Experimental and numerical study of disk specimen with rough structural plane under splitting test

Abstract

To study the effects of the anisotropic matrix and structural planes on the splitting strength and failure mode of rocks, Brazilian splitting tests were carried out with seven different loading angles on specimens of rock-like materials with rough structural planes. The surface strains of the samples during the failure process were monitored and analysed with the help of a high-speed camera and digital image correlation (DIC) technology. The test results showed that the Brazilian splitting strength (BSS) decreased gradually with an increased loading angle. According to the crack morphology, the samples showed three failure modes, and the structural plane and the loading angle (θ) had an important effect on the failure mode. When θ < 75°, the sample failure was mainly affected by the matrix, and when θ > 75°, the sample failure was mainly controlled by the structural plane. The numerical simulation of the sample with a structural plane was carried out by the PFC^2D particle flow program, the micro parameters were calibrated using a back propagation (BP) neural network model. The internal cracks of the sample under a splitting load were mainly matrix tensile microcracks and structural plane shear microcracks, and the tensile microcracks in the side with the weak matrix appeared significantly earlier than those in the side with the strong matrix. With increasing loading angle, the proportion of tensile microcracks in the matrix increased, while the proportion of shear microcracks in the matrix decreased, especially in the weak matrix. The microcracks at the structural plane mainly changed from tensile microcracks to shear microcracks, and the development degree of microcracks along the structural plane was more significant than that on the weak matrix with increasing loading angle. The results of the study can provide a reference for rock stability evaluation and utilization.

Keywords

Structural plane Brazilian test failure mode particle flow code BP neural network

1 Introduction

The existence of a structural plane in a rock mass results in discontinuity and anisotropy, both of which adversely affect the strength and stability of the rock mass. Engineering practice shows that the instability failure of the surrounding rock in a semi-coal roadway mainly depends on the development and expansion of the joints or cracks in the coal rock mass, and the structural plane of the fractured rock mass determines the mechanical properties of the rock mass [1, 2]. In 1973, Barton [3] derived the empirical formula of the peak shear strength of rock mass discontinuity through experiments and first proposed the joint roughness coefficient (JRC). On this basis, domestic and foreign research on the roughness of rock mass structural surfaces gradually changed from qualitative descriptions to quantitative evaluations and from empirical formulas to theoretical deductions [4, 5].

As a simple and quick method to test the tensile strength of rock, the Brazilian splitting test has been widely used at home and abroad [6, 7]. On the basis of the joint roughness theory, remarkable achievements have been made in the use of Brazilian splitting tests on structural samples. For example, Hou et al. [8] conducted the Brazilian splitting test with different bedding angles to study the anisotropy characteristics of the tensile strength and failure morphology of black shale. Vervoort et al. [9] conducted splitting tests on 9 rock types and summarized the failure types into three modes characterized as the central zone, noncentral zone and plane. Since it is difficult to accurately observe the cracking process from crack initiation to crack propagation in laboratory tests, relevant scholars use numerical simulation software to simulate jointed rock. For example, Nezhad et al. [10] studied the effects of bedding and matrix on the tensile strength and fracture toughness of shale by combining the Brazilian splitting test and finite element simulation. Li et al. [11] carried out Brazilian splitting tests of carbonaceous slate with different bedding angles and used a cohesive force model for numerical analysis to explore the influence of weak structure on its destructive state. Meng et al. [12] conducted Brazil splitting tests on prepared rock-like samples and, combined with the finite element bond zone model, discussed the influence of the strength of the bedding plane on the tensile properties of the samples. At present, most of the studies on the tensile properties of Brazilian discs with structural samples are limited to the discontinuities of rock masses with the same lithology on both sides, and the strength characteristics of discontinuities of rock masses with different properties are rarely discussed. There are a large number of anisotropic structural planes with different lithologies on either side of the natural rock mass. With the continuous development in the field of deep engineering, the necessity of safety and stability is increasing daily. Therefore, it is of great significance to study the tensile properties of rock-like materials with rough structural planes.

With the continuous progress of 3D printing technology, a large number of scholars have explored the feasibility of its application in Rock mechanics. Jiang et al. [13] explored the use of 3D printing technology to prepare rock like samples for mechanical experiments, and found that the mechanical properties of the samples were significantly different from those of natural rocks. They proposed to use hard brittle materials similar to natural rocks as printing materials for 3D printing; Sharafisafa et al. [14] used a combination of quasi-static compression loading tests and digital image correlation (DIC) technology to study the effect of defect filling materials on the deformation and failure behaviour of 3D printed rock Brazilian plates containing defects; Wang et al. [15] established a physical model of fracture network with different joint roughness coefficients based on 3D printing technology, and analysed the Shear strength and failure mode of each group of specimens by conducting indoor direct shear tests. The above research indicates that compared with traditional rock preparation methods with structural planes, 3D printing technology has the advantages of repeatable preparation of samples with identical structural planes and easier determination of joint roughness. At the same time, model testing of cement mortar samples can to some extent overcome the difficulty of natural rock sampling, providing a more suitable method for studying the mechanical properties of rock masses with structural planes.

Therefore, this article uses a 3D printed design of rock like joint moulds to prepare anisotropic rock like disk specimens with rough structural planes on both sides, which are poured from cement mortar. Brazilian splitting tests were carried out on disk samples with rough structural planes under different loading angle condition. With the help of digital image correlation technology (DIC) [16–20], the influence of anisotropic rock mass and loading angles on the mechanical properties and failure evolution of the samples was analysed. At the same time, based on the numerical simulation software (PFC), the BP neural network model was established to quickly and accurately calibrate the micro mechanical parameters, Further exploration was conducted from a macro and micro perspective on the initiation and propagation process of its failure cracks as a function of the loading angle.

2 Test materials and procedures

2.1 Sample making and material characteristics

According to the JRC estimation formula proposed by Xie et al. [21], this paper obtains the roughness of the joint profile through the fractal dimension D of the joint, and the fractal dimension D of the joint can be calculated according to two statistical parameters L and h (L and h are the average base length and average height of the joint roughness, respectively), namely: $D = log 4 / log {2 [1 + cos (arctan (2 h / L))]}$ (1) $JRC = 85.267 {(D - 1)}^{0.5679}$ (2)

This test highlights the influence of joint roughness on the samples. The joint roughness JRC is set at 20, the average base length of the joint roughness is 5 mm, and the average height is 1.81 mm. The design diagram of the joint surface curve is shown in Fig. 1(a). Acrylonitrile-butadiene-styrene plastic (ABS) is used as the material for the three-dimensional structural plane mould. The printed mould (see Fig. 1(b)) has the advantages of high strength and a smooth joint surface and is not easily stained or corroded by the rock-like materials.

Fig. 1

Schematic diagram of mould.

In this study, a self-designed mould was used to prepare the cement mortar samples. By constantly adjusting the mix ratio of cement, standard sand and water, rock-like materials with close to real rock mechanical properties was produced, in which the matrix on the left was Class A material and the matrix on the right was Class B material. The physical and mechanical parameters of the prepared rock-like materials are shown in Table 1.

Table 1

Physical and mechanical parameters of rock-like specimen

Kind of species	Cement: sand: water	Density/kg.m^–3	UCS/MPa	BTS/MPa	E (GPa)
A	1 : 1:0.25	2.27	80.88	9.24	15.79
B	1 : 2:0.32	2.16	65.61	8.13	17.72

The sample preparation process is shown in Fig. 2. A standard disc sample with a diameter of 50 mm and a thickness of 25 mm was prepared. All samples were stored in a dry environment at room temperature to ensure that the parallelism between the upper and lower surfaces was controlled within 0.5 mm, and the flatness of the surface was controlled within 0.1 mm, meeting the standards recommended by the International Society of Rock Mechanics (ISRM) [6].

Fig. 2

Sample making flow chart: a fabrication of matrix A; b curing for the prepared rock-like specimen; c cutting; d polishing.

2.2 Test equipment and loading program

The Brazilian splitting test was carried out in a DRTS-500 rock mechanics test system [22] independently developed by China University of Mining and Technology (Beijing). The loading diagram of the sample is shown in Fig. 3(a). In this experiment, seven different loading angles were selected with θ values of 0°, 15°, 30°, 45°, 60°, 75° and 90° (θ definitions are shown in Fig. 3(b)) to study the influence of various loading angles on the mechanical properties of anisotropic rock-like samples. At the same time, to reduce the dispersion of the test results, three samples were arranged for each loading angle to be tested. The loading of the sample was controlled by displacement, and the rate of loading was 0.1 mm/min, loaded until specimen damage occurs. To explore the deformation law of the sample in the loading process, the failure process of the sample was recorded, and the surface strain field of the sample was analysed in combination with DIC technology at a later stage. The camera resolution for recording was 2592 pixels×1944 pixels, the acquisition rate was 8 images/s, and the LED light source was placed at the side of the camera. The digital images collected were also checked to have sufficient brightness and contrast for testing purposes. See Fig. 4 for the final macro photos of the sample.

Fig. 3

Loading and monitoring systems of Brazilian test.

Fig. 4

Macrograph of samples.

3 Analysis of experimental results

3.1 Axial load–displacement curve

Figure 5 shows the load-displacement curve obtained from the test. Figure 5 shows that the peak loads of samples under different loading angles differed greatly, but the curve forms were similar to some extent, which can be roughly divided into three stages: compaction, elasticity and failure. In the initial loading stage, due to the existence of pores or microcracks in the specimen, the indenter will produce local compaction deformation at the contact point of the specimen surface, and the curve shows an upwards concave trend after the contact area increased with increasing load. As the load continued to increase to the peak strength, the load and displacement of the specimen after compaction showed a linear increasing trend, showing good elastic characteristics, and the slope of the specimen gradually decreased with increasing loading angle. When the peak strength was reached, the curve fell vertically to a slope almost parallel to the load axis, and the sample lost its bearing capacity, and showed significant brittle failure characteristics. The mechanical properties of the specimen were similar to those of real rock and can be applied to rock mechanics tests.

Fig. 5

Axial loading force (P) versus displacement (d) curves of specimen under Brazilian test.

3.2 Analysis of Brazilian splitting strength

The tensile strength of the sample can be calculated by the elastic solution equation [6]: $σ_{t} = \frac{2 P_{max}}{π Dh}$ (3) where σ_t is the tensile strength, P_max is the peak load of the sample when it is damaged, D is the diameter of the sample, and h is the thickness of the sample. Equation (3) is a tensile strength formula established based on the plane stress, which assumed that the sample was an anisotropic homogeneous material and cracks propagated from the centre of the sample. The sample used in this study clearly did not conform to this assumption, but the strength obtained by using this equation can represent the tensile property of the sample, so Equation (3) is still used in this study to calculate the strength. However, the calculated strength is the Brazilian splitting strength (BSS) rather than the tensile strength of the sample [25, 26].

As shown in Fig. 6, the loading angle has a significant influence on the BSS of the samples. The splitting strength did not decrease significantly when the loading angle ranged from 0–15°. From 15° to 60°, the splitting strength decreased gradually increased to 5.70%, 9.23% and 11.83%, respectively. When the loading angle was 60–75°, the strength curve sharply dropped, reaching 45.91%. At a loading angle of 75–90°, the decreasing trend was no longer clear, and the splitting strength tended to become stabilized. The maximum value was 6.34 MPa when θ = 0°. When θ = 90°, the splitting strength was 2.26 MPa. The BSS decreased with increasing loading angle θ. The main reason is that the transverse deformation caused by compression is restrained by the rough structural plane to some extent. However, with increasing loading angle, the resistance of the structural plane to deformation gradually weakened, which resulted in a gradual decrease in BSS.

Fig. 6

Variation of Brazilian splitting strength with loading angles.

3.3 Failure mode analysis

The failure of quasibrittle materials is a process in which internal microcracks appear, expand, and eventually develop into macroscopic cracks. The failure morphology of some samples under different loading angles is shown in Fig. 7, where the light coloured area represents matrix B with weak strength characteristics, and the dark coloured area represents matrix A with strong strength characteristics (same as Fig. 5). By referring to the previous classification methods [24–27] and observing the failure cracks of the samples in Fig. 7, the three typical failure modes are divided into:

Matrix failure mode: the crack is basically parallel to the vertical loading direction and located in the central part of the sample.

Failure mode of the structural plane: cracks appear on the structural plane, as shown in Fig. 7(f)-(g).

Mixed failure mode: matrix failure is mixed with structural plane failure, as shown in Fig. 7(a)–(e).

Fig. 7

Failure mode of specimen after Brazilian test.

Table 2 shows the transverse strain nephogram of the sample under different loading angles. Different colours distinguish the transverse strain size, in which red is positive, indicating tensile strain, and black is negative, representing compressive strain.

Table 2

Lateral strain evolution of specimen under Brazilian test

θ (°)	0.2P	0.4P	0.8P	P
(a) 0
(b) 15
(c) 30
(d) 45
(e) 60
(f) 75
(g) 90

The specimen first generates tensile strain at the upper loading end. With increasing load, the tensile strain area at both ends develops towards the centre of the specimen, and finally passes through the specimen. When θ = 0°, the sample finally presents a mixed failure mode of matrix failure and structural plane failure, but the central part of the sample is the first to be pulled apart, and the weak matrix body is the first to fail. The main crack constantly expanding to the upper and lower ends makes the sample exist in a nonequilibrium state, which significantly weakens the tensile shear strength and instantaneously reduces the anti-slip ability of the sample. As a result, the secondary cracks continue to expand along the direction of the structural plane, and shear slip failure occurs. When θ = 15°, 30°, 45°, and 60°, the failure mode of the sample is similar to that when θ = 0°. Although a certain degree of shear failure will occur along the structural plane, the main reason for the failure of the sample is still the tensile crack through the sample. When θ = 75°, the structural plane of the sample is weaker than that of the matrix on both sides, and shear failure along the structural plane appears but does not break along the centreline of the sample. When θ = 90°, the failure mode of the sample is also affected by the structural plane. In this case, the structural plane is vertical and bears most of the vertical load, and the sample finally shows a structural plane failure mode.

In conclusion, the matrix strength and structural plane are both important factors affecting the failure mode of samples. With increasing loading angle, the factor determining the final failure mode shifts from the matrix strength to the structural plane strength. When θ < 75°, the failure of the sample is mainly affected by the matrix. When θ > 75°, the failure of the sample is mainly controlled by the structural plane.

4 Numerical simulation

4.1 PFC^2D model construction

Particle Flow Code (PFC^2D) numerical simulation software can directly simulate crack initiation and propagation without the application of a complex constitutive model. It has gradually become an important method to study the influence of material microscopic properties on the macroscopic response. In this paper, the failure mechanisms of anisotropic rock-like materials with rough structural planes under different loading angles are analysed from the microscopic level by means of the PFC^2D software.

The simulated model size of the sample is the same as that of the actual physical sample, and a disc sample with a diameter of 50 mm is used. The sample was composed of 34,850 particle units with a particle size range of 0.3–0.5 mm. Random particles were generated within the specified range by using fish language, and samples with close contact between particles were formed by calculating the equilibrium.

The model established in this study included two parts: a rock-like matrix and a structural plane. The rock-like matrix portion was the particle set using the parallel bonding model [28, 29], while the structural plane was simulated using the particle set weakening parallel bonding. The model generation process was as follows (Fig. 8):

Generating the basic model: Construct a particle aggregate as the matrix portion.

Generating the structural plane: The bonding strength and stiffness of the contact between particles in the structural plane region are weakened to obtain the structural plane of the sample model.

Matrix model parameter assignment: assign meso-mechanical parameters to the matrix and structural plane.

Fig. 8

PFC^2D numerical simulation.

4.2 Calibration of microscopic parameters

There is no specific correspondence between the microscopic parameters involved in PFC to characterize particles and the macroscopic mechanical parameters of rock like materials. Therefore, it is crucial to select a set of microscopic parameters that can appropriately reflect the mechanical behaviour of the sample. Previous researchers used the trial and error method [30] for calibration, and improved the consistency between numerical simulation results and experimental results through multiple trial and error methods. Considering the complexity of the relationship between macro and micro parameters in this study, a neural network method was attempted to establish a prediction model for micro parameters and further complete the calibration of micro parameters.

4.2.1 Orthogonal design of experiments

The use of orthogonal experiments can scientifically design the combination of microscopic parameters, ensuring reliable experimental results while solving the problem of excessive number of experiments. The micromechanical parameters describing the parallel bonding model used in this study include: bonding radius coefficient $\bar{λ}$ , Particle elastic modulus E_c, particle normal/tangential stiffness ratio k_n/k_s, adhesive elastic modulus ${\bar{E}}_{c}$ , adhesive normal/tangential stiffness ratio ${\bar{k}}_{n} / {\bar{k}}_{s}$ , particle friction coefficient μ, Normal bonding strength σ_c and tangential bonding strength ${\bar{τ}}_{c}$ . Among them, $\bar{λ}$ , E_c, k_n/k_s, ${\bar{E}}_{c}$ and ${\bar{k}}_{n} / {\bar{k}}_{s}$ determine the deformation properties of materials and have great influence on macroscopic mechanical parameters such as elastic modulus E and Poisson’s ratio ν; μ, ${\bar{σ}}_{c}$ and ${\bar{τ}}_{c}$ determines the strength properties of the material, and have great influence on macroscopic mechanical parameters such as uniaxial compressive strength σ_c and uniaxial tensile strength σ_t. To reduce the difficulty of calibration, it is necessary to make appropriate assumptions, Referring to previous research results [31, 32], the following assumption is made: particle density ρ take 2700 kg/m³; The particle radius ratio R_max/R_min is taken as 1.66, and the minimum particle size R_min = 0.3 mm; $\bar{λ}$ is 1; The particle contact modulus and stiffness ratio are the same as those of the parallel bonding model, E_c = ${\bar{E}}_{c}$ , k_n/k_s = ${\bar{k}}_{n} / {\bar{k}}_{s}$ .

The orthogonal test design is shown in Table 3. Uniaxial compression numerical tests and splitting numerical tests are carried out for different combinations of mesoscopic parameters, and macro-parameters of rock-like samples corresponding to each group of mesoscopic parameters are obtained through numerical tests. Five mesomechanical parameters were selected as output layer parameters, namely E_c, k_n/k_s, μ, ${\bar{σ}}_{c}$ and ${\bar{τ}}_{c}$ . Four macroscopic mechanical parameters were selected as input layer parameters, namely, E, ν, σ_c and σ_t. 100 sets of data were collected as sample base for the subsequent research on the prediction of mesoscopic parameters by neural network.

Table 3
Orthogonal test design table of micro parameters of PFC^2D model

Microscopic parameters Factor level

1 2 3 4 5

E_c/Gpa 5 10 15 20 25

k_n/k_s 0.4 0.8 1.2 1.6 2

μ 0.2 0.4 0.6 0.8 1

${\bar{σ}}_{c}$ /MPa 6 12 18 24 30

${\bar{τ}}_{c}$ /Mpa 5 10 15 20 25

Microscopic parameters	Factor level
E_c/Gpa	5	10	15	20	25
k_n/k_s	0.4	0.8	1.2	1.6	2
μ	0.2	0.4	0.6	0.8	1
${\bar{σ}}_{c}$ /MPa	6	12	18	24	30
${\bar{τ}}_{c}$ /Mpa	5	10	15	20	25

4.2.2 Construction of neural network prediction model

Artificial neural network is a mathematical model that can perform Massively parallel processing, distributed storage and linear operation according to the behavior characteristics of animal neural network. Artificial neural networks are becoming increasingly mature with the advancement of technology, playing an important role in predicting rock mechanics parameters, mining stability, slope deformation, and other geotechnical fields. The back propagation (BP) neural network is a multilayer Feedforward neural network trained according to the error back propagation algorithm. As one of the most widely used network architectures, the main structure of BP neural network consists of input layer, hidden layer, and output layer, among which the hidden layer can have multiple layers, the network structure is shown in Fig. 9. Cybenko [33] believes that a continuous Feedforward neural network with only one internal hidden layer and any continuous S-shaped nonlinearity can well approximate any decision region. Therefore, this study establishes a three-layer BP neural network with a hidden layer to calibrate the mesoscopic parameters of rock-like samples.

Fig. 9

Schematic diagram of BP model network structure.

Based on the database of samples obtained in the previous section and the determined parameters of the BP neural network model, the data samples are first preprocessed, and the first 95 sets of data in the database are used as training data samples, while the remaining 5 sets are used as test samples. Due to the different dimensions of sample data, in order to improve the efficiency and accuracy of network learning, the data should be normalized. Similarly, the various indicators predicted by the BP neural network model need to undergo reverse normalization processing. Equations (4) and (5) respectively refer to the linear transformation function used for normalization and denormalization processing: $a_{n} = \frac{a - a_{min}}{a_{max} - a_{min}}$ (4)

In the equation: a_n is the normalized processing value; a is the original value of the sample; a_max and a_min are the original maximum and minimum values of the sample, respectively. $b = b_{n} (b_{max} - b_{min}) + b_{min}$ (5)

In the equation: b is the value of the anti normalization treatment; b_n is the original value of the simulation results; b_max and b_min are the original maximum and minimum values of the predicted results, respectively.

From the front section, the input layer is set with 4 neurons, and the output layer is set with 5 neurons. The number of neurons in the hidden layer has a great impact on the prediction results, and it is difficult to determine [34, 35], so it is crucial to select an appropriate number of neurons. Based on this, this study refers to previous research results [36], first determines a range according to empirical Equation (6), and then selects the most appropriate number of neurons through comparison. $n = \sqrt{{n_{i} + n}_{0}} + α$ (6)

In the equation: n, n_i and n₀ represent the number of neurons in the hidden layer, input layer, and output layer, respectively, α is an integer between 0 and 10.

The calculated number of hidden layer neurons ranges from 3 to 13. The mean square error value between the expected value and the inversion output value was used to measure the generalization ability of the network, and the number of neurons was finally determined to be 7, as shown in Fig. 10. Therefore, the neural network structure for calibrating the mesoscopic parameters of rock-like samples is 4-7-5.

Fig. 10

Comparison of the number of neurons selected.

The network training function uses the trainlm function, which uses the Levenberg-marquardt algorithm. The hidden layer transfer function of the network uses the tansig function, and the output layer transfer function uses the logsig function. The network performance function is the Mean squared error function mse. The maximum number of iterations is 1000, the learning rate is 0.01, and the minimum target error is 0.01. The error changes during the training process are shown in Fig. 11. As shown in Fig. 11, when the training frequency reaches 200, the model error basically reaches the specified range, and the established BP neural network model meets the accuracy requirements.

Fig. 11

The evolution of error during the training process of BP neutral model.

4.2.3 Reliability verification of BP neural network calibration method

To further verify the reliability of the BP neural network model, 5 sets of test samples were used to test the predictive inversion ability of the neural network, and it was imported into the corresponding microscopic parameters of the established prediction model. Input the inversion results into PFC for numerical simulation, calculate the required macroscopic mechanical parameters based on the simulation results, and compare and analyse them with the macroscopic mechanical parameter test sample values imported into the network model.

The accuracy of the inversion results can be verified by Equation (7) to measure the accuracy of the inversion network system. The definition of precision T is $T = (1 - \frac{| x - X |}{X}) \times 100 %$ (7)

In the equation, x is the simulated macro parameter value obtained by inputting the inverted micro parameters into the PFC for numerical simulation testing, and X is the sample value for macro parameter testing.

The accuracy calculation values of each macroscopic parameter are shown in Fig. 12. As shown in Fig. 12, the accuracy of the test sample inversion results is above 92%, indicating that the established neural network model has a high prediction accuracy value. The established neural network model has high prediction accuracy and a reliable model structure. See Fig. 13 for comparison of Stress–strain curve obtained from numerical simulation and laboratory test under splitting condition. Since there is no microcrack or gap in the PFC2D parallel bonding model, there is no compaction stage in the simulation loading process, and the Stress–strain curve is straight. Therefore, it was observed that there is a certain gap between the simulated curve and the actual experimental curve, but the slopes of the two are approximately equal in the elastic stage. Figure 14 shows a comparison between indoor experiments and PFC simulation of BSS changes with loading angle. From Fig. 14, it can be seen that the simulated BSS values are close to the experimental results. At the same time, the final fracture mode of the sample simulated under different loading angles is in good agreement with the test results, which fully indicates that the micro parameters (see Table 4) inversed from the actual macro mechanical parameters are relatively accurate, and the established Grain flow model is in good agreement with the mechanical properties of the actual rock like sample, indicating that the BP neural network method is feasible to calibrate the micro parameters of the parallel bond model.

Fig. 12

Calculation accuracy values of macro parameters for test samples.

Fig. 13

Comparison of tensile strength of specimen versus axial strain between experiment and PFC simulation.

Fig. 14

Comparison of BSS variation with respect to the loading angles between experiment and PFC simulation.

Table 4

Micro-parameters in PFC^2D

Micro-Parameter	Value
	A	B	Bedding-plane
Ratio of normal to shear stiffness of the particle,kn/ks	1	1	1
Young’s modulus of the particle, Ec(GPa)	18	16.5	5.4
Particle friction coefficient,μ	0.5	0.5	0.15
Young’s modulus of the parallel bond, pb_Ec (GPa)	18	16.5	5.4
Ratio of normal to shear stiffness of the parallel bond, pb_kn/pb_ks	1	1	1
Parallel-bond normal strength, pb_sn (MPa)	26	24	7.8
Parallel-bond shear strength, pb_ss (MPa)	20	16	6

4.3 Failure behaviour analysis

Table 5 shows the fracture morphology, microcrack distribution and microscopic force field of the disc model after failure occurs. In the microcrack distribution image, the blue segment is the tensile microcrack, and the green segment is the shear microcrack. In the microscopic field image, blue represents compressive stress, and red represents tensile stress.

Table 5
Failure behaviour analysis of specimen in micro-level in PFC^2D

θ (°) Fracture morphology Microcrack distribution Microscopic force field

(a)0

(b)15

(c)30

(d)45

(e)60

(f)75

(g)90

As shown in Table 5, the fracture morphology of the model is similar to the experimental results shown in Fig. 7. Based on the analysis of the microcrack distribution and microscopic force field, when the loading angle is small, the main reason for the failure of the sample is tensile failure of the matrix, and the number of tensile microcracks in the weak matrix is greater than that in the strong matrix, indicating that the weak matrix fails before the strong matrix; that is, the weak matrix fails before the strong matrix fails fully. Under the condition of a large loading angle, the failure of the structural plane is the direct cause of the failure of the sample. In this case, there are more shear microcracks located at the structural plane, and the sample is more likely to crack along the structural plane. With changes to the loading angle, the distribution of contact force in the sample constantly changes. Except for the two loading points, the contact force of the sample is mainly concentrated around the loading axis of the sample. When θ ≤ 45°, the contact force is larger near the loading axis and is less concentrated at the structural plane, indicating that it is less affected by the structural plane. When θ ≥ 60°, the contact force concentrates clearly near the structural plane, and the distribution of the contact force changes near the loading axis. The reason why the splitting strength of the sample varies with the loading angle is that the contact force between particles is concentrated to different degrees near the structural plane. The compressive stress is concentrated in the central part of the sample and the structural plane along the loading direction, and the tensile stress is dispersed in the fracture zone on both sides of the microcrack.

4.4 Effect of microcrack evolution on the failure mechanism

The experimental and simulation results confirm that there are obvious differences in the strength and failure mode of rock-like specimens during the Brazil splitting test under different loading angles. To study the damage evolution during the failure process, six kinds of microcracks were recorded and counted in the model: strong matrix tensile microcrack (SMC-T), strong matrix shear microcrack (SMC-S), weak matrix tensile microcrack (WMC-T), weak matrix shear microcrack (WMC-S), structural plane tensile microcrack (SPC-T) and structural plane shear microcrack (SPC-S).

Figure 15 shows the evolution curve of the number of microcracks and stress with strain when the loading angles are 0°, 30° and 60°. According to Fig. 15, the evolution of microcracks can be roughly divided into three stages according to the number and growth rate of the microcracks:

Fig. 15

Tensile stress and micro-crack evolution of disk specimen during the loading.

Stage I: No microcracks appear, and the stress-strain curve is nearly linear.

Stage II: Microcracks start to occur with a slow growth rate. The stress fluctuation shows that there is some irreversible damage in the model.

Stage III: The number of microcracks increases rapidly and accumulates to a high level. A large number of microcracks are connected, and macroscopic cracks appear, which leads to model failure.

As shown in Fig. 15, SMC-S and WMC-S were less prevalent than SMC-T and WMC-T, and SPC-S was more numerous. This indicated that the main fracture modes of the model in the Brazil splitting test were rock-like matrix tensile fracture and structural plane shear fracture. On the other hand, the evolution of microcracks varied with different loading angles. As shown in Fig. 15a and b, when the loading angle increased from 0° to 30°, the number of tensile microcracks in the matrix and structural plane on both sides increased slowly in stage II until in stage III, when the number of tensile microcracks in the matrix on both sides increased rapidly and exceeded the number of shear microcracks, but the number of tensile microcracks in the structural plane was still less than the number of shear microcracks. As shown in Fig. 15c, when the loading angle was increased to 60°, the structural plane microcracks increased rapidly at the beginning of stage II, and the tensile microcracks were more numerous than the shear microcracks until stage III was reached, and the matrix microcracks on both sides were similar to the low loading angle.

Due to space limitations, only specimens with loading angles of 30° were selected for stress field and microcrack evolution analysis. Three marker points were drawn, and the crack distribution diagram at the corresponding time of the marker points is shown in Fig. 16. In Fig. 16, point a is the initiation point of the microcrack, at which there was almost no obvious microcrack, and the compressive stress and tensile stress were distributed along the axial direction. Point b is the point of peak stress. A large number of microcracks spread from the centre of the disc along the structural plane facing both ends. Microcracks appeared at the loading points up and down and gradually propagated to the centre of the disc. When point c finally failed, the microcrack expanded along the axial direction to form a fracture, the tensile crack appeared in the centre of the sample, and the tensile stress decreased. It can be seen that a large number of microcracks appeared only after the peak stress was achieved, while before the peak stress, the axial tensile stress field was still obvious. Because the microcracks increased rapidly after the peak stress was reached, axial macroscopic fracture occurred suddenly. The newly added microcracks were mainly concentrated in the structural plane and the matrix on both sides, and the number of microcracks generated in the weak matrix was much larger than that in the strong matrix, indicating that the model damage changed from being concentrated in the structural plane to being concentrated in the weak matrix.

Fig. 16

Stress and micro-crack evolution of specimen in PFC2D.

To better understand the influence of microcrack types on the fracture morphology of rock-like models, the proportion of matrix and structural plane cracks on both sides to the total number of cracks under different loading angles was calculated, as shown in Fig. 17. As shown in Fig. 17, with increased loading angle, the proportion of microcracks in the matrix on both sides decreased, while the proportion of microcracks in the structural plane increased. The difference in the proportion of microcracks between the weak and strong matrices gradually decreased, which indicated that the development degree of microcracks along the direction of the structural plane was more significant than that of the weak matrix with increased loading angle.

Fig. 17

Proportional variation of microcrack type with loading angle.

To further study the influence of loading angles on the types of microcracks in the model, the variation in the proportion of tensile and shear microcracks in the matrix and structural plane on both sides as a function of the loading angles is plotted in Fig. 18. As shown in Fig. 18(a) and (b), the microcracks in the matrix are mainly tensile cracks, and the proportion of tensile microcracks in the matrix increases with increasing loading angle, while the proportion of shear microcracks decreases, especially in the weak matrix. As the loading angle increases, the microcracks at the structural plane change from consisting largely of tensile microcracks to consisting largely of shear microcracks, and the proportion of tensile and shear microcracks is approximately equal to that when the loading angle is at 45°.

Fig. 18

The proportion of shear and tensile microcracks on both sides of the matrix and structural plane varies with loading angle.

5 Discussion

The anisotropic matrix and loading angle have a significant impact on the mechanical properties and deformation characteristics of rock like disk specimens, and this study briefly discusses these effects.

During the experiment, it was found that the sample had a higher splitting strength at a lower loading angle, and as the loading angle increased, the splitting strength continued to decrease. This is because the compressive stress borne by the structural plane continuously decreases with the increase of loading angle, and the shear stress continuously increases, resulting in a continuous decrease in the contribution of the structural plane to the strength of the specimen (the ability to resist normal loads). For example, when the loading angle is 90°, the direction of the structural plane basically coincides with the direction of the axial force. The structural plane mainly bears horizontal tensile stress, and due to the extremely low bonding force at the structural plane, it is easy to divide into two parts along the structural plane. The structural plane has no obstruction effect on the transverse deformation of the specimen, directly leading to the direct failure of the specimen after bearing extremely low loads. As the loading angle decreases, the compressive stress borne by the structural plane continues to increase. When the loading angle is 0°, the maximum normal stress borne by the sample results in a significant increase in frictional force between structural planes, which greatly suppresses the lateral deformation of the sample and improves its strength and stiffness. In summary, the impact of loading angle on the specimen can be summarized as follows: high loading angle significantly weakens the specimen strength, while low loading angle enhances the specimen strength by constraining lateral deformation.

The biggest difference between anisotropic structural planes and isotropic structural planes is the difference in matrix properties on both sides. This study, combined with numerical simulation, believes that the difference in failure mechanism between anisotropic structural planes and isotropic structural planes is mainly influenced by strength differences, and the influence of matrix strength on the sample is mainly twofold; One is that when the sample is at a low loading angle, the main reason for sample failure is matrix failure, where the weak side matrix is destroyed before the strong side matrix, and when the weak side matrix fails, the strong side matrix is not fully destroyed, resulting in lower sample strength; Secondly, when the sample is at a high loading angle, the main reason for the failure of the sample is the failure of the structural plane. At this time, the strength of the structural plane is mainly provided by the surface bonding force of the structural plane. The mortar used to prepare the weak side matrix has a lower strength, resulting in a lower surface bonding force, making the sample prone to cracking along the structural plane.

6 Conclusion

In this paper, a series of Brazilian splitting tests were carried out to study the splitting strength and failure mode of anisotropic rock-like discs with rough structural planes. Combined with the DIC system, the surface strain of the samples during the failure process under different loading angles was recorded and analysed. A sample model with a rough structural plane was established by using PFC^2D software, and the evolution of microcracks at the microscopic level of the model under different loading angles was analysed.

The sample showed a significant brittleness failure characteristic, and with increasing loading angle, the BSS decreased, approximately along a quadratic function.

Combined with the observation and failure mode of DIC surface deformation, the failure mode of the sample was affected by the combined action of loading angle and matrix strength. At low loading angles, the failure mode of the sample was mainly affected by the matrix. When the loading angle was large, the failure of the sample was mainly controlled by the structural plane.

A BP neural network prediction model has been established to calibrate the micro parameters in the parallel bonding model. The calibration results show that the accuracy of the simulated macro parameters of rock like materials is above 92%, and the overall error of the simulated values is small, indicating that the neural network prediction model is more reliable in reverse performing micro parameters. The neural network calibration method can be used as an efficient and convenient modeling method in parallel bonding models.

There was good agreement between the numerical simulation and the experimental results. The main fracture modes of the samples under the Brazil splitting strength test were rock-like matrix tensile fracture and structural plane shear fracture. With increasing loading angle, the proportion of tensile microcracks in the weak matrix obviously increased, while the proportion of shear microcracks decreased. The microcracks at the structural plane mainly changed from tensile microcracks to shear microcracks.

Footnotes

Acknowledgments

This research was supported by the National Natural Science Foundation of China (Grant No. 52274148) and the Fundamental Research Funds for the Central Universities (Grant No. 2022XJLJ01).

Conflicts of interest

The authors declare that they have no conflicts of interest.

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θ (°)	Fracture morphology	Microcrack distribution	Microscopic force field
(a)0
(b)15
(c)30
(d)45
(e)60
(f)75
(g)90

Experimental and numerical study of disk specimen with rough structural plane under splitting test

Abstract

Keywords

1 Introduction

2 Test materials and procedures

2.1 Sample making and material characteristics

3.1 Axial load–displacement curve

4.1 PFC2D model construction

4.2.1 Orthogonal design of experiments

Table 3 Orthogonal test design table of micro parameters of PFC2D model Microscopic parameters Factor level 1 2 3 4 5 Ec/Gpa 5 10 15 20 25 kn/ks 0.4 0.8 1.2 1.6 2 μ 0.2 0.4 0.6 0.8 1 σ ¯ c /MPa 6 12 18 24 30 τ ¯ c /Mpa 5 10 15 20 25

Table 5 Failure behaviour analysis of specimen in micro-level in PFC2D θ (°) Fracture morphology Microcrack distribution Microscopic force field (a)0 (b)15 (c)30 (d)45 (e)60 (f)75 (g)90

6 Conclusion

Footnotes

Acknowledgments

Conflicts of interest

References

4.1 PFC^2D model construction

Table 3
Orthogonal test design table of micro parameters of PFC^2D model

Microscopic parameters Factor level

1 2 3 4 5

E_c/Gpa 5 10 15 20 25

k_n/k_s 0.4 0.8 1.2 1.6 2

μ 0.2 0.4 0.6 0.8 1

${\bar{σ}}_{c}$ /MPa 6 12 18 24 30

${\bar{τ}}_{c}$ /Mpa 5 10 15 20 25

Table 5
Failure behaviour analysis of specimen in micro-level in PFC^2D

θ (°) Fracture morphology Microcrack distribution Microscopic force field

(a)0

(b)15

(c)30

(d)45

(e)60

(f)75

(g)90