Fracture Process and Failure Mode of Brazilian Discs with Cracks of Different Angles: A Numerical Study

Abstract

In order to determine the effect of internal cracks on the tensile failure of materials, a hybrid finite–discrete element method was used to analyze the Brazilian disc test with cracks of different angles. When the pre-crack angle is between 0° and 60°, the wing crack is initiated from the pre-crack end. When the pre-crack is 90°, the crack initiated from the pre-crack center. When the pre-crack angle is between 0° and 60°, the maximum principal stress and plastic strain are concentrated at the pre-crack end. When the pre-crack angle is 90°, the maximum principal stress and plastic strain are concentrated in the pre-crack center. As the crack angle increased from 0° to 90°, the failure mode of Brazilian discs with cracks transits from splitting into two parts to splitting into four parts. The influence of crack length is further studied. When the crack length is less than 5 mm, the crack angle has little influence on the disc failure mode; Brazilian discs with cracks of different angles undergoes splitting failure along the loading axis. When the crack length is larger than 5 mm, the crack angle has a great effect on the disc failure mode.

1. Introduction

Brittle materials such as concrete and rock are the main components in mining engineering and civil engineering, and the failure of brittle materials is a complex evolution process [1,2,3,4]. Because the tensile strength of brittle materials is much smaller than their compressive strength, the failure of brittle materials is mainly caused by tensile fracture propagation in most cases [5,6,7]. In order to obtain the tensile strength of brittle materials, ISRM recommends the Brazilian disc test as an indirect method for tensile testing [8], which has the advantages of simple sample preparation and testing procedures. According to Hondor’s analytical solution [9], a large portion of a Brazilian disc is in a tensile stress state in the horizontal direction.

Since the Brazilian disc test was proposed by Carneiro and Akazawa [10,11], many scholars have carried out various studies on the Brazilian disc test [12,13,14,15,16,17,18,19]. Hudson and Swab et al. [20,21] found that the crack initiation point and maximum tensile strain of Brazilian discs tended to be far away from the center of the disc in a plate-loaded Brazilian disc tensile test. Li et al. [22] proposed a theory of maximum tensile strain for judging the initiation point of the Brazilian disc. This could explain the phenomenon wherein the cracking point deviates from the center in some Brazilian disc tests. In addition, Li et al. [23] observed the tensile strain of Brazilian discs under different loading configurations through digital image correlation techniques, and discussed the effectiveness of loading configurations. It was found that curved jaw loading was best for the Brazilian disc test, and that small-diameter rod loading was not suitable for the Brazilian disc test.

Brittle materials such as rock mass and concrete usually contain joints and fissures, which often dominate the failure process of brittle materials [24,25,26,27,28,29]. Therefore, many scholars have studied the effect of cracks on rock failure under various conditions [30,31,32,33,34]. In order to study the effect of cracks on the tensile failure of brittle materials, some scholars have carried out the Brazilian disc test with cracks [35,36,37]. Luo et al. [38] studied the Brazilian disc test with different angles of cracks and found that with an increase in the crack angle, the peak load of the disc first decreased and then increased. Xiao et al. [31] studied cuboid granite with cracks of different angles under coupled static and dynamic loads, and found that with a decrease in crack angle, the dynamic strength and combined static and dynamic strength of granite decreased. In physical testing, it is difficult to obtain some important information such as stress field, strain field and damage evolution of discs with cracks. The numerical methods can make up for these deficiencies. For example, Haeri et al. [39] simulated the failure process of prefabricated holes and cracks in Brazilian discs by PFC^3D and found that the main failure mode remained unchanged in tensile failure with an increase in ball diameter. Chang et al. [40] carried out numerical calculations on Brazilian discs with cracks with different angles using the embedded viscous CZM method, and obtained the effect of crack angles on the crack propagation of discs.

Among the existing studies on Brazilian discs with cracks, some studies focused on the analysis of mode I fracture and mode II fracture toughness, and some studies focused on the effects of crack angle on the failure process, failure mode and peak load of the disc [41,42,43,44]. The evolution of the stress field and strain field in the fracture process of Brazilian discs with cracks of different angles is important for analyzing the effect of cracks on the tensile failure and fracture behavior of brittle materials. In this study, the stress field, damage strain field and fracture process of Brazilian discs with cracks are analyzed. The influence of crack angle and length on the Brazilian disc is further discussed, and can explain some phenomena regarding indirect tensile strength differences in materials caused by internal defects.

2. Numerical Methods and Models

2.1. Numerical Methods

Among the commonly used numerical calculation methods, the finite element method is difficult to simulate the fracture process of materials with, and the discrete element method has some defects in simulating the continuum. In recent years, a hybrid finite element and discrete element method (FDEM) has been developed and applied. The method divides the numerical model into finite element domains and inserts new cracks into the finite elements by introducing fracture mechanics criteria to realize the fracture process of continuum [45,46,47,48]. A hybrid program called ELFEN simulates the fracture process of materials by increasing the strain of materials during loading [49,50,51,52]. The method determines the internal element and inter-element fractures according to the failure plane, which makes the fracture process more realistic.

The crack initiation, propagation and coalescence of the model in the hybrid program are the result of the increase in tensile strain. In addition to the tensile stress field, the model also expands in the orthogonal direction due to the Poisson effect under the compressive stress field. The CAI’s research shows that the shear failure of brittle materials under uniaxial compression is caused by the propagation and coalescence of tensile micro-cracks in materials [53]. The traditional Mohr–Coulomb failure criterion exaggerates the tensile strength of materials. The Rankine failure criterion is suitable to judge tensile failure. The Mohr–Coulomb with Rankine failure criterion is used to better judge the tensile and shear fracture behavior of brittle materials in this study. A detailed introduction to this numerical method can be seen in earlier studies [52,54].

2.2. Brazilian Disc with Crack Models

Seven Brazilian disc models with cracks of different angles were built, as shown in Figure 1. The disc diameter (D) is 50 mm, the disc thickness (T) is 25 mm, the crack length (L) is 25 mm, the crack width is 0.5 mm and the crack is located in the disc’s center. The crack angle $\mathsf{\alpha }$is defined as the angle between the crack’s direction and the loading direction; $\mathsf{\alpha }$ ranges from 0° and 90°. Velocity loading was selected as the loading type. The loading velocity was 1 mm/s, and the strain rate was 0.02 s⁻¹, which constituted quasi-static loading.

The granite material in Li’s research [23] was chosen as the disc material in the model. The uniaxial compression strength was 171.7 MPa and the indirect tensile strength was 12.8 MPa. In this study, the friction angle for the material was set to 28° and the cohesion was set to 51 MPa. The material properties for the model are shown in

Table 1. Material property of model.

Mechanical Parameters	Brazilian Disc	Loading Platen
Young’s modulus (E, GPa)	43.2	211.00
Poisson’s ratio (ν)	0.23	0.29
Shear modulus (G, GPa)	17.5	-
Density (ρ, Ns2/mm4)	2.81 × 109	7.84 × 109
Cohesion (c, MPa)	51	-
Friction angle ($\phi$)	28°	-
Tensile strength (${\sigma }_t$, MPa)	12.8	-
Fracture energy (Gf, N/mm)	0.01	-
Normal penalty (Pn, N/mm2)	43,200	211,000
Tangential penalty (Pt, N/mm2)	4320	21,100
Friction (γ)	0.1	0.1
Mesh element size (mm)	0.8	0.8

. The normal penalty is generally 1.0 times the elastic modulus, and the tangential penalty is generally 0.1 times the normal penalty. The mesh element size was 0.8mm, and the diameter of discs was 62 times the mesh size. The influence of mesh size on Brazilian discs with cracks is shown in in Appendix A.

3. Results

3.1. Load–Displacement Curve

Figure 2 shows the load–displacement curves for the Brazilian discs with cracks of different angles. The seven curves include the initial compaction stage, elastic stage and post-peak stage. The initial compaction stage is shorter and the elastic stage is longer. When the crack angle is between 0° and 45°, as can be seen from Figure 2a, the peak load and curve slope of the disc decrease gradually as the crack angle increases. When the crack angle is between 45° and 90°, as can be seen from Figure 2b, the angle increases, both the peak load and peak displacement of disc gradually decrease and the slope of the curve increases slightly.

The peak load and peak displacement of the Brazilian disc with the 0° crack are 5.34 kN and 0.041 mm, respectively. The peak load and peak displacement of the Brazilian disc with the 45° crack are 3.19 kN and 0.030 mm, respectively. The peak load and peak displacement of the Brazilian disc with the 90° crack are 5.15 kN and 0.045 mm, respectively. The Brazilian disc with the 0° crack has the largest peak load, the Brazilian disc with the 90° crack has the largest peak displacement, and the Brazilian disc with the 45° crack has the smallest peak load and peak displacement.

3.2. Fracture Process

Figure 3, Figure 4, Figure 5, Figure 6, Figure 7, Figure 8 and Figure 9 show the crack propagations for the Brazilian discs with cracks of different angles. The failure process includes crack initiation, crack propagation, crack penetration, and failure. Figure 3 shows the fracture process of the Brazilian disc with the 0° crack. The crack starts at the pre-crack end and spreads along the loading path toward the disc end and penetrates the disc. Finally, the disc is split into two halves. Figure 4, Figure 5 and Figure 6 show the fracture processes of Brazilian discs with 15°, 30° and 45° cracks, respectively. The fracture processes of these three types of discs with cracks are similar. It expands in the direction of the loading point and penetrates the disc and, finally, the disc splits into two halves.

Figure 7 shows the fracture process of the Brazilian disc with the 60° crack. The crack starts at the pre-crack end and propagates towards the loading point and penetrates the disc. After the crack penetrated the disc, a third crack developed at the pre-crack tip and expanded to the left and, finally, the disc failed into three parts. Figure 8 shows the fracture process of the Brazilian disc with the 75° crack. The crack initiation point is at about 3 mm from the pre-crack tip, and propagates toward the loading point and penetrates the disc. After the crack penetrated the disc, two cracks were generated at both the pre-crack ends and expanded along the direction of the pre-crack and, finally, the disc failed into four parts. Figure 9 shows the fracture process of the Brazilian disc with the 90° crack. The crack starts at the pre-crack center and propagates along the loading path towards the disc end and penetrates the disc. After the crack penetrates the disc, two cracks are generated on the left and right sides of the disc and expand along the direction of the pre-crack and, finally, the disc fails into four uniform fan shapes.

3.3. Stress Field Evolution

Figure 10, Figure 11, Figure 12, Figure 13, Figure 14, Figure 15 and Figure 16 show the evolution of the maximum principal stress field for the Brazilian discs with cracks of different angles. Figure 10 shows the evolution of the maximum principal stress field of the Brazilian disc with the 0° crack. The maximum principal stress is first concentrated at the pre-crack end and increases with the increase in the external load. After crack initiation, the region of maximum principal stress moves to the disc edge along the loading path. Figure 11, Figure 12, Figure 13 and Figure 14 show the evolution of maximum principal stress fields of Brazilian discs with 15°, 30°, 45° and 60° cracks, respectively. The evolution of the maximum principal stress fields of these four types of discs with cracks are similar. The maximum principal stress is first concentrated at the pre-crack end. After crack initiation, the region of maximum principal stress moves with the direction of crack propagation. It is worth noting that before the crack penetrates the disc, the maximum principal stress concentration area is transferred to the disc edge in the direction of the pre-crack.

Figure 15 and Figure 16 show the evolution of maximum principal stress fields of Brazilian discs with 75° and 90° cracks, respectively. When the pre-crack angle is 75°, the maximum principal stress is first concentrated between the pre-crack end and the center. When the pre-crack angle is 90°, the maximum principal stress is first concentrated in the pre-crack center. After crack initiation, the maximum principal stress region moves to the disc edge along the direction of crack propagation. The maximum principal stress concentration region also appears at the disc edge in the direction of the pre-crack. After the initial crack penetrated the disc, the secondary crack originated in the maximum principal stress concentration area on the disc edge, extended to the pre-crack end and penetrated the disc again.

3.4. Plastic Strain Zone

Figure 17, Figure 18, Figure 19, Figure 20, Figure 21, Figure 22 and Figure 23 show the plastic strain evolution for the Brazilian discs with cracks of different angles. Figure 17 shows the plastic strain evolution of the Brazilian disc with the 0° crack. The plastic strain is initiated at the pre-crack end and propagates along the loading path towards the disc end. The crack initiates inside the plastic strain and is less than the length of the plastic strain. The plastic strain zone finally penetrates the disc along the loading direction, which indicates that the damage area of the Brazilian disc with the 0° crack is mainly confined to the loading path.

Figure 18, Figure 19 and Figure 20 show the plastic strain evolution of Brazilian discs with 15°, 30° and 45° cracks, respectively. The plastic strain evolution patterns for these three types of discs with cracks are similar. The plastic strain is initiated at the pre-crack end and propagates towards the loading point at the disc end. The cracks are initiated within the plastic strain and propagate along with the plastic strain. The plastic strain zone eventually reaches the loading point at the disc end and penetrates the disc. The damage area is mainly the area near the macroscopically propagated crack. Figure 21 shows the plastic strain evolution of the Brazilian disc with the 60° crack. The plastic strain is initiated at the pre-crack end and propagates towards the loading point at the disc end. After the initial plastic strain penetrates the disc, the third plastic strain zone is initiated on the left side of the pre-crack, which indicates that when the pre-crack angle is greater than 45°, the damage degree begins to increase.

Figure 22 shows the plastic strain evolution of the Brazilian disc with the 75° crack. The plastic strain initiates from about 3 mm away from the pre-crack tip and propagates towards the loading point at the disc end. After the initial plastic strain penetrates the disc, the third and fourth plastic strain zones are initiated on the left and right sides of the pre-crack and, finally, penetrate the disc, and the damage degree of disc is further increased. Figure 23 shows the plastic strain evolution of the Brazilian disc with the 90° crack. The plastic strain is initiated at the pre-crack center and propagates along the loading path towards the disc end. After the initial plastic strain penetrates the disc from the vertical direction, the third and fourth plastic strain zones are initiated on the left and right sides of the pre-crack and, finally, penetrate the disc from the horizontal direction.

4. Discussion

4.1. Failure Mode Transition

Figure 24 shows the failure mode transition for the Brazilian discs with cracks of different angles, and the crack angles range from 0° and 90°. When the crack angle is between 0° and 45°, the Brazilian disc fails in two halves. When the crack angle is 60°, the Brazilian disc fails in three parts. When the crack angle is 75°, the Brazilian disc fails in four non-uniform sectors. When the crack angle is 90°, the Brazilian disc fails in four uniform sectors. It can be seen that with the increase in the crack angle, the failure of the disc transitions from splitting into two halves to a uniform four-sector failure. The increase in crack angle leads to an increase in the damage degree of the disc. As shown in Figure 25, a similar failure mode transition was observed in Luo’s physical test [38]. In Figure 26, the peak load of disc decreases first and then increases with the increase in crack angle, similar to Figure 2, which indicates that the simulation results in this paper are relatively reliable.

4.2. Effect of Crack Length

In addition to crack angle, the crack length is also an important factor affecting material fracture. Figure 27, Figure 28 and Figure 29 show the failure modes of Brazilian discs with cracks of different lengths and of 0°, 45° and 90° angles, respectively. It can be seen that 0° cracks with different lengths have little effect on the failure mode of Brazilian discs, and the disc always splits into two halves along the loading axis. When the length of a 45° crack is less than 5 mm, the disc splits along the loading axis. When the crack length is greater than 5 mm, the main crack of disc deviates from the loading axis. When the length of a 90° crack is less than 10 mm, the disc splits into two halves along the loading axis. When the crack length is greater than 10 mm, the secondary cracks are generated in the horizontal direction, and the disc is broken into three parts or four uniform sectors.

When the crack length is less than 5 mm, the Brazilian discs with cracks of different angles always split into two half-discs along the loading axis, and the crack angle has little influence on the disc failure mode. When the crack length is greater than 5 mm, the failure modes of Brazilian discs with cracks of different angles are significantly different. Figure 30 shows the peak load of Brazilian discs with cracks of different lengths. It can be seen that the increase in crack length leads to a rapid decrease in the peak load of discs. When the crack length is less than 5 mm, the crack angle has little influence on the peak load of the disc. When the crack length is greater than 5 mm, the crack angle has greater influence on the peak load of the disc, and the peak load of discs with 45° cracks is the smallest.

It can be seen from the above analysis that when the crack length is less than 5 mm, Brazilian discs with cracks of different angles undergo splitting failure along the loading axis. This is consistent with the failure mode of the intact Brazilian disc sample, but the peak load is lower than the peak load of the intact Brazilian disc sample (the crack length is 0). This shows that micro-cracks with different directions in brittle materials such as rocks are one of the reasons for the discreteness of the Brazilian disc test results, although these micro-cracks do not change the final failure mode of the Brazilian disc.

5. Conclusions

The main conclusions are as follows:

(1)

A hybrid finite–discrete element method was used to analyze the Brazilian disc test with cracks of different angles. When the pre-crack angle is between 0° and 60°, the wing crack initiates from the pre-crack end, propagates toward the loading point and penetrates the disc. When the pre-crack angle is 75°, the crack initiation point is about 3 mm from the pre-crack tip. When the pre-crack angle is 90°, the crack starts from the pre-crack center and propagates to the disc end along the loading path. After the crack penetrates the disc, two secondary cracks are generated on the left and right sides of disc and, finally, the disc fails into four uniform sectors;

(2)

When the angle of the pre-crack is between 0° and 60°, the maximum principal stress is first concentrated at the pre-crack end. After crack initiation, the maximum principal stress region moves toward the loading point with crack propagation. When the pre-crack is at 75°, the maximum principal stress is first concentrated between the pre-crack end and the center. When the pre-crack is at 90°, the maximum principal stress is first concentrated in the crack center. After crack initiation, the maximum principal stress moves to the disc edge along the direction of crack propagation, and the maximum principal stress also appears at the disc edge in the direction of the pre-crack;

(3)

When the angle of the pre-crack is between 0° and 60°, the plastic strain starts at the pre-crack end and propagates towards the loading point. The crack initiates within the plastic strain and propagates with the plastic strain. The damage area is mainly the area near the macroscopically propagated crack. When the pre-crack angle is 75°, the plastic strain initiates from about 3 mm from the pre-crack tip. When the pre-crack angle is 90°, the plastic strain starts from the pre-crack center and propagates to the disc end along the loading axis. After the initial plastic strain penetrates the disc, the third and fourth plastic strain zones are activated and finally penetrate the disc from the horizontal direction;

(4)

The influence of pre-crack length is further studied. When the crack length is less than 5 mm, Brazilian discs with cracks of different angles always split into two half-discs along the loading axis, and the crack angle has little influence on the disc failure mode. When the crack length is larger than 5mm, the crack angle has a great effect on the disc failure mode. The increase in crack length leads to a rapid decrease in the peak load of the disc. When the crack length is less than 5 mm, Brazilian discs with cracks of different angles undergo splitting failure along the loading axis. This is consistent with the failure mode of the intact Brazilian disc sample, but the peak load is lower than that of the intact Brazilian disc sample.