11] study the influence of load distribution on tensile strength based on theoretical and experimental methods and give the minimum loading angle to meet the crack initiation at the center point of the end face. The head-disc contact angle affects the test results in arc loading mode. Kourkoulis et al. [12–15] study the contact problem in the Brazilian splitting test considering the contact problem between the head and the disc, and the result shows that the head-disc contact condition is the factor of affecting the stress state inside the disc. Yu et al. [16, 17] obtain the stress state and failure process of specimens in Brazilian discs under different arc loading angles. Bahaaddini et al. [18] find that with the increase of contact loading angle, the compressive stress in the contact area decreases and the degree of stress concentration decreases. Yuan and Shen [19] point out that the elastic modulus of the disc and the ratio of compressive strength to tensile strength are the main factors affecting the stress distribution near the contact point. The specimen with a smaller elastic modulus and a longer contacting range, which affects the stress distribution near the contact of the disc and then affects the tensile strength. However, no quantitative results are given.The above-given research studies show that the head-disc contact stress transfer mode has an important influence on the crack initiation position and failure mode of the disc. However, the determination of rock tensile strength does not consider the influence of elastic modulus on the contact load distribution, resulting in the deviation between the theoretical and experimental result. Therefore, the calculation formula of rock tensile strength needs to be modified. In this paper, Brazilian splitting experiments and uniaxial compression experiments were carried out on different types of rocks, and the influence of elastic modulus on peak contact load and maximum contact angle was analyzed. The head-disc contact load distribution of Brazilian splitting experiments was studied by simulation software. The rock types applicable to different types of contact load distribution were analyzed, and the tensile strength correction formula and correction coefficient of different types of contact load distribution were theoretically derived. These works will guide the Brazilian splitting experiment to obtain a more accurate tensile strength of rocks.


## 2. Arc Loading Brazilian Splitting Test on Soft and Hard Rock

### 2.1. Experimental Design

In order to compare and analyze the test results of rocks with different modulus, rocks in the tests were taken from different hard and soft rock mining areas in the east and west of China. As shown in Figure1, five kinds of rock specimens are mudstone, white sandstone, green sandstone, red sandstone, and marble. The uniaxial compression specimens are processed into a cylinder with a diameter of 50 mm and a height of 100 mm, and samples in the Brazilian splitting test are processed into a disc with a diameter of 50 mm and a thickness of 25 mm. According to the rock mechanics test standard, the loading rate of these experiments is 0.5 MPa/s.Figure 1
Indoor experimental samples. (a) Uniaxial compression samples. (b) Brazilian disc samples.
(a)(b)The purpose of the uniaxial compression test is to obtain the elastic modulus and Poisson’s ratio of rock samples. Two strain gauges are affixed to the transverse and axial direction of the specimen. In order to ensure the accuracy, the strain gauges are bonded to the tested piece with a quick-drying adhesive, which has high tensile strength, to ensure that the strain gauges are closely connected with the rock during loading, so this bonding method has little influence on the measurement accuracy. Four groups of strain gauges were placed in the perpendicular diameter direction as shown in Figure2(a). In the Brazilian splitting test, in order to obtain the strain at the center of the rock disc end face, a strain gauge was placed in the horizontal direction. Meanwhile, a strain gauge was placed at the head-disc contact position to determine the change rule of the head-disc load transfer as shown in Figure 2(b).Figure 2
Diagram of strain gauge sticking position. (a) Uniaxial compression test. (b) Brazilian splitting test.
(a)(b)


### 2.2. Experimental Data Processing

#### 2.2.1. Mechanical Parameters

According to the uniaxial compression test, the elastic modulusE and Poisson’s ratio μ of rock samples are calculated as shown in Table 1.Table 1
Mechanical parameters of rock samples.
MaterialElastic modulusE (GPa)Poisson’s ratio (μ)Mudstone7.350.236White sandstone13.430.262Green sandstone20.620.283Red sandstone27.400.274Marble53.580.298


#### 2.2.2. Head-Disc Contact Angle

In order to calculate the head-disc contact angle when the rock is fractured, the geometric relationship between the head-disc contact angle and the vertical displacement of the head is shown in Figure3, in which A and A′ represent the edge point of the contact area between the head and the disc, respectively. Thus, the head-disc contact angle 2α is determined as(1)2α=2arccosR−∆yR,where R is the disc radius, and ∆y is the head displacement. Based on equation (1) and the test result, the maximum contact angle of different samples is calculated and the results are listed in Table 2.Figure 3
Schematic diagram of head displacement.Table 2
Maximum head-disc contact angle.
MaterialMudstoneWhite sandstoneGreen sandstoneRed sandstoneMarbleElastic modulusE (GPa)7.3513.4320.6227.4053.58Maximum contact angle (°)31.527.725.323.119.0Figure4 shows the relationship between the maximum head-disc contact angle and the elastic modulus of rock under splitting failure. The maximum contact angle decreases with the increase of the elastic modulus of the rock. The relationship of them can be fitted as(2)2αE=−0.0068E2−0.6851E+36.1.Figure 4
Changing law between elastic modulus and maximum head-disc contact angle.


#### 2.2.3. Head-Disc Contact Load

The peak contact load between the head and disc is shown in Figure5, which is calculated according to the strain value at the center of the head-disc contact area. The load sharply increases with the increase of the elastic modulus. In the initial stage of loading, the changing rate of the contact peak load of the five samples basically remains unchanged. With the increasing of applied load, the power changing rate of the contact load decreases.Figure 5
Peak value of head-disc contact load.


## 2.1. Experimental Design

In order to compare and analyze the test results of rocks with different modulus, rocks in the tests were taken from different hard and soft rock mining areas in the east and west of China. As shown in Figure1, five kinds of rock specimens are mudstone, white sandstone, green sandstone, red sandstone, and marble. The uniaxial compression specimens are processed into a cylinder with a diameter of 50 mm and a height of 100 mm, and samples in the Brazilian splitting test are processed into a disc with a diameter of 50 mm and a thickness of 25 mm. According to the rock mechanics test standard, the loading rate of these experiments is 0.5 MPa/s.Figure 1
Indoor experimental samples. (a) Uniaxial compression samples. (b) Brazilian disc samples.
(a)(b)The purpose of the uniaxial compression test is to obtain the elastic modulus and Poisson’s ratio of rock samples. Two strain gauges are affixed to the transverse and axial direction of the specimen. In order to ensure the accuracy, the strain gauges are bonded to the tested piece with a quick-drying adhesive, which has high tensile strength, to ensure that the strain gauges are closely connected with the rock during loading, so this bonding method has little influence on the measurement accuracy. Four groups of strain gauges were placed in the perpendicular diameter direction as shown in Figure2(a). In the Brazilian splitting test, in order to obtain the strain at the center of the rock disc end face, a strain gauge was placed in the horizontal direction. Meanwhile, a strain gauge was placed at the head-disc contact position to determine the change rule of the head-disc load transfer as shown in Figure 2(b).Figure 2
Diagram of strain gauge sticking position. (a) Uniaxial compression test. (b) Brazilian splitting test.
(a)(b)


## 2.2. Experimental Data Processing

### 2.2.1. Mechanical Parameters

According to the uniaxial compression test, the elastic modulusE and Poisson’s ratio μ of rock samples are calculated as shown in Table 1.Table 1
Mechanical parameters of rock samples.
MaterialElastic modulusE (GPa)Poisson’s ratio (μ)Mudstone7.350.236White sandstone13.430.262Green sandstone20.620.283Red sandstone27.400.274Marble53.580.298


### 2.2.2. Head-Disc Contact Angle

In order to calculate the head-disc contact angle when the rock is fractured, the geometric relationship between the head-disc contact angle and the vertical displacement of the head is shown in Figure3, in which A and A′ represent the edge point of the contact area between the head and the disc, respectively. Thus, the head-disc contact angle 2α is determined as(1)2α=2arccosR−∆yR,where R is the disc radius, and ∆y is the head displacement. Based on equation (1) and the test result, the maximum contact angle of different samples is calculated and the results are listed in Table 2.Figure 3
Schematic diagram of head displacement.Table 2
Maximum head-disc contact angle.
MaterialMudstoneWhite sandstoneGreen sandstoneRed sandstoneMarbleElastic modulusE (GPa)7.3513.4320.6227.4053.58Maximum contact angle (°)31.527.725.323.119.0Figure4 shows the relationship between the maximum head-disc contact angle and the elastic modulus of rock under splitting failure. The maximum contact angle decreases with the increase of the elastic modulus of the rock. The relationship of them can be fitted as(2)2αE=−0.0068E2−0.6851E+36.1.Figure 4
Changing law between elastic modulus and maximum head-disc contact angle.


### 2.2.3. Head-Disc Contact Load

The peak contact load between the head and disc is shown in Figure5, which is calculated according to the strain value at the center of the head-disc contact area. The load sharply increases with the increase of the elastic modulus. In the initial stage of loading, the changing rate of the contact peak load of the five samples basically remains unchanged. With the increasing of applied load, the power changing rate of the contact load decreases.Figure 5
Peak value of head-disc contact load.


## 2.2.1. Mechanical Parameters

According to the uniaxial compression test, the elastic modulusE and Poisson’s ratio μ of rock samples are calculated as shown in Table 1.Table 1
Mechanical parameters of rock samples.
MaterialElastic modulusE (GPa)Poisson’s ratio (μ)Mudstone7.350.236White sandstone13.430.262Green sandstone20.620.283Red sandstone27.400.274Marble53.580.298


## 2.2.2. Head-Disc Contact Angle

In order to calculate the head-disc contact angle when the rock is fractured, the geometric relationship between the head-disc contact angle and the vertical displacement of the head is shown in Figure3, in which A and A′ represent the edge point of the contact area between the head and the disc, respectively. Thus, the head-disc contact angle 2α is determined as(1)2α=2arccosR−∆yR,where R is the disc radius, and ∆y is the head displacement. Based on equation (1) and the test result, the maximum contact angle of different samples is calculated and the results are listed in Table 2.Figure 3
Schematic diagram of head displacement.Table 2
Maximum head-disc contact angle.
MaterialMudstoneWhite sandstoneGreen sandstoneRed sandstoneMarbleElastic modulusE (GPa)7.3513.4320.6227.4053.58Maximum contact angle (°)31.527.725.323.119.0Figure4 shows the relationship between the maximum head-disc contact angle and the elastic modulus of rock under splitting failure. The maximum contact angle decreases with the increase of the elastic modulus of the rock. The relationship of them can be fitted as(2)2αE=−0.0068E2−0.6851E+36.1.Figure 4
Changing law between elastic modulus and maximum head-disc contact angle.


## 2.2.3. Head-Disc Contact Load

The peak contact load between the head and disc is shown in Figure5, which is calculated according to the strain value at the center of the head-disc contact area. The load sharply increases with the increase of the elastic modulus. In the initial stage of loading, the changing rate of the contact peak load of the five samples basically remains unchanged. With the increasing of applied load, the power changing rate of the contact load decreases.Figure 5
Peak value of head-disc contact load.


## 3. Numerical Simulation of Brazilian Splitting Test considering Modulus Effect

Through the test, the maximum head-disc contact angle and the peak contact load of the five rock samples are obtained. However, the distribution of the head-disc contact load could not be obtained. Therefore, it is necessary to carry out the numerical simulation.


### 3.1. Numerical Model

The simulation model of the Brazilian splitting test is established in FLAC3D, as shown in Figure 6. The diameter of the arc head is 75 mm, the diameter of the disc is 50 mm, and the thickness is 25 mm. The disc is divided into 15,000 hexahedral elements and 4000 elements in the head. The head is set to a linear elastic body, and the disc adopts an elastic-plastic body. The failure rule obeys the M-C yield criterion. The coordinate origin is established at the center point of the disc end face, and the vertical directions are y axis, x axis along the horizontal direction, and z axis along the thickness direction of the disc. There are no displacements in the x–z plane and only vertical deformation in the y direction. Surface load is applied on the upper and lower faces of the head.Figure 6
Calculation model and grid meshing.In order to compare with the test results, the elastic modulus and Poisson’s ratio of the rock specimens are set to be consistent with the parameters in the Brazilian splitting test, as shown in Table3.Table 3
Parameter settings of Brazilian splitting numerical calculation.
MaterialElastic modulusE (GPa)Tensile strengthσt (MPa)Poisson’s ratio (μ)Cohesion (MPa)Friction angleφ (°)Head210—0.25——Mudstone7.351.590.2362.625White sandstone13.433.220.2624.538Green sandstone20.624.170.2833.830Red sandstone27.446.080.2743.635Marble53.588.930.29850.070


### 3.2. Distribution of Contact Load

The distribution of head-disc contact load during the splitting failure is presented in Figure7. The maximum head-disc contact angle decreases with the increase of elastic modulus. The load-displacement curve of the sample is directly obtained by the FLAC3D simulation software, and the peak point on the curve is the peak contact load. When the elastic modulus increases from 7.35 GPa to 13.43 GPa, the peak contact load increases from 17.5 MPa to 41 MPa. However, when the elastic modulus increases from 13.43 GPa to 20.62 GPa, the peak contact load increases less than 1.5 times, which indicates that the smaller the elastic modulus is, the greater the impact on the peak contact loads.Figure 7
Head-disc contact load distribution.Supposeqθ is the distributed load transmitted by the pressure head to the disc. Jose et al. [20] and Markides and Kourkoulis [21], respectively, put forward different load distribution form of sinusoidal type and parabolic type I which are shown in Figure 8; the load distribution expression is expressed as(3)qθ=Pmaxcosθ−cosα2Rtsinα−αcosα,(4)qθ=3Pmax4πRtsinα1−sin2θsin2α,where R is the disc radius, Pmax is breaking load, t represents disc thickness, and 2α is the head-disc maximum contact angle.Figure 8
Diagram of contact load distribution. (a) Sinusoidal distribution [20]. (b) Parabolic type I distribution [21].
(a)(b)Brazilian discs with different elastic modulus have different head-disc contact angles during splitting failure. For the convenience of analysis, dimensionless processing is processed for Figure7 to obtain Figure 9. The horizontal axis represents the relative position of the head-disc contact angle. The points with abscissa of −1 or 1 represent the contact boundaries on the left and right sides, and the coordinate value on the vertical axis is the relative value of the distributed load.Figure 9
Normalization of head-disc contact load distribution.The distribution of contact load varies with the elastic modulus. According to equations (3) and (4), the distribution of contact load is close to the parabolic curve which is defined as parabolic type I. Meanwhile, the distribution of contact load is very close to the sinusoidal curve which is defined as a sinusoidal type. However, the distribution of contact load with an elastic modulus of 27.44 GPa and 53.58 GPa could not be simply expressed by parabolic type I or sinusoidal type. The contact load distribution of hard rock which is defined as parabolic type II can be fitted as follows:(5)qθ=3Pmax4tRα3α2−θ2.


## 3.1. Numerical Model

The simulation model of the Brazilian splitting test is established in FLAC3D, as shown in Figure 6. The diameter of the arc head is 75 mm, the diameter of the disc is 50 mm, and the thickness is 25 mm. The disc is divided into 15,000 hexahedral elements and 4000 elements in the head. The head is set to a linear elastic body, and the disc adopts an elastic-plastic body. The failure rule obeys the M-C yield criterion. The coordinate origin is established at the center point of the disc end face, and the vertical directions are y axis, x axis along the horizontal direction, and z axis along the thickness direction of the disc. There are no displacements in the x–z plane and only vertical deformation in the y direction. Surface load is applied on the upper and lower faces of the head.Figure 6
Calculation model and grid meshing.In order to compare with the test results, the elastic modulus and Poisson’s ratio of the rock specimens are set to be consistent with the parameters in the Brazilian splitting test, as shown in Table3.Table 3
Parameter settings of Brazilian splitting numerical calculation.
MaterialElastic modulusE (GPa)Tensile strengthσt (MPa)Poisson’s ratio (μ)Cohesion (MPa)Friction angleφ (°)Head210—0.25——Mudstone7.351.590.2362.625White sandstone13.433.220.2624.538Green sandstone20.624.170.2833.830Red sandstone27.446.080.2743.635Marble53.588.930.29850.070


## 3.2. Distribution of Contact Load

The distribution of head-disc contact load during the splitting failure is presented in Figure7. The maximum head-disc contact angle decreases with the increase of elastic modulus. The load-displacement curve of the sample is directly obtained by the FLAC3D simulation software, and the peak point on the curve is the peak contact load. When the elastic modulus increases from 7.35 GPa to 13.43 GPa, the peak contact load increases from 17.5 MPa to 41 MPa. However, when the elastic modulus increases from 13.43 GPa to 20.62 GPa, the peak contact load increases less than 1.5 times, which indicates that the smaller the elastic modulus is, the greater the impact on the peak contact loads.Figure 7
Head-disc contact load distribution.Supposeqθ is the distributed load transmitted by the pressure head to the disc. Jose et al. [20] and Markides and Kourkoulis [21], respectively, put forward different load distribution form of sinusoidal type and parabolic type I which are shown in Figure 8; the load distribution expression is expressed as(3)qθ=Pmaxcosθ−cosα2Rtsinα−αcosα,(4)qθ=3Pmax4πRtsinα1−sin2θsin2α,where R is the disc radius, Pmax is breaking load, t represents disc thickness, and 2α is the head-disc maximum contact angle.Figure 8
Diagram of contact load distribution. (a) Sinusoidal distribution [20]. (b) Parabolic type I distribution [21].
(a)(b)Brazilian discs with different elastic modulus have different head-disc contact angles during splitting failure. For the convenience of analysis, dimensionless processing is processed for Figure7 to obtain Figure 9. The horizontal axis represents the relative position of the head-disc contact angle. The points with abscissa of −1 or 1 represent the contact boundaries on the left and right sides, and the coordinate value on the vertical axis is the relative value of the distributed load.Figure 9
Normalization of head-disc contact load distribution.The distribution of contact load varies with the elastic modulus. According to equations (3) and (4), the distribution of contact load is close to the parabolic curve which is defined as parabolic type I. Meanwhile, the distribution of contact load is very close to the sinusoidal curve which is defined as a sinusoidal type. However, the distribution of contact load with an elastic modulus of 27.44 GPa and 53.58 GPa could not be simply expressed by parabolic type I or sinusoidal type. The contact load distribution of hard rock which is defined as parabolic type II can be fitted as follows:(5)qθ=3Pmax4tRα3α2−θ2.


## 4. Correction of Theoretical Tensile Strength of Brazilian Disc Splitting

### 4.1. Analytical Solution of Brazilian Splitting Stress under Arc Loading

Stresses under Sinusoidal contact load is expressed as follows [20]:(6)σx=−Pmax2cos2αsinα+sinα−3αcosα3πRtsinα−αcosα,σy=Pmax2cos2αsinα−5sinα+3αcosα3πRtsinα−αcosα.For parabolic type I, the stress solution in the disc is given as follows [21]:(7)σx=−Pmax−45sin2α+90αcos2α+60α2sin2α+16α516πRtα5,σy=−Pmax45sin2α−90αcos2α−60α2sin2α+16α516πRtα5.For parabolic type II, the stress calculation formula of the center point of the end face are established as follows.


#### 4.1.1. Stress of Boundary Point in Brazilian Disc

In Figure10, take the tiny arc length dl corresponding to the contact angle dθ, and the tiny force of the disk per unit thickness is dF=qθdl=qθRdθ. Under the action of dF, the two radial stresses at any point M on the disc boundary are calculated, respectively,(8)dσr1=−2qθcosθ1cosθ+sinθ1sinθRπr1dθ,dσr2=−2qθcosθ2cosθ+sinθ2sinθRπr2dθ,where θ1,θ2 are the angle between the upper and lower infinitesimal and the M point connection and the vertical direction, radian; r1,r2 are the distances of AM and BM.Figure 10
Stress analysis on the Brazilian disc boundary.According to the coordinate transformation formula in elastic mechanics, the stressdσn and dστ in NM direction and τ direction of point M can be obtained as(9)dσn=dσr1cos2π2−θ2+dσr2cos2−π2−θ1=−2dFπcosθ1cosθ+sinθ1sinθsin2θ1r1+cosθ2cosθ+sinθ2sinθsin2θ2r2dστ=dσr1cosπ2−θ2sinπ2−θ2+dσr2cos−π2−θ1sin−π2−θ1=−2dFπcosθ1cosθ+sinθ1sinθsinθ2cosθ2r1−cosθ2cosθ+sinθ2sinθsinθ1cosθ1r2.In∆MAN and ∆MBN:(10)r1=2Rcosπ2−θ2=2Rsinθ2,r2=2Rcosπ2−θ1=2Rsinθ1.Substitute equation (10) into equation (9) yields(11)dσn=−dFπRcosθsinθ1+θ2+sinθ1−cosθ1+θ2,dστ=0.In∆AMB(12)∠AMB=π−θ1+θ2,r1sinθ2=2Rcosθsinθ1+θ2,sinθ1+θ2=cosθ.Substitute equation (12) into equation (11), and we receive(13)dσn=−dFπRcos2θ+sinθ.Integrate equation (13) along the contact area of the disc under load distribution. The stress at any point M on the boundary can be determined as(14)σn=∫−αRαRdσn=3Pmax4πtRα3sin2αsin2α−12sin2α+12α+18sin4α,στ=0.From equation (14), there exists a constant stress on the disc boundary. Since the disc boundary is free, it is necessary to apply the equal magnitude and opposite direction radial stress solution of the semi-infinite plane to the stress field in equation (13) to meet the actual free boundary conditions.


#### 4.1.2. Stresses of Inner Point in Brazilian Disc

As shown in Figure11, the stresses at any point M in the disc can also be deduced from equation (8). By using the coordinate transformation formula, the microelement stresses component in the disc in the rectangular coordinate system are calculated as(15)dσx=−2dFπcosθ1cosθ+sinθ1sinθsin2θ1r1+cosθ2cosθ+sinθ2sinθsin2θ2r2,dσy=−2dFπcosθ1cosθ+sinθ1sinθcos2θ1r1+cosθ2cosθ+sinθ2sinθcos2θ2r2,dτxy=−2dFπcosθ1cosθ+sinθ1sinθsinθ1cosθ1r1−cosθ2cosθ+sinθ2sinθsinθ2cosθ2r2,Figure 11
Stress analysis at any point in Brazilian disc.The integration is carried out along the contact area of the disc under the load distribution, then the tensile stress at any pointM in the disc can be determined as(16)σx=−3Pmax4πtα3∫−ααf1x,y,θ⋅α2−θ2dθ+fαR,σy=−3Pmax4πtα3∫−ααf2x,y,θ⋅α2−θ2dθ+fαR,τxy=−3Pmax4πtα3∫−ααf3x,y,θ⋅α2−θ2dθ,where(17)f1x,y,θ=Rcosθ−ycosθ+x+Rsinθsinθx+Rsinθ2x+Rsinθ2+Rcosθ−y22+Rcosθ+ycosθ+x+Rsinθsinθx+Rsinθ2x+Rsinθ2+Rcosθ+y22,f2x,y,θ=Rcosθ−ycosθ+x+RsinθsinθRcosθ−y2x+Rsinθ2+Rcosθ−y22+Rcosθ+ycosθ+x+RsinθsinθRcosθ+y2x+Rsinθ2+Rcosθ+y22,f3x,y,θ=Rcosθ−ycosθ+x+RsinθsinθRcosθ−yx+Rsinθx+Rsinθ2+Rcosθ−y22+Rcosθ+ycosθ+x+RsinθsinθRcosθ+yx+Rsinθx+Rsinθ2+Rcosθ+y22,fα=sin2αsin2α−12sin2α+12α+18sin4α.According to equation (16), the stresses at the central point (0, 0) are expressed as(18)σx=3Pmax4πtRα3sin2αsin2α+52α+18sin4α−32sin2α−αsin2α,σy=3Pmax4πtRα3−3sin2αsin2α+12α−38sin4α+12sin2α−4αsin2α,τxy=0.


### 4.2. Correction of Rock Tensile Strength considering Modulus Effect

Combining equations (6), (7), and (18), the tensile stress of arc loading is determined as(19)σt=−Pmax−45sin2α+90αcos2α+60α2sin2α+16α516πRtα5,softrock,−Pmax2cos2αsinα+sinα−3αcosα3πRtsinα−αcosα,mediumhardrock,3Pmax4πtRα3sin2αsin2α+52α+18sin4α−32sin2α−αsin2αhardrock.According to the test results, the peak of the head-disc contact load and contact angle are related to the elastic modulus. Therefore, equation (19) should be modified. Considering the influence of elastic modulus, the correction coefficient k related to elastic modulus is introduced, which is defined as(20)k=σt2Pmax/πdt.Substitute equation (19) into equation (20), and the correction coefficient k is calculated as(21)k=45sin2α−90αcos2α−60α2sin2α−16α516α5,softrock3αcosα−2cos2αsinα−sinα3sinα−αcosα,mediumhardrock,34α3sin2αsin2α+52α+18sin4α−32sin2α−αsin2α,hardrock.In equation (21), the maximum contact angle 2α is a function of elastic modulus according to equation (2). Thus, the correction coefficient and correction formula of the Brazilian splitting test considering the elastic modulus effect are obtained. Compared with the previous Brazilian splitting experiment, the same correction formula is used to calculate the tensile strength of all types of rocks, so that the tensile strength of some types of rocks are quite different from the actual value; in this paper, the corresponding correction formula and correction coefficient are used to calculate the rock of different hardness, and the resulting tensile strength is closer to the real value, which is more reliable.


### 4.3. Model Verification

In order to verify the reliability of the corrected formula, the Brazilian splitting test results with different elastic modulus are carried out by literatures [9, 10, 22, 23] and these studies are selected to determine the tensile strength, and the relative errors between the calculating value and the test value are listed in Table 4.Table 4
Comparison of calculating results of tensile strength by different methods.
Elastic modulusE (GPa)7.3510.213.4320.6223.4527.43745.253.58Maximum contact angle (°)33.831.73025.52723212019Tensile strength test (MPa)1.592.243.224.174.56.086.267.248.93Wang and Cao [9]1.932.693.774.584.816.476.177.047.75Relative error (%)21.4420.0016.969.726.806.38−1.40−2.76−13.21Huang et al. [10]1.912.653.744.554.786.446.396.897.92Relative error (%)20.3018.2016.119.056.205.842.00−4.83−11.27Wang et al. [22]1.922.673.764.574.796.466.097.017.69Relative error (%)21.0519.0016.679.496.406.20−2.77−3.18−13.87Satoh [23]1.942.603.774.584.826.486.076.737.91Relative error (%)21.8116.0017.249.937.006.56−3.00−7.04−11.44In this study (MPa)1.632.273.294.264.446.196.367.18.77Relative error (%)2.521.202.172.16−1.401.811.60−1.931.79According to Table4, when the elastic modulus of the rock and the maximum contact angle are the same and when the elastic modulus is less than or equal to 27.4 GPa, Wang et al. obtain a large correction value of tensile strength; when the modulus of elasticity is greater than 27.4 GPa, the correction value of tensile strength obtained by them is small, and the absolute value of the relative error of the obtained elastic modulus first decreases and then increases with the increase of the elastic modulus, and its discreteness is large. The mean absolute values of relative errors of the four articles by Huang et al. are 11.11%, 10.96%, 10.96%, and 10.42%, respectively. All the tensile strength correction values obtained in this paper are always close to the tensile strength experimental values, and the absolute relative error value is always within 3%. Therefore, equation (19) for the tensile strength of rock proposed in this paper has high accuracy and good stability.


## 4.1. Analytical Solution of Brazilian Splitting Stress under Arc Loading

Stresses under Sinusoidal contact load is expressed as follows [20]:(6)σx=−Pmax2cos2αsinα+sinα−3αcosα3πRtsinα−αcosα,σy=Pmax2cos2αsinα−5sinα+3αcosα3πRtsinα−αcosα.For parabolic type I, the stress solution in the disc is given as follows [21]:(7)σx=−Pmax−45sin2α+90αcos2α+60α2sin2α+16α516πRtα5,σy=−Pmax45sin2α−90αcos2α−60α2sin2α+16α516πRtα5.For parabolic type II, the stress calculation formula of the center point of the end face are established as follows.


### 4.1.1. Stress of Boundary Point in Brazilian Disc

In Figure10, take the tiny arc length dl corresponding to the contact angle dθ, and the tiny force of the disk per unit thickness is dF=qθdl=qθRdθ. Under the action of dF, the two radial stresses at any point M on the disc boundary are calculated, respectively,(8)dσr1=−2qθcosθ1cosθ+sinθ1sinθRπr1dθ,dσr2=−2qθcosθ2cosθ+sinθ2sinθRπr2dθ,where θ1,θ2 are the angle between the upper and lower infinitesimal and the M point connection and the vertical direction, radian; r1,r2 are the distances of AM and BM.Figure 10
Stress analysis on the Brazilian disc boundary.According to the coordinate transformation formula in elastic mechanics, the stressdσn and dστ in NM direction and τ direction of point M can be obtained as(9)dσn=dσr1cos2π2−θ2+dσr2cos2−π2−θ1=−2dFπcosθ1cosθ+sinθ1sinθsin2θ1r1+cosθ2cosθ+sinθ2sinθsin2θ2r2dστ=dσr1cosπ2−θ2sinπ2−θ2+dσr2cos−π2−θ1sin−π2−θ1=−2dFπcosθ1cosθ+sinθ1sinθsinθ2cosθ2r1−cosθ2cosθ+sinθ2sinθsinθ1cosθ1r2.In∆MAN and ∆MBN:(10)r1=2Rcosπ2−θ2=2Rsinθ2,r2=2Rcosπ2−θ1=2Rsinθ1.Substitute equation (10) into equation (9) yields(11)dσn=−dFπRcosθsinθ1+θ2+sinθ1−cosθ1+θ2,dστ=0.In∆AMB(12)∠AMB=π−θ1+θ2,r1sinθ2=2Rcosθsinθ1+θ2,sinθ1+θ2=cosθ.Substitute equation (12) into equation (11), and we receive(13)dσn=−dFπRcos2θ+sinθ.Integrate equation (13) along the contact area of the disc under load distribution. The stress at any point M on the boundary can be determined as(14)σn=∫−αRαRdσn=3Pmax4πtRα3sin2αsin2α−12sin2α+12α+18sin4α,στ=0.From equation (14), there exists a constant stress on the disc boundary. Since the disc boundary is free, it is necessary to apply the equal magnitude and opposite direction radial stress solution of the semi-infinite plane to the stress field in equation (13) to meet the actual free boundary conditions.


### 4.1.2. Stresses of Inner Point in Brazilian Disc

As shown in Figure11, the stresses at any point M in the disc can also be deduced from equation (8). By using the coordinate transformation formula, the microelement stresses component in the disc in the rectangular coordinate system are calculated as(15)dσx=−2dFπcosθ1cosθ+sinθ1sinθsin2θ1r1+cosθ2cosθ+sinθ2sinθsin2θ2r2,dσy=−2dFπcosθ1cosθ+sinθ1sinθcos2θ1r1+cosθ2cosθ+sinθ2sinθcos2θ2r2,dτxy=−2dFπcosθ1cosθ+sinθ1sinθsinθ1cosθ1r1−cosθ2cosθ+sinθ2sinθsinθ2cosθ2r2,Figure 11
Stress analysis at any point in Brazilian disc.The integration is carried out along the contact area of the disc under the load distribution, then the tensile stress at any pointM in the disc can be determined as(16)σx=−3Pmax4πtα3∫−ααf1x,y,θ⋅α2−θ2dθ+fαR,σy=−3Pmax4πtα3∫−ααf2x,y,θ⋅α2−θ2dθ+fαR,τxy=−3Pmax4πtα3∫−ααf3x,y,θ⋅α2−θ2dθ,where(17)f1x,y,θ=Rcosθ−ycosθ+x+Rsinθsinθx+Rsinθ2x+Rsinθ2+Rcosθ−y22+Rcosθ+ycosθ+x+Rsinθsinθx+Rsinθ2x+Rsinθ2+Rcosθ+y22,f2x,y,θ=Rcosθ−ycosθ+x+RsinθsinθRcosθ−y2x+Rsinθ2+Rcosθ−y22+Rcosθ+ycosθ+x+RsinθsinθRcosθ+y2x+Rsinθ2+Rcosθ+y22,f3x,y,θ=Rcosθ−ycosθ+x+RsinθsinθRcosθ−yx+Rsinθx+Rsinθ2+Rcosθ−y22+Rcosθ+ycosθ+x+RsinθsinθRcosθ+yx+Rsinθx+Rsinθ2+Rcosθ+y22,fα=sin2αsin2α−12sin2α+12α+18sin4α.According to equation (16), the stresses at the central point (0, 0) are expressed as(18)σx=3Pmax4πtRα3sin2αsin2α+52α+18sin4α−32sin2α−αsin2α,σy=3Pmax4πtRα3−3sin2αsin2α+12α−38sin4α+12sin2α−4αsin2α,τxy=0.


## 4.1.1. Stress of Boundary Point in Brazilian Disc

In Figure10, take the tiny arc length dl corresponding to the contact angle dθ, and the tiny force of the disk per unit thickness is dF=qθdl=qθRdθ. Under the action of dF, the two radial stresses at any point M on the disc boundary are calculated, respectively,(8)dσr1=−2qθcosθ1cosθ+sinθ1sinθRπr1dθ,dσr2=−2qθcosθ2cosθ+sinθ2sinθRπr2dθ,where θ1,θ2 are the angle between the upper and lower infinitesimal and the M point connection and the vertical direction, radian; r1,r2 are the distances of AM and BM.Figure 10
Stress analysis on the Brazilian disc boundary.According to the coordinate transformation formula in elastic mechanics, the stressdσn and dστ in NM direction and τ direction of point M can be obtained as(9)dσn=dσr1cos2π2−θ2+dσr2cos2−π2−θ1=−2dFπcosθ1cosθ+sinθ1sinθsin2θ1r1+cosθ2cosθ+sinθ2sinθsin2θ2r2dστ=dσr1cosπ2−θ2sinπ2−θ2+dσr2cos−π2−θ1sin−π2−θ1=−2dFπcosθ1cosθ+sinθ1sinθsinθ2cosθ2r1−cosθ2cosθ+sinθ2sinθsinθ1cosθ1r2.In∆MAN and ∆MBN:(10)r1=2Rcosπ2−θ2=2Rsinθ2,r2=2Rcosπ2−θ1=2Rsinθ1.Substitute equation (10) into equation (9) yields(11)dσn=−dFπRcosθsinθ1+θ2+sinθ1−cosθ1+θ2,dστ=0.In∆AMB(12)∠AMB=π−θ1+θ2,r1sinθ2=2Rcosθsinθ1+θ2,sinθ1+θ2=cosθ.Substitute equation (12) into equation (11), and we receive(13)dσn=−dFπRcos2θ+sinθ.Integrate equation (13) along the contact area of the disc under load distribution. The stress at any point M on the boundary can be determined as(14)σn=∫−αRαRdσn=3Pmax4πtRα3sin2αsin2α−12sin2α+12α+18sin4α,στ=0.From equation (14), there exists a constant stress on the disc boundary. Since the disc boundary is free, it is necessary to apply the equal magnitude and opposite direction radial stress solution of the semi-infinite plane to the stress field in equation (13) to meet the actual free boundary conditions.


## 4.1.2. Stresses of Inner Point in Brazilian Disc

As shown in Figure11, the stresses at any point M in the disc can also be deduced from equation (8). By using the coordinate transformation formula, the microelement stresses component in the disc in the rectangular coordinate system are calculated as(15)dσx=−2dFπcosθ1cosθ+sinθ1sinθsin2θ1r1+cosθ2cosθ+sinθ2sinθsin2θ2r2,dσy=−2dFπcosθ1cosθ+sinθ1sinθcos2θ1r1+cosθ2cosθ+sinθ2sinθcos2θ2r2,dτxy=−2dFπcosθ1cosθ+sinθ1sinθsinθ1cosθ1r1−cosθ2cosθ+sinθ2sinθsinθ2cosθ2r2,Figure 11
Stress analysis at any point in Brazilian disc.The integration is carried out along the contact area of the disc under the load distribution, then the tensile stress at any pointM in the disc can be determined as(16)σx=−3Pmax4πtα3∫−ααf1x,y,θ⋅α2−θ2dθ+fαR,σy=−3Pmax4πtα3∫−ααf2x,y,θ⋅α2−θ2dθ+fαR,τxy=−3Pmax4πtα3∫−ααf3x,y,θ⋅α2−θ2dθ,where(17)f1x,y,θ=Rcosθ−ycosθ+x+Rsinθsinθx+Rsinθ2x+Rsinθ2+Rcosθ−y22+Rcosθ+ycosθ+x+Rsinθsinθx+Rsinθ2x+Rsinθ2+Rcosθ+y22,f2x,y,θ=Rcosθ−ycosθ+x+RsinθsinθRcosθ−y2x+Rsinθ2+Rcosθ−y22+Rcosθ+ycosθ+x+RsinθsinθRcosθ+y2x+Rsinθ2+Rcosθ+y22,f3x,y,θ=Rcosθ−ycosθ+x+RsinθsinθRcosθ−yx+Rsinθx+Rsinθ2+Rcosθ−y22+Rcosθ+ycosθ+x+RsinθsinθRcosθ+yx+Rsinθx+Rsinθ2+Rcosθ+y22,fα=sin2αsin2α−12sin2α+12α+18sin4α.According to equation (16), the stresses at the central point (0, 0) are expressed as(18)σx=3Pmax4πtRα3sin2αsin2α+52α+18sin4α−32sin2α−αsin2α,σy=3Pmax4πtRα3−3sin2αsin2α+12α−38sin4α+12sin2α−4αsin2α,τxy=0.


## 4.2. Correction of Rock Tensile Strength considering Modulus Effect

Combining equations (6), (7), and (18), the tensile stress of arc loading is determined as(19)σt=−Pmax−45sin2α+90αcos2α+60α2sin2α+16α516πRtα5,softrock,−Pmax2cos2αsinα+sinα−3αcosα3πRtsinα−αcosα,mediumhardrock,3Pmax4πtRα3sin2αsin2α+52α+18sin4α−32sin2α−αsin2αhardrock.According to the test results, the peak of the head-disc contact load and contact angle are related to the elastic modulus. Therefore, equation (19) should be modified. Considering the influence of elastic modulus, the correction coefficient k related to elastic modulus is introduced, which is defined as(20)k=σt2Pmax/πdt.Substitute equation (19) into equation (20), and the correction coefficient k is calculated as(21)k=45sin2α−90αcos2α−60α2sin2α−16α516α5,softrock3αcosα−2cos2αsinα−sinα3sinα−αcosα,mediumhardrock,34α3sin2αsin2α+52α+18sin4α−32sin2α−αsin2α,hardrock.In equation (21), the maximum contact angle 2α is a function of elastic modulus according to equation (2). Thus, the correction coefficient and correction formula of the Brazilian splitting test considering the elastic modulus effect are obtained. Compared with the previous Brazilian splitting experiment, the same correction formula is used to calculate the tensile strength of all types of rocks, so that the tensile strength of some types of rocks are quite different from the actual value; in this paper, the corresponding correction formula and correction coefficient are used to calculate the rock of different hardness, and the resulting tensile strength is closer to the real value, which is more reliable.


## 4.3. Model Verification

In order to verify the reliability of the corrected formula, the Brazilian splitting test results with different elastic modulus are carried out by literatures [9, 10, 22, 23] and these studies are selected to determine the tensile strength, and the relative errors between the calculating value and the test value are listed in Table 4.Table 4
Comparison of calculating results of tensile strength by different methods.
Elastic modulusE (GPa)7.3510.213.4320.6223.4527.43745.253.58Maximum contact angle (°)33.831.73025.52723212019Tensile strength test (MPa)1.592.243.224.174.56.086.267.248.93Wang and Cao [9]1.932.693.774.584.816.476.177.047.75Relative error (%)21.4420.0016.969.726.806.38−1.40−2.76−13.21Huang et al. [10]1.912.653.744.554.786.446.396.897.92Relative error (%)20.3018.2016.119.056.205.842.00−4.83−11.27Wang et al. [22]1.922.673.764.574.796.466.097.017.69Relative error (%)21.0519.0016.679.496.406.20−2.77−3.18−13.87Satoh [23]1.942.603.774.584.826.486.076.737.91Relative error (%)21.8116.0017.249.937.006.56−3.00−7.04−11.44In this study (MPa)1.632.273.294.264.446.196.367.18.77Relative error (%)2.521.202.172.16−1.401.811.60−1.931.79According to Table4, when the elastic modulus of the rock and the maximum contact angle are the same and when the elastic modulus is less than or equal to 27.4 GPa, Wang et al. obtain a large correction value of tensile strength; when the modulus of elasticity is greater than 27.4 GPa, the correction value of tensile strength obtained by them is small, and the absolute value of the relative error of the obtained elastic modulus first decreases and then increases with the increase of the elastic modulus, and its discreteness is large. The mean absolute values of relative errors of the four articles by Huang et al. are 11.11%, 10.96%, 10.96%, and 10.42%, respectively. All the tensile strength correction values obtained in this paper are always close to the tensile strength experimental values, and the absolute relative error value is always within 3%. Therefore, equation (19) for the tensile strength of rock proposed in this paper has high accuracy and good stability.


## 5. Conclusion

In this paper, the head-disc contact angle and the distribution of contact load in the Brazilian disc splitting test considering the modulus effect are studied by combining laboratory test, theoretical analysis, and numerical simulation, and the calculating formula of tensile strength is modified. The main conclusions are listed as follows:(1)
The elastic modulus of rock has an important influence on the contact behaviour between the head and the disk. The peak value of the interface load increases with the increase of the elastic modulus, but the load distribution range decreases; on the contrary, the peak value decreases as the disk becomes soft. The contact area between the two becomes larger due to the increase of disc deformation.(2)
According to the range of elastic modulus of the disc, the distribution of contact load is simplified into three types: parabolic type I, sinusoidal type, and parabolic type II distribution, which are suitable for soft rock, medium-hard rock, and hard rock, respectively.(3)
Considering the influence of rock lithology, the calculation method of disc tensile strength proposed in this paper has higher calculation accuracy than the original method. The error between the tensile strength obtained by the correction formula and the experimental value is within 3%.



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*Source: 6632624-2023-03-03.xml*