1 Introduction

Brittle fracture of rock occurs due to the initiation and propagation of crack induced by stresses. The crack damage of rocks is a precursor to brittle failure. The brittle fracture ultimately leads to borehole collapse, reservoir fracture and instability of underground engineering such as tunnel, cavern, slope, mine and other engineering structures [1]. The test of large-scale indoor brittle rock failure and instability represents that stress–strain curve includes the crack closure stage, the linear elastic deformation stage, the stable crack propagation stage and the unstable crack propagation stage. The important stress threshold values are related to crack closure stress, crack initiation stress, crack damage stress (dilatancy stress) and peak stress [1,2,3,4]. Moreover, the crack initiation stress represents the compaction of the original cracks inside the rock and the initiation and expansion of new cracks. Generally, the crack initiation stress corresponds to 30%-50% of the peak stress [5], which is the threshold value for the stable growth of rock cracks, indicating the end of the elastic deformation stage. The dilatancy stress is the initial stress of the unstable propagation of rock cracks, which generally corresponds to 60%-80% of the peak stress [6]. Chen and Kang [7] believe that the rock crack propagation and failure occurred at the dilatancy point. It will be better and more accurate to use dilatancy stress as the long-term strength of rock.

Therefore, it is essential to study the characteristics of crack initiation stress and dilatancy stress for determining the strength criterion of rock, which is significantly important for analyzing the progressive failure process of rock and predicting rock failure. Zhang et al. [8] studied the relationship between the initiation stress, dilatancy stress and peak stress of rock under uniaxial loading in different directions such as vertical, parallel and oblique. It was found that the rock anisotropy plays a significant role in the crack propagation under uniaxial loading. Liu et al. [9] determined the marble stress threshold value under different confining pressures, established the characteristic strength envelope and further subdivided the traditional damage area into failure area, damage area and disturbance area. Zhang et al. [10] studied that the crack initiation stress and dilatancy stress of marble under uniaxial loading were 0.4 ~ 0.5 and 0.8 times of peak stress, respectively. Zhu et al. [11] studied the crack initiation stress of granite under different confining pressures. The elastic stage became longer with the increases of confining pressure and the rate of crack propagation slowed down at the starting crack growth stage. Based on the fracture mechanics theory, the initiation criterion of cracks was deduced. Huang et al. [12] studied the effects of different loading rates on stress–strain curves, failure modes, strength, elastic modulus, deformation modulus and energy dissipation under uniaxial compression. They found that crack initiation stress, dilatancy stress, peak stress, a ratio of crack initiation stress, dilatancy stress and peak stress increased with the increase in the loading rate.

The above analysis shows that the three mechanical parameters, i.e., crack initiation stress, dilatancy stress and peak stress, during progressive rock failure are of great significance in the analysis of internal crack propagation mechanism and strength failure mechanism in the rock. Previous studies on rock stress thresholds have focused on the determination of σci, σcd, σci/σmax, σcd/σmax of different rocks in loading processes. However, very little research is conducted on the correlation between crack initiation stress, dilatancy stress and mechanical parameters such as elastic modulus and porosity. In fact, the strength of rock is closely related to elastic modulus and porosity. Moreover, it shows that there is a strong positive correlation between rock uniaxial compressive strength and elastic modulus. The relationship function varies for different rocks [3, 4, 13,14,15]. The previous research shows a relationship for limestone, E = 0.374σc, R2 = 0.784 and breccia, E = 0.438σ 0.675c , R2 = 0.949 [3]. The uniaxial compressive strength decreases with the increase in the rock porosity. The influence of porosity on rock strength is mainly reflected due to the presence of micro-fissures in the rock, and stress concentration tends to occur on the edge of micro-fissures under the action of external forces, which leads to the initiation and development of cracks. Therefore, micro-fissures are the main source of stress concentration and have a strong impact on the strength of rock [16,17,18]. Palchik and Hatzor [19] studied the porous anisotropic sandstone and their research findings show that porosity has a significant impact on the uniaxial compressive strength of rock, and the influence of lattice boundary of particles can be almost ignored. Uniaxial compressive strength is a function of elastic modulus and porosity.

However, what is the relationship between crack initiation stress, dilatancy stress, elastic modulus and porosity? Can elastic modulus and porosity predict the crack initiation stress and dilatancy stress of rock? Based on this, the uniaxial loading test of sandstone samples was carried out in this study to explore the variation law of crack initiation stress and dilatancy stress with elastic modulus and porosity, and to analyze the internal causes. The prediction model of dilatancy stress based on porosity, peak stress, peak strain, crack initiation stress, crack initiation strain and elastic modulus is established using an artificial neural network (ANN). The research results can provide a reference for related rock mechanics test and rock engineering deformation and failure mechanism research and will also give specific guidelines significance for the efficient design of excavation and support in deep rock engineering.

2 Experimental Design

The processing size of the rock sample is 70 × 70 × 140 mm. The two ends of the specimen were carefully ground with a grinder and sandpaper so that the parallelism of the upper and lower surfaces was within 0.05 mm and the flatness of the surface was within 0.02 mm. The electrohydraulic servo universal testing machine of model C64.106* with a maximum load capacity of 1000 kN was used as an experimental loading device. The testing machine was set to load at an equal displacement of 0.1 mm/min and the collection rate of 10 times/s, since the rock’s macro-failure imposes no effects on rock components and the related microstructures. Sandstone samples after tests were thoroughly evaluated under a scanning electron microscope (Quanta™ 250 SEM) for elemental and microstructure analysis. The sandstone samples pieces after loading were placed on the sample stage of the SEM, and the sample chamber was vacuumed; next, the sample stage was moved and rotated, and the fragments were observed under the SEM at a magnification factor of 200 and 1500, respectively. The clear pictures of the pieces with distinct structural characteristics were finally taken and recorded, while some satisfactory samples from tested sandstone were also selected for rock elemental analysis. Different samples from the same tested sandstone sample (denoted as piece A and piece B) were selected and magnified by 200 times and 1500 times for rock microstructures and elemental analysis.

3 Mechanical Parameters

3.1 Elasticity Modulus

There are three methods to calculate rock elastic modulus: 1) tangent modulus, 2) secant modulus and 3) average modulus [20, 21]. In this paper, the average modulus is used to calculate the elastic modulus in the elastic phase. Assuming that the stress–strain formula is σ = f(ε), the final modulus of elasticity is obtained by the following formula:

$$ E = \frac{{f(\varepsilon_{1} ) - f(\varepsilon_{2} )}}{{\varepsilon_{1} - \varepsilon_{2} }} $$
(1)

where E refer to elastic modulus, f(ε1) and f(ε2) are the stress values corresponding to the starting and ending points of the elastic phase, respectively.

3.2 Porosity

It is the ratio between the sums of all available pore spaces in a rock sample to the volume of the rock sample. Porosity is obtained by calculating the dry density and particle density of the rock sample [6], and its calculation formula is

$$ n = \left[ {1 - \left( {\frac{\rho }{{G_{s} }}} \right)} \right] \times 100\% $$
(2)

In Eq. (2), ρ is the dry density of rock sample, g/cm3, and Gs is particle density, 2.6 g/cm3.

3.3 Crack Initiation Stress and Dilatancy Stress

In this paper, the stiffness–stress curve is used to determine the crack initiation stress and dilatancy stress of rocks [22, 23], and the stiffness–stress curve of rock sample A1 is analyzed.

As shown in Fig. 1, the stage AB is the crack closure stage in the uniaxial loading process of rock sample A1. When the stiffness–stress curve intersects with the strain–stress curve, it indicates that the crack closure stage is over. When the stiffness continues to increase and reaches an inflection point C (the extreme point of stiffness–stress curve curvature), the elastic stage ends. This point is called a crack initiation point of rock samples. The stiffness curve increased and reached point D (the sudden point of the rigid stress curve), which is called dilatancy point, and corresponding stress is dilatancy stress. The stage CD is the stable crack propagation stage. Furthermore, with an increase in loading, the curve reached point E, which is called peak stress. The stage DE is the unstable crack propagation stage. At this stage, the rock sample gives out the noticeable sound; the cracks rapidly develop, propagate and combine each other to form the macroscopic fracture surface, whereas the rock sample remains intact.

Fig. 1
figure 1

Results of uniaxial compression tests for sample A1

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3.4 Artificial Neural Network

ANN models are nowadays widely used for prediction purposes in complex problem of engineering, which is manually difficult and unable to solve [24,25,26,27,28,29,30]. It is accepted as one of the most efficient meta-modeling methods based on its flexibility, to be used for solutions of simple to complex nature problems in different fields of engineering [31,32,33,34]. The significant feature of ANNs is to build soft sensors that can easily predict the critical quantities without measuring them [34]. ANN models are trained, tested and validated based on inputs and outputs data for the prediction of the required quantities and diminishes the time and money required for the experiments. ANN does not require any prior knowledge about the nature of the problem, which is considered one of the key benefits of it [35]. This makes ANN as efficient tool to be applied for accurate prediction of the dilatancy stress by practitioners, who have not expertise in numerical modeling and programming. In this research, ANN with a multilayer perceptron (MLP) is applied for the prediction of dilatancy stress.

The accuracy and performance of any classification and task in ANN need a supervised learning algorithm for the training process. A set of examples used in the training of networking by backpropagation (BP) algorithm to connect and link the nodes and find the parametric function, also known as weight insufficient ways. The mean square error (MSE) is reduced iteratively to minimize the error between the actual and predicted output, and even training helps to identify each iteration weight [36]. The BP algorithm is used to train different types of networks. The previous work reveals that the BP algorithm considers and assumes random value, which is used in activation function of neural network operation to calculate the output. To achieve minimum error, the weight value will be modified and will repeat the process until the minimum result is achieved [37]. The model needs to be trained, and many researchers have given detail about how the best training is achieved [38,39,40]. In this research, the ANN performance is evaluated by the coefficient of determination (R2), root mean square error (RMSE) and mean absolute error (MAE).

4 Experimental Results

4.1 Microstructures and Component Analysis Results

Previous experimental results indicate that rock’s microstructure and components generally affect the rock strength to certain degree under uniaxial compressive test [41]. Hence, it is necessary to observe and illustrate microstructures and components of the present sandstone samples to provide a solid foundation for expounding relation between the rock microstructure and components and strength in depth.

  1. (1)

    Microstructures of the sandstone samples

Figure 2a shows the microstructures of the sandstone samples after the magnification by 200 times. Overall, the samples had uneven surfaces, on which a lot of irregularly shaped holes can be obviously observed. In addition, some short micro-fractures with rough and irregular edges also can be noted on the surface of the piece. Figure 2b shows the microstructures of the sandstone samples after the magnification by 1500 times. In the right part of samples, honeycomb structure can be observed; different crystal particles with different sizes were staggered in distribution. Moreover, various sizes of micro-holes also existed among crystal particles and showed no visible crack development. In the left part of the sample, the crystal had smooth surface, angular edges and some interconnected trans crystalline cracks were formed; additionally, a lot of micro-holes were distributed around the crystal.

Fig. 2
figure 2

Microstructure of sandstone specimens

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  1. (2)

    Components of the sandstone samples

Table 1 shows the component analysis result, such as statistics of various elements, mass percentage, percentage elements and error of the mass percentage of each element. Table 1 shows that Si and O occupy highest percentages in piece A, with the sum of their mass percentages of up to 92.10% and the sum of their number percentages of up to 94.96%, followed by Al and Fe, and only a small amount of F, K, Ti, Cl, Ca and Mg can be observed. The measurement errors of the mass percentages of O and Si were also high (5.51% and 1.52%, respectively), while the errors of the mass percentages of other elements were low, and all were below 0.30%.

Table 1 Composition of sandstone samples
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4.2 Peak Stress

To reflect the reliability of data to a certain extent, the relationship between uniaxial compressive strength, elastic modulus and porosity of sandstone specimens was discussed. Figure 3 shows the relationship curve between peak stress and modulus of elasticity, with the correlation coefficient 0.89. Figure 4 shows the relationship curve between peak stress and porosity having correlation coefficient 0.79.

Fig. 3
figure 3

Peak strength and elastic modulus

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Fig. 4
figure 4

Peak strength and porosity

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Furthermore, uniaxial compressive strength is positively correlated with elastic modulus, while it is negatively correlated with porosity. This is because the elastic modulus reflects the stiffness of the rock, which is reflected in the contact stiffness between particles and the stiffness of the rock matrix. Macroscopically, it reflects the ability of rock to resist the deformation. The higher the elastic modulus, more will be the ability of rock to resist compression deformation and the higher will be uniaxial compressive strength. Porosity is the measurement index to measure the volume of pore space in rock, which is a reflection of the initial micro-cracks, micro-holes and tension cracks in the rock. To a certain extent, it can reflect the initial damage degree of rock. The larger the porosity of the rock means, the more the internal hole and crack development and ultimate reduction in the compressive strength.

4.3 Crack Initiation Stress

Figure 5 shows the relationship between the crack initiation stress and the elastic modulus. The power function, exponential function and polynomial were used for fitting, the fitting results were y = 1.56x1.25, y = − 5.03 + 3.28x and y = exp(1.27 + 0.28x − 0.007x2), and their corresponding correlation coefficients were 0.84, 0.77 and 0.81, respectively. In this paper, the power function having high correlation coefficient is selected to characterize the relationship between the crack initiation stress and the elastic modulus of rock.

Fig. 5
figure 5

Crack initiation stress and elastic modulus

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The relationship curve between crack initiation stress and porosity, as shown in Fig. 6. The power function, quadratic function and primary function were used for fitting, the fitting results were y = 107x−0.75, y = 48.5 − 3.0x-0.012x2 and y = 49.18 − 3.18x, and the corresponding correlation coefficients were 0.74, 0.77 and 0.81, respectively. In this paper, the first-order function with a high correlation coefficient is selected to characterize the relationship between rock crack initiation stress and porosity.

Fig. 6
figure 6

Crack initiation stress and porosity

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4.4 Dilatancy Stress

Figure 7 represents the relationship between the dilatancy stress and elastic modulus. The power function, exponential function and polynomial were used for fitting, the fitting results were y = 1.28x1.55, y = −18.6 + 6.52x and y = exp(1.1 + 0.38x − 0.005x2), and the corresponding correlation coefficients were 0.84, 0.81 and 0.84, respectively. In this paper, the power function with high correlation coefficient is selected to characterize the relationship between rock dilatancy stress and elastic modulus.

Fig. 7
figure 7

Dilatancy stress and elastic modulus

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The relationship between dilatancy stress and porosity is shown in Fig. 8. Different functions were utilized such as power function, quadratic function and primary function are used for fitting, the fitting results are y = 62.8x0.78, y = 94.6 − 7.79x + 0.09x2 and y = 89.9 − 6.39x, and the corresponding correlation coefficients are 0.77, 0.83 and 0.84, respectively. In this paper, the first-order function with high correlation coefficient is used to characterize the relationship between rock dilatancy stress and porosity. The crack initiation stress and the dilatancy stress of the rock sample become more significant as the elastic modulus increases and become shorter as the porosity increases. From the energy point of view, the failure of the rock is carried out with the minimum energy consumption under the corresponding constraints. The larger the porosity, the smaller the energy required for the damage. Therefore, as the porosity increases, the stress required to initiate crack initiation and unsteady propagation of the rock will decrease.

Fig. 8
figure 8

Dilatancy stress and porosity

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4.5 Stress Ratio

The effects of elastic modulus and porosity on σci/σmax and σcd/σmax were investigated. Dimensionless parameters σci/σmax and σcd/σmax reflect the relative positions of crack initiation stress and dilatancy stress on stress–strain curves during rock failure, respectively. Because the absolute value of characteristic stress varies greatly in the process of rock failure under different elastic modulus and porosity, the influence of elastic modulus and porosity on the process of rock failure can be more clearly reflected by using relative parameters.

Figure 9 shows the variation trend of σci/σmax and σcd/σmax with elastic modulus and porosity. As shown in Fig. 9, σci/σmax almost does not change with the change of elastic modulus and porosity, and its value is about 0.443. σcd/σmax change with the increase in the elastic modulus and shows an uptrend line (positive slope), while with the increase in the porosity a down trend line (negative slope) is noted. It reflects that under condition of high porosity or low elastic modulus, the proportion of the stable crack propagation stage will decrease the loading in the whole process of rock sample. In contrast, the proportion of the unstable crack propagation stage in the whole loading process of rock sample will increase, and more internal damage is expected.

Fig. 9
figure 9

Stress ratio with porosity and elastic modulus

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4.6 Failure Mode

The sandstone selected in this test is hard brittle sandstone, which usually breaks down immediately after peak stress and accompanies with sound. The stress–strain curve shows a rapid decline after peak stress. The low-porosity rock sample have high uniaxial compressive strength is high, and most rock samples are axial splitting i.e., tension failure near parallel to the axial direction, as shown in Fig. 10a. The specimens will not be destroyed immediately after axial splitting, but have a specific bearing capacity until the complete formation of a tensile failure crack inside the rock occurred. Shear failure occurs in most of rock samples with medium porosity. Compared with rock samples with low porosity, the failure of samples is more violent and tends to be rapid after the peak strength. Meanwhile, the failure process of rock samples with medium porosity takes a relatively short time. The end effect limits the further development of tensile cracks. The failure mode shows as a large area falling off on the periphery of the rock samples, and finally, shear failure occurs, as shown in Fig. 10b. The tensile shear composite failure occurs in most rock samples with high porosity, as shown in Fig. 10c. These rock samples with high porosity are more prone to the propagation of primary cracks and the generation and development of new cracks (tensile cracks and shear cracks). The propagation of primary cracks can prompt more new cracks growth and is prone to local failure, which significantly reduces the strength of rock samples.

Fig. 10
figure 10

Experimental visible light

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In conclusion, with the increase in the porosity, the crack initiation stress and dilatancy stress decreases, but the proportion of the unstable crack propagation stage in the whole process of rock loading increases. In addition, the plasticity of the rock increases with the increase in the porosity, which is manifested as the tensile failure is weak, while shear failure is strengthened. The mode of the failure varies with porosity, i.e., when porosity is low, medium and high, the failure will be tensile, shear and tensile shear composite failure, respectively.

5 Artificial Neural Networks

5.1 Design and Training of Neural Networks

ANN model was designed in MATLAB software for dilatancy stress. The model was developed using six input variables includes elastic modulus, porosity, crack stress, crack strain, peak stress, peak strain and output layer having one output variable i.e., dilatancy stress. The descriptive statistics of the data is given in Table 2.

Table 2 Descriptive statistics of data
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The schematic of the ANN topology is shown in Fig. 11. The Levenberg–Marquardt algorithm was used for BP network as a training function. In the literature, it is revealed that network, training of neural network and architecture need algorithm training process. Different algorithms are available for ANN, but the most valuable and efficient is Levenberg–Marquardt algorithm suggested by [42]. He conducted a detailed study on types of learning algorithms available for ANN training. Rao and Kumar [43] research shows that the Levenberg–Marquardt is more efficient and takes less time for convergent to achieve optimum epoch and give better result as compared to other algorithms. For each model, the neuron was selected for the hidden layer on trial-and-error methods manually.

Fig. 11
figure 11

Architecture topology of ANN for dilatancy stress

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5.2 Network Phases and Regression Model

The ANN dilatancy stress models all phases, i.e., training, validation, testing and regression values are shown in Fig. 12. Figure 12 shows good regression for training, validation and testing values between predicted and measured dilatancy stress. A good coefficient of determination value (0.96) between predicted and calculated dilatancy stress was found, as shown in Fig. 12.

Fig. 12
figure 12

ANN phases of training, validation, testing and regression coefficient for dilatancy stress

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5.3 Network Performance and Accuracy

The network performance and accuracy was evaluated by the maximum R2 and minimum MSE and RMSE values. The MSE and RMSE values decrease as the number iteration increases by increasing neuron numbers for the hidden layer. The optimum value of the dilatancy stress is achieved by trial-and-error methods of neuron number. In the present study, optimum dilatancy stress value is achieved at five neurons in the hidden layer with R2 (0.96%), MAE (0.18981 MPA) and RMSE (0.17016 MPa) as shown in Table 3 and Fig. 13, respectively. The calculated and measured dilatancy stress is shown in Fig. 14.

Table 3 Correlation between actual vs predicted following mean absolute error and root mean square error using different ANN architectures
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Fig. 13
figure 13

a Relation between RMSE and R2 with neuron variation, b relation between MAE and R2 with neuron variation

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Fig. 14
figure 14

Calculated and ANN dilatancy stress verse number test

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6 Discussion

(1) The sandstone samples selected in this paper are from the same engineering area, but the crack initiation stress and dilatancy stress of rock are quite different. On the one hand, there are differences in the structure of the rock itself; on the other hand, it is related to the geological environment in which the rock exists. In a high-stress environment, when the core is drilled, the rock is released from the initial stress state to the complete stress state, which is equivalent to the tensile stress. The larger the initial stress, the larger the tensile strain and the greater the initial damage. The initial damage caused by excavation unloading of rock under low-stress condition is smaller than that caused by excavation unloading [44]. In summary, the porosity of the rock varies greatly. Therefore, in the future, when evaluating the crack initiation stress and dilatancy stress of underground engineering rock, multi-point sampling should be carried out according to the occurrence conditions of surrounding rock to obtain comprehensive strength parameters.

(2) Szwedzicki [45] once made a speculation about the relationship between failure mode of samples and ultimate compressive strength. The ultimate compressive strength of rocks is a function of their failure mode. The ultimate compressive strength is graded high, medium and low in the order as; tensile failure intensity, tensile shear composite failure intensity and shear failure intensity, respectively. However, in this test, it is found that the compressive strength of tensile failure is relatively high, followed by shear failure rock sample, and the failure mode of rock sample with small strength is tension shear composite type, and the corresponding relationship between failure mode and compressive strength is not strict. The research results in this paper are not reliable with the test results of Wang et al. [46]. It may be that the selected rock types are different, and the distribution of pore fractures and mineral composition in different types of rocks are different. It is also possible that the loading method is different. In this test, the loading method of 0.1 mm/min is adopted, while Wang et al. [46] conduct the sandstone loading test with different strain rates.

(3) Based on the fact that σci/σmax does not change with the change of elastic modulus and porosity, and its value is about 0.443, it is proposed to evaluate the crack initiation stress by measuring the peak strength of rock samples. In addition, an innovative ANN model is proposed to predict the dilatancy stress of rock. The prediction method of crack initiation stress and dilatancy stress is proposed only for the sandstone test studied in this paper. The porosity and elastic modulus vary in this research work range from 3.51% - 11.8% and 4.55-12.9 GPa, respectively. The prediction model is not suitable for the case of cyclic loading and unloading. The prediction method is based on the statistics of laboratory test results of a large number of rock samples. When we predict the initiation stress and dilatancy stress, it is not necessary to test the size, morphology and distribution characteristics of microparticles in the rock; neither do we need to carry out microscopic tests such as electron microscope scanning. Only the physical and mechanical parameters such as elasticity, porosity and peak strength measured by macroscopic tests can be used to predict the crack initiation stress and dilatancy stress.

(4) The rock dilatancy stress is closely related to the defect in structure, such as pore and crack. The existence of these defects in structures changes the mechanical properties of rock. The higher the porosity, the lower the dilatancy stress value. In addition, the dilatancy phenomenon is common in the process of rock failure under load. The dilatancy point is the turning point of rock volume from compression to expansion. It is also the starting point of the unstable development of rock fissures. After the dilatancy point, the fissures developed rapidly until the rock is damaged. Therefore, the dilatancy stress value can be used not only as one of the indicators to evaluate the development degree of pore and fracture in the rock, but also as one of the early warning stresses of rock failure, which may deserve further attention.

7 Conclusions

The following conclusions are drawn from the research:

  1. 1.

    The crack initiation stress and dilatancy stress of rock are power function relationship with its elastic modulus, but negative linear correlation with its porosity.

  2. 2.

    σci/σmax does not change with the change of elastic modulus and porosity, and its value is about 0.443. σcd/σmax increased with the increase in the elastic modulus, while it decreased with the increase in the porosity.

  3. 3.

    When the porosity is low, most of the rock samples are tensile failure; when the porosity is medium, most of the rock samples encountered shear failure; when the porosity is high, most of the rock samples faced tensile shear composite failure.

  4. 4.

    The ANN prediction model of dilatancy stress shows the coefficient of correlation is 0.96, with minimum MAE 0.189 and RMSE 0.170.

  5. 5.

    The characteristics of rock crack initiation stress and dilatancy stress with respect to elastic modulus and porosity were analyzed in this paper. On this basis, a new method ANN is proposed to predict dilatancy stress. The research results lay a theoretical and experimental foundation for determining the long-term strength of the rock, predicting the failure and instability of the rock, analyzing the damage process of the rock and studying the weight of each index on the damage.