Q.Zhang·X.P.Zhang·P.Q.Ji

Abstract The Brazilian test is a widely used method for determining the tensile strength of rocks and for calibrating parameters in bonded particle models(BPM s).In previous studies,the Brazilian disc has typically been trimmed from a compacted rectangular specimen.The present study shows that different tensile strength values are obtained depending on the compressive loading direction.Several measures are proposed to reduce the anisotropy of the disc.The results reveal that the anisotropy of the disc is significantly influenced by the compactibility of the specimen from which it is trimmed.A new method is proposed in which the Brazilian disc is directly generated with a particle boundary,effectively reducing the anisotropy.The stiffness(particle and bond)and strength(bond)of the boundary are set at less than and greater than those of the disc assembly,respectively,which significantly decreases the stress concentration at the boundary contacts and prevents breakage of the boundary particle bonds.This leads to a significant reduction in the anisotropy of the disc and the discreteness of the tensile strength.This method is more suitable for carrying out a realistic Brazilian test for homogeneous rock-like material in the BPM.

Keywords Bonded-particle model·Brazilian disc·Anisotropy ·Compactibility ·Particle boundary

1 Introduction

The Brazilian disc test is a simple, effective and efficient testing method commonly used to obtain mechanical parameters of intact rock, including tensile strength, fracture toughness,and elastic modulus. The method has gained popularity since it was first introduced by the International Society for Rock Mechanics(ISRM)[1]in 1978 as a standard approach for determining the indirect tensile strength of rock masses.As research has continued,additional factors influencing tensile strength have been considered within the framework of the Brazilian test,such as differences in the elastic properties of rock under tension and compression[2,3],length-to-diameter ratio[4,5],size effect and boundary conditions[6],and loading platen shapes[7].Numerical simulation has been found to be an effective supplementary technique for studying the cracking processes and failure mechanisms of Brazilian discs[8–13]and rectangular specimens[14–17].The bonded-particle model(BPM),a discrete element method,is a common approach for studying the mechanisms of crack initiation and propagation in rock and rock-like material[18–23].The BPM can directly generate compression-induced tensile cracks to reproduce the behavior of hard rock,and thus the model has been used extensively to study the cracking processes in Brazilian tensile tests.For example,Potyondy and Cundall[24]explored the relationship between Brazilian tensile strength obtained by the BPM and mode I fracture toughness. In another work, Cai and Kaiser [25] employed a finite element method/discrete element method(FEM/DEM)coupled approach to simulate the fracturing processes in Brazilian tensile tests.A work by Yang and Huang[26]used a two-dimensional Particle Flow Code(PFC2D)to study the tensile failure behavior of jointed rock mass disc specimens,while Ghazvinian et al.[27]used the PFC2D to evaluate mixed-mode crack propagation in low brittle rock-like materials under Brazilian tensile tests.In addition,Zhang and Wong[21]used the PFC2D to examine the cracking processes in the Brazilian tensile test under different loading rates.

The Brazilian test is commonly used to calibrate the tensile strength of specimens in the BPM.For the simulation of generally homogeneous rock such as granite or marble,the disc is typically considered a homogeneous specimen,and the isotropic stress of the assembled disc should be uniform.However,the present study indicates that the stress distribution in the Brazilian disc is non-uniform,as generated by the method in the PFC manual.Several measures are investigated to reduce the anisotropy of the specimen.The aim of the present study,therefore,is to devise a new method for generating an isotropic Brazilian disc.

2 Modeling a Brazilian disc in the BPM

2.1 Bonded-particle model

The bonded-particle model,a basic model found in the PFC commercial software package,has been widely used to simulate rock material behavior in fields such as m ining,underground excavation,hydraulic fracturing,and slope sliding.Two types of BPM are available in the PFC:the contact bond model and the parallel bond model.The former can transmit normal and shear forces between adjacent particles and is generally used to study soil behavior[28,29].The latter can transmit both force and moment between adjacent particles,and is frequently used to study rock and rock-like material behavior[30–33].The model takes the rock behaviors as a cemented granular material of complex-shaped grains.When the force applied on a bond exceeds its normal or shear strength,the bond will break and a micro-crack is formed.The BPM can directly generate compression induced tensile cracks to reproduce the behavior of rock-like material;hence,it is an effective tool for simulating the Brazilian test.Figure 1 shows a typical stress–strain curve of a Brazilian test for a brittle rock material in the BPM.The Brazilian disc is composed of gray particles as shown in Fig.2.Two vertical lines(gray)on the two sides of the disc represent the loading platens.Cracking processes and force distribution at different loading stages(marked by points A,B,C,D,and E)are plotted in Fig.2.In the upper group,the white and red segments represent micro-tensile and microshear cracks,respectively.It is clear that the cracks initiate and extend to split the disc along the loading direction.At the early stage,no micro-cracks are present(Fig.2a).At the stress level of the peak load,some discrete micro-cracks appear within the Brazilian disc,which largely consist of micro-tensile cracks(white)(Fig.2b).As the load continues(post-peak),the cracks extend towards the loading boundaries(Fig.2c,d).At the end of loading,a macro-crack path forms,and the Brazilian disc is split into two parts,which consist predominantly of micro-tensile cracks(white),indicating that bond breaks within the Brazilian disc are induced by tensile stress.This phenomenon is confirmed by the force distribution within the Brazilian disc in the bottom group. Thediscrete straight segments represent parallel bond forces,and the black and red segments represent compressive and tensile forces,respectively.The thickness and orientation of the segments indicate the force magnitude and direction,respectively.At the loading boundaries,compressive forces are dominant(black),while inside the Brazilian disc,especially along the loading direction,the force distribution is predominantly tensile(red).The Brazilian disc is split by the tensile failure,and the tensile strength can be calculated by the peak compressive force.This shows that the cracking processes and force distribution of the Brazilian disc in the BPM are comparable to those obtained by theoretical analysis[34,35]and laboratory experiments[36,37].

Fig.1 Stress–strain curve of a Brazilian test in the BPM

2.2 The specimen genesis procedure

In a study by Potyondy and Cundall[24],the authors presented a specimen genesis procedure and calibration process for the Brazilian tensile test.This method is widely used to calibrate tensile strength when modeling rock material in the BPM,as shown in Table 1.The specimen genesis procedure is described in the following five steps:

(1)Compact initial assembly:A rectangular specimen(length H,width W)composed of arbitrarily placed particles is generated,which is confined by four frictionless walls.Under zero friction,the specimen is rearranged and the assembly can reach static equilibrium.

(2)Install specified isotropic stress:To achieve a specified isotropic stress,σo(defined as the average direct stresses),the radii of all particles are reduced uniformly.

Fig.2 Cracking processes(upper group)and force distribution(bottom group)corresponding to the five loading stages marked in Fig.1.The white and red segments denote micro-tensile and micro-shear cracks,respectively.Tensile and compressive forces are shown in red and black,respectively.The orientation and thickness of the lines represent force direction and magnitude,respectively

(3)Delete “ floating”particles:A large number of “ floating”particles(defined as fewer than three contacts)are contained in models.To obtain a more densely bonded assembly,it is necessary to eliminate the “ floating”particles.

(4)Install parallel bonds:Parallel bonds are installed throughout the assembly between all particles.Then all particles are assigned a friction coefficient of μ.

(5)The specimen is trimmed into a circular shape with a diameter equal to W.The Brazilian disc is in contact with the lateral walls.

The specimen is loaded by moving the lateral walls towards each other at velocity Vp,as shown in Fig.3.To obtain the Brazilian tensile strength,a peak compressive force(F)acting on the loading platens is recorded during the test.The tensile stress σtis calculated by:

where R and t are the radius and thickness of the Brazilian disc,respectively.

Theoretically,the force distribution in the Brazilian disc is uniform,and the locked-in forces are low.If the loading platens are rotated at different angles,such as 15°,60°,and 90°,the stress–strain curves and the peak tensile stress should be comparable.To investigate the isotropy of the modeled Brazilian disc,the loading walls are rotated at angles of 15°,30°,45°,60°,75°,and 90°,as shown in Fig.4.

2.3 Model parameters

In the present study,the mechanical properties of Belgian“blue”limestone are calibrated.The uniaxial compressive strength and tensile strength of the limestone are 100–130 MPa and about 17 MPa,respectively,based on laboratory experiments conducted by Lavrov et al.[38].The rectangular specimen is 113 mm in width and 226 mm in length,and consists of about 10000 particles.The particle radius follows a uniform distribution ranging from a minimum of Rmin=0.65mm to a maximum of Rmax=1.0mm.The micro-parameters of the BPM are given in Table 2.The Brazilian disc,which contains about 3900 particles,has a diameter of 113 mm,and is trimmed from a rectangular specimen(Method 1 in Table 3).A loading rate of 0.01 m/s is adopted in the Brazilian tensile test to carry out a quasi-static loading[21].The uniaxial compressive strength and tensile strength of the specimen obtained from Method 1 are approximately 95.58 and 17.56 MPa,respectively.Unless otherwise stated,all four models in the present study use the same micro-parameters,as listed in Table 2.

3 Measures to reduce the anisotropy of a Brazilian disc

A method,defined as Method 1 in Table 3,is used to determine the tensile strength of a Brazilian disc,as suggested by the manual[39].To eliminate the effect of arbitrarily generated particle assembly with different particle radii and micro-strength values,ten sets of random numbers are used for modeling the specimen.Figure 5 shows the ten curves of peak force versus different rotation angles.Note that if the fracture surface is not parallel to the compressive loading direction,the Brazilian tensile test is considered invalid.Thepeak force of the invalid test is not plotted in Fig.5.For example,there are two invalid tests in the curve of random number R3(marked by an open triangle),corresponding to rotation angles of 15°and 45°.It is clear that different random numbers correspond to different peak forces.The average peak forces corresponding to different rotation angles are marked by solid pentagrams.The peak forces versus rotation angle of 0°are comparable,about 3.2×106N,corresponding to different random numbers.However,when the rotation angle is greater than 15°,the peak force decreases significantly,and

the average force is about 1.6×106N.The reason may be that the compactibility of rectangular specimens in the width direction is greater than that in other directions.Thus the Brazilian tensile strength shows significant anisotropy as the loading walls are rotated.

Table1 References using Brazilian test to calibrate tensile strength of specimens in the BPM

Fig.3 Sketch of the Brazilian disc trimmed from a rectangular specimen and the loading of a Brazilian tensile test

To reduce the anisotropy of the Brazilian disc,two methods are carried out.One method generates a square specimen with dimensions of 113mm×113mm,which is defined as Method 2 in Table 3.The Brazilian disc is tangential to the loading walls.The compactibility of the disc is improved in the direction of 90°compared with that in Method 1.Figure 6 shows the curves of peak force versus rotation angles of ten sets of random numbers.The variations in peak force are large,corresponding to different random numbers.The average peak forces of rotation angles 0°and 90°are comparable,about 3.14×106N and 3.23×106N,respectively.When the rotation angle is between 15°and 75°,the average force decreases to about 1.4×106N,which is about half that in the horizontal or vertical direction,indicating that the tensile strength is closely related to the compactibility of the Brazilian disc.If the compactibility of the disc is greater in the loading direction,the peak forces are greater and less scattered.

Fig.4 The loading walls(gray lines)rotated at different angles of a 0°,b 15°,c 30°,d 45°,e 60°,f 75°,and g 90°

Table2 Micro-parameters of the model

In another method(defined as Method 3 in Table 3),the Brazilian disc is trimmed from a larger square specimen(226mm×226mm),with a diameter of 113 mm.A large number of invalid tests are obtained with this method,as shown in Fig.7,and the peak forces present a discrete distribution corresponding to different random numbers,especially for a rotation angle of0°.There is a large difference in peak force between the random numbers R5(0.83×106N)and R10(3.04 × 106N)versus a rotation angle of 0°.To assess the discreteness of peak forces among the four methods,standard deviations are calculated,which show that discreteness increases as the standard deviation increases(Table 4).In Method 1,the peak forces with the greatest discreteness correspond to rotation angles of 45°and 60°,and in Method 2,the greatest discreteness corresponds to a rotation angle of 75°.Thus it appears that the discreteness of the peak force is related to the compactibility of the Brazilian disc in the loading direction.For Method 3,the largest value is 0.68,corresponding to an angle of 0°.The peak forces of R5 and R10 do not significantly affect the average peak force,which changes from 1.65×106to 1.56×106N as the data are removed.To achieve an average of these ten values that is sufficient and effective,the peak forces of R5 and R10 versus a rotation angle of 0°are removed.The average forces corresponding to different rotation angles are comparable,about 1.6×106N.Although the average Brazilian tensile strengths obtained with Method 3 are comparable,the discreteness of the peak forces is large compared with those obtained in Method 4,which will be discussed later in this section.In addition,an excessive number of invalid Brazilian tests occur with the use of Method 3.To better assess the anisotropy of the Brazilian disc specimen,an anisotropic ratio is listed in Table 3,defined as the ratio of the maximum average force to minimum average force.The anisotropic ratios are 2.48(3.30/1.33),2.56(3.23/1.26),and 1.19(1.60/1.34),corresponding to Methods 1,2,and 3,respectively.The ratio of Method 3 is much lower than that of Method 1 or Method 2,in which the compactibility of the Brazilian disc is improved.This demonstrates that the compactibility of the specimen has a significant effect on the anisotropy of the Brazilian disc,as shown by comparison with the average peak forces of the three methods.

The PFC2D manual[39]provides another algorithm,which is more effective in achieving a specified isometric stress distribution for the specimen.The force chain of a specimen modeled using the new algorithm is more homogeneous than those modeled with Methods 1,2,and 3,as shown in Fig.8.To exam ine this effect,a ratio of maximum isotropic stress to minimum isotropic stress varying from 1.05 to 3.50 is used for the simulation.Note that the same random number is used for different ratios.A Brazilian disc with a diameter of 113 mm is trimmed from a square specimen(113mm×113mm).The curves of the peak force with different ratios versus rotation angles are shown in Fig.9.The peak forces are relatively low when the rotation angle ranges from 15°to 75°,whereas they are comparable when the rotation angles are 0°and 90°.This phenomenon is similar to that observed in Method 2,in which the peak forces in the horizontal and vertical directions are larger and less scattered than those in other directions.This phenomenon indicates that the force chain distribution of the specimen does not have an obvious effect on the anisotropy.

To eliminate the effect of compactibility,another method is carried out,defined as Method 4 in Table 3.In Method 4,the particle boundary is initially generated with a diameter of 113 mm(blue particles in Fig.10).Arbitrarily placed particles,at half their final size,are then generated without overlap to fill the circle vessel.The particle radii increase to their final values and the assembly is allowed to rearrange under zero friction.This is carried out in steps until the ratio of maximum unbalanced force(or average unbalanced force)of all balls to maximum contact force(or average contact force)is less than 0.01,and an acceptable equilibrium status is reached for the assembly.“Floating”particles,defined as those having fewer than three contacts,are then deleted.When the contact force distribution is rebalanced,parallel bonds are installed between all particles.The microparameters follow the parameters listed in Table2,which are carried out in steps until static equilibrium is achieved.In this way,the compactibility of the Brazilian disc is comparable in all directions.To ensure that no breakage occurs throughout the loading process,a greater bond strength is used for the boundary particles.This value is initially set to two or three times that of the disc particles, but the bonds are easily broken at the early loading stage due to the local stress concentration.To avoid breakage,the bond strength of the boundary particles should be at least five times that of the disc particles.Therefore,a value of six times the strength of the disc particle is adopted for the boundary particle in the present study.In the case of large boundary particle strength values,if the stiffness of the boundary particles is set equal to or larger than that of the disc,the tensile strength obtained with the test is strongly related to the boundary particles.Finally,the stiffness of the boundary particles(blue particles)is set to 1/5 of the Brazilian disc(gray particles),which can significantly decrease the stress concentration at the boundary contacts.Note that if the stiffness of the boundary particles is too small(less than 1/100),the difference in stiffness between the loading platens and the boundary particles istoo large,and the platens easily traverse the boundary particles during the loading process,resulting in a non-uniform application of the stress of the loading plate on the loading boundary.The suggested stiffness varies from 1/5 to 1/20.Figure 11 shows the curves of peak force versus rotation angle of ten sets of random numbers.The peak force is random without an obvious trend corresponding to different rotation angles.The average peak forces for different rotation angles are approximately 2.0×106N,which are marked by solid pentagrams.The maximum average force is about 2.20×106N,corresponding to rotation angles of 45°and 75°.The minimum average force is about 1.98×106N,corresponding to a rotation angle of 60°.The anisotropy ratio is 1.11.The Brazilian disc modeled with this method is close to isotropic.Meanwhile,most of the fracture surfaces are parallel to the compressive loading direction.Few invalid tests are obtained with this method compared to Method 3.In addition,the discreteness of the peak forces is the lowest by comparison with the standard deviations of the four different methods listed in Table 4.This method is effective in reducing the anisotropy of the Brazilian disc and the discreteness of the tensile strength,thus enabling a more accurate determination of the tensile strength for homogeneous rock-like materials.

Table3 Four methods for generating the Brazilian disc(anisotropic ratio is the maximum average force to minimum average force)

Fig.5 Curves of peak forces versus different rotation angles.Ri(i=1−10)represents the ith random number.“Average”represents the average peak force corresponding to a rotation angle with different random numbers(Method 1 in Table 3)

Fig.6 Curves of peak force versus rotation angle of different random numbers(R1−R10)(Method 2 in Table 3)

Fig.7 Curves of peak force versus rotation angle for the large square specimen.R1−R10 represents different random numbers.Note that the data within the dashed circle are not used to calculate the average peak force(Method 3 in Table 3)

Table4 Standard deviation of peak forces of ten sets of random numbers corresponding to different rotation angles for different methods

Simulation and analysis of the four methods reveals that the compactibility of the specimen has a significant effect on the anisotropy of a Brazilian disc(Methods 1 and 2),and the boundary condition of the disc is closely related to the discreteness of strength and whether the Brazilian test of the generated disc is valid(Method 3).In Method 4,the compactibility of the Brazilian disc is generally comparable in all directions,thus overcoming the shortcomings of the discs generated with Methods 1,2,and 3.A relatively flat contact surface between boundary particles and loading platen is easy to achieve using a soft boundary(low particle stiffness),which results in a significant reduction in the local stress concentration,while exerting no effect on the crack propagation path within the Brazilian disc.To ensure that no breakage of the boundary occurs through out the loading process,larger bond strength is set for the boundary particles.In this way,Method 4 is superior to the other methods and more suitable for carrying out a realistic Brazilian test for homogeneous rock-like materials in the BPM.

Fig.8 Force chain distribution of specimens modeled by different methods.The thickness of the black lines represents force magnitude.a Force chain distribution of specimens modeled by Method 2.b Force chain distribution of specimens with a ratio equal to 1.25(the ratio is the maximum isotropic stress to minimum isotropic stress)

4 Discussion

Fig.9 Curves of peak force versus rotation angles with different ratios ranging from 1.05 to 3.50

Fig.10 Brazilian disc generated within particle boundary.Blue particles denote the particle boundary;dark gray particles indicate the Brazilian disc

Fig.11 Curves of peak force versus rotation angles corresponding to different random numbers(R1−R10)(Method 4 in Table 3)

Fig.12 Stress–strain curve of an invalid Brazilian test obtained with Method 3.Four points(A,B,C,and D)corresponding to different peak stresses of loading stages

Fig.13 Crack initiated corresponding to the stress level of point A(in Fig.12).The balls in contact with the walls are shown in green.Tensile and shear micro-cracks are shown by white and red lines,respectively

In Method 3,a considerable number of invalid tests are obtained,and the reasons are discussed in this section.Figure 12 shows the stress–strain curve of the rotation angle of 15°obtained with Method 3.To determine the cracking processes in the Brazilian test, five points are monitored on the stress–strain curve.Four points(A,B,C,D)represent different peak stresses of loading stages.The curve terminates at point E,which indicates a post-peak stress measuring 0.7 times the corresponding peak stress.Figure 13 shows the cracking processes corresponding to point A.There are five micro-cracks initiated at this stage:three micro-tensile cracks(white line)and two micro-shear cracks(red line).There are three particles(green ball)in contact with the left loading wall and one particle(green ball)in contact with the right loading wall.Therefore,the stress concentration occurs on the right loading wall.The cracks are formed around the particle which is in contact with the right loading wall.Figure 14b shows the crack propagation path corresponding to the second peak stress(point B in Fig.12).When the stress reaches the third peak(point C in Fig.12),the Brazilian disc specimen is broken near the boundary.After that,a central diametrical crack is formed,as shown in Fig.14d.At the end of loading,the disc specimen is split into two halves(Fig.14e).The specimen fails before the central diametrical crack is formed.In laboratory experiments,when axial stress reaches the first peak(point A in Fig.12),axial loading will stop.The peak stress obtained in this way does not correspond to the actual Brazilian tensile strength.Therefore,the Brazilian test is considered invalid.In Method 3,the Brazil-ian disc is trimmed from a large square specimen,and the boundary surface is irregular,so stress concentration tends to occur on the boundary.In Method 4,the boundary surface of the Brazilian disc is smooth(a flat contact surface formed)so that fewer invalid Brazilian tests are obtained.

Fig.14 Crack propagations corresponding to different stress levels(B,C,D,and E marked in Fig.12)

5 Conclusions

In the present study,the loading platens of a Brazilian disc are rotated at different angles(15°,30°,45°,60°,75°,and 90°),revealing that strong anisotropy is present with existing methods for generating Brazilian discs.Alternative measures are proposed to reduce the anisotropy. The major conclusions are as follows:

1.The compactibility of the specimen trimmed into a Brazilian disc plays a significant role in the anisotropy,while the force chain distribution of the specimen has no obvious effect.

2.The new method with a particle boundary is effective in reducing the anisotropy of the Brazilian disc.The compactibility of the Brazilian disc is comparable in all directions.The average tensile strengths corresponding to different rotation angles are comparable,and the discreteness of the tensile strength is the lowest.In addition,most of the fracture surfaces in the Brazilian tests are parallel to the loading direction,and few invalid tests occur with this method.

AcknowledgementsSupport provided by the National Basic Research Program of China(2015CB258500,2015CB058102,2014CB046904)is gratefully acknowledged.