1 Introduction

Deep buried tunnels often suffer brittle instability and failure during the excavation process under high geological stress conditions (Liu et al. 2018). During this process, the surrounding rocks partially begin to break with crack initiation and growth. The deformation and failure of brittle rocks is a gradual process involving the initiation, growth and aggregation of cracks (Martin and Chandler 1994). The deformation and failure process of brittle rocks has been widely concerned by many researchers (Dai et al. 2015; Yang 2016; Yang and Hu 2019; Yang et al. 2018; Zhang and Wong 2013; Zhang et al. 2017; Zhou et al.2015). Researchers studied the failure of brittle rocks and divided the stress–strain curves of brittle rocks into five stages: stage I-crack closure; stage II-linear elastic deformation; stage III-crack initiation and stable crack growth; stage IV-unstable crack growth; and stage V-failure and post-peak behavior (Wawersik and Brace 1971; Eberhardt et al. 1999; Hoek and Bieniawski 1965; Scholz 1968; Zhou et al. 2014; Cheng et al. 2016; Zhang et al. 2011; Hallbauer et al. 1973; Tapponnier and Brace 1976). These five stages of the stress–strain curve are divided by four stress thresholds: crack closure stress (σcc), crack initiation stress (σci), crack damage stress (σcd) and peak strength (σf). The crack closure stress (σcc) indicates the closing of the preexisting microcracks within the rock; the crack initiation stress (σci) indicates the beginning of newborn microcracks within the rock; the crack damage stress (σcd) characterizes the coalescing of the nascent microcracks, which means the specimen loads into the unstable crack growth stage.

To accurately obtain the stress thresholds, many methods have been proposed by researchers. Brace et al. (1966) studied the crack initiation and growth of granite, marble and aplite. These authors stated that the crack initiation stress is the deflected point from linearity and the onset of dilatancy in the stress–volumetric strain curve. Lajtai (1974)noted that lateral strain was more sensitive than axial strain in response to fracture development. Lajtai (1974) proposed to define the point where the lateral strain curve deviates from the linear section as the initiation stress. Martin and Chandler (1994) explored the failure of Lac du Bonnet granite and obtained the stress thresholds by calculating the total volumetric strain and the crack volumetric strain. Eberhardt et al. (1998) and Zhao et al. (2012) explored the relationship between crack initiation and growth and acoustic emission in brittle rock uniaxial compression. These studies indicated that the properties of acoustic emission response were significantly different before and after the crack initiation stress and the crack damage stress. The moving point regression method was also developed to establish the crack closure stress and the crack initiation stress. Nicksiar and Martin (2012) introduced the lateral strain response method to identify the crack initiation stress in igneous rocks, sedimentary rocks and metamorphic rocks. These studies showed that the crack initiation stress of those rocks occurs at 0.42–0.47 of the peak stress under uniaxial compression.

Due to the unloading after excavation/drilling and stress redistribution, the surface of the tunnel is the stress concentration region. As shown in Fig. 1, the axial stress \(\sigma_{\text{r}}\) reduced to 0 after tunnel excavation while the tangential stress \(\sigma_{\theta }\) increased to a certain value. So, the stress concentration region on the tunnel surface is approximately under uniaxial compression. Hence, the reliable stress thresholds under uniaxial compressive test are critically important for evaluating the failure state and designing the support of surrounding rock. Nevertheless, it is difficult to precisely determine the stress thresholds. At present, there is no unified solution method worldwide. In the present study, siliceous siltstone and limestone are used as examples to carry out stress–strain measurement and acoustic emission evaluation during uniaxial compression tests and to compare and analyze several methods for obtaining reliable stress thresholds.

Fig. 1
figure 1

Stress condition on the tunnel surface after excavation

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2 Experimental Procedure

The uniaxial compressive tests were carried out on specimens of siliceous siltstone and limestone in the laboratory of Wuhan University, China. The siliceous siltstone and limestone were taken from the tunnels of the two water transfer projects in Xinjiang Uygur Autonomous Region and Changchun city, China, respectively. The specimens were carefully prepared by cutting and polishing as cylindrical specimens with ɸ 50 mm × H 100 mm. The average density and P wave velocity of siliceous siltstone and limestone are shown in Table 1. The tests were conducted by increasing the strain at 0.001 mm/s on the test specimen using the RMT-301 rock mechanics testing system housed at the Wuhan University. Four displacement sensors were utilized to monitor the deformation and failure behavior of the specimens. The acoustic emission signals generated by the initiation and propagation of internal microcracks in specimen loading were monitored by the test PCI-2 multichannel AE detection system and the NANO-30 type AE sensors (Zhang and Zhang 2017). The frequency of the AE sensors was in the range of 100–400 kHz. The threshold was set at 45 dB, and a gain of 40 dB was used in the experiments. Before the experimental test, the ends of the specimens were greased to reduce the effects of end confinement. The experimental setup for strain measurements (displacement sensors) and AE monitoring (AE sensors) is shown in Fig. 2.

Table 1 Physical and mechanical properties of tested specimens
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Fig. 2
figure 2

Experimental setup for uniaxial compressive strength tests with acoustic emission monitoring of a siliceous siltstone and b limestone

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3 Methods of Obtaining the Rock Stress Thresholds

3.1 Crack Volumetric Strain Method (CVS)

Martin and Chandler (1994) performed a series of damage-controlled tests on specimens of Lac du Bonnet granite. These authors suggested that the volumetric strain (\(\varepsilon_{\text{v}}\)) of rock compression can be divided into two parts: elastic volumetric strain (\(\varepsilon_{\text{ve}}\)) and crack volumetric strain (\(\varepsilon_{\text{vc}}\)). The elastic modulus E and Poisson’s ratio μ are calculated by elastic segment (stage II). Elastic volumetric strain is calculated using formula (1):

$$\varepsilon_{\text{ve}} = \frac{\Delta V}{{\Delta V_{\text{elastic}} }} = \frac{1 - 2\mu }{E}\left( {\sigma_{1} - \sigma_{3} } \right)$$
(1)

where ΔV is the change in specimen volume and ΔVelastic is the change inelastic volume.

Under uniaxial compression, σ3 = 0. Then, the volumetric strain is calculated using the axial strain and lateral strain:

$$\varepsilon_{\text{v}} = \Delta V/V \approx \varepsilon_{\text{axial}} + 2\varepsilon_{\text{lateral}}$$
(2)

where \(\varepsilon_{\text{axial}}\) is the axial strain and \(\varepsilon_{\text{lateral}}\) is the lateral strain.

The crack volumetric strain is calculated using formula (3):

$$\varepsilon_{\text{vc}} = \varepsilon_{\text{v}} - \varepsilon_{\text{ve}}$$
(3)

The crack initiation stress (σci) is the starting point of the stress–strain curve from the crack initiation stage to the stable crack growth stage. The loading strength corresponds to the crack initiation stress (σci) when the crack volumetric strain curve changes from zero to a negative value. The lateral strain–axial stress curve changes from linear to nonlinear from the crack initiation stress (σci) as shown in Fig. 3. The newborn cracks do not increase unless the axial load is increased. The crack initiation stress (σci) is difficult to determine using the stress–strain curve, especially for specimens with many preexisting cracks. The crack damage stress (σcd) is the reversal point in the volumetric strain curve. The volumetric strain reaches its maximum value and turns from compression to dilation, which represents the beginning of the unstable crack growth stage. It shows a significant increase in the strain rate in the axial strain–stress curve. The main reason is that the adjacent tensile cracks are connected with each other and the shear band is gradually formed, which eventually leads to the macroscopic damage. The results of the crack volumetric strain method are shown in Table 2.

Fig. 3
figure 3

Curves of axial stress, total volumetric strain and calculated crack volumetric strain versus axial strain obtained from a siliceous siltstone and b limestone. Stages I, II, III and IV represent crack closure, elastic deformation, crack initiation and stable crack growth, unstable crack growth to failure, respectively

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Table 2 Result of the stress thresholds obtained by the crack volumetric strain method
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The crack closure stress (σcc) is the axial stress where the crack volumetric strain approaches stability. The axial stress reaches the crack initiation stress (σci) when the crack volumetric curve decreases to a negative value. The crack damage stress (σcd) is the reversal point of the volumetric strain curve. These characteristic stresses (σcc, σci, σcd, σf) of the siliceous siltstone and limestone are marked in Fig. 3. The stress thresholds of these specimens are 22% (σcc/σf), 47% (σci/σf), 98% (σcd/σf) of peak stress (siliceous siltstone), and 30% (σcc/σf), 50% (σci/σf), 98% (σcd/σf) of peak stress (limestone), respectively. However, the value of crack closure stress (σcc) and crack initiation stress (σci) will be clearly changed by small fluctuations of Poisson’s ratio. Nevertheless, this result shows that the nonlinear change in lateral strain makes the measurement of Poisson’s ratio complicated (1998). Different Poisson’s ratios corresponding to the crack volume curves are shown in Fig. 4.

Fig. 4
figure 4

Crack volumetric strains calculated by different values of Poisson’s ratio for siliceous siltstone

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It can be seen from the figure that the volumetric strain curve of the cracks solved by different Poisson’s ratios is approximately similar. The differences are the position of the curves approaching the zero point (σcc) and the position of the deviation from the zero point (σci). In Fig. 4, when different values of Poisson’s ratio are taken, the points of σcc and σci are significantly different. In general, the larger the value of Poisson’s ratio that is calculated, the larger the crack initiation stress is. Table 3 shows the results of different values of Poisson’s ratio for solving σcc and σci. Therefore, when the fracture volumetric strain method is used to solve the crack initiation stress, it is necessary to objectively determine the value of Poisson’s ratio that is relatively correct to avoid large errors due to differences in the value of Poisson’s ratio.

Table 3 Result of the stress thresholds obtained by different values of Poisson’s ratio
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3.2 Lateral Strain Response Method (LSR)

Nicksiar and Martin (2012) proposed the LSR to obtain the crack initiation stress (σci). These authors obtained the value by calculating the difference between the lateral strain and the reference line (Fig. 5). Then, polynomial fitting is performed on the value, and the peak point of the fitting curve corresponds to the crack initiation stress (σci). First, by analyzing the axial stress–total volumetric strain curve, the maximum point in the volumetric strain curve is obtained, which corresponds to the crack damage stress (σcd). In the axial stress–lateral strain curve, the crack damage stress point and zero point are selected as a reference line. At the same axial stress level, the actual lateral strain is subtracted from the reference line to solve the lateral strain difference, and the relationship between the lateral strain difference and the axial stress is plotted. By polynomial fitting of the data points, the maximum value of the fitting curve is the crack initiation stress (σci).

Fig. 5
figure 5

Principle of the lateral strain response method

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The key to determine the crack initiation stress by the lateral strain response method is to identify the crack damage stress. Utilizing the lateral strain response method to obtain the crack initiation stress, the determination of the maximum value of the lateral strain difference is unique in its use of data fitting to determine the extreme value, and human error is avoided. As seen from Fig. 6, the method to calculate the crack initiation stress (σci) is relatively objective. The crack initiation stress (σci) is 64% (siliceous siltstone) and 55% (limestone) of the peak stress. As seen from Fig. 6, the value obtained using the LSR is much larger than that using the CVS. When the specimen starts loading, the slope of the initial lateral strain curve of the specimen is larger than the slope of the reference line. The lateral strain difference peak point is the point where the slope of the tangent strain curve is the same as the slope of the reference line, so the elastic deformation section must be below the peak point of the lateral strain difference. This method does not explain the physical meaning, and the crack initiation stress is larger than the above CVS result.

Fig. 6
figure 6

Curves of lateral strain difference for a siliceous siltstone and b limestone

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3.3 Moving Point Regression Method (MPR)

Eberhardt et al. (1998) proposed a new method to obtain the crack closure stress (σcc) and the crack initiation stress (σci). The volumetric stiffness (Fig. 7) is calculated from the volumetric strain divided by the axial stress. As the cracks close, the volumetric stiffness rapidly increases. The volumetric strain and volumetric stiffness curves during crack closure are non linear, followed by a linear region, and the slope represents the rate of change. This breakthrough of the slope represents a shift from crack closure to an approximately linear elastic behavior, and a further slope mutation represents the transition from a linear segment to the crack initiation and stable crack growth segment. It is well known that the reversal point of the volumetric strain curve represents the loading of the rock specimen into the unstable crack growth segment. This reversal point indicates the point at which the volumetric stiffness curves from positive to negative.

Fig. 7
figure 7

Volumetric stiffness curves of a siliceous siltstone and b limestone obtained by the moving point regression technique

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As we can see in Fig. 7a, from the beginning of loading to approximately 53 MPa, the specimen is in the crack closure segment, and the volumetric stiffness curve increases with irregular fluctuation. After this loading phase, the curve enters the linear segment. When the loading force reaches approximately 110 MPa, the curve begins to change irregularly. This point is the onset of stable crack growth. Until the point of volumetric stiffness changes from positive to negative (crack damage stress), the specimen loads into the unstable crack growth segment.

The final stress thresholds were 24% (σcc/σf) and 50% (σci/σf) (siliceous siltstone), 30% (σcc/σf) and 50% (σci/σf) (limestone). Compared with the crack volumetric strain method, the method of the MPR to obtain the stress threshold avoids the influence of the corresponding deformation analysis of rock physical and mechanical parameters, such as Poisson’s ratio. The disadvantage of the MPR method is that it still needs the key inflection point in the artificial analysis curve. The MPR method has not yet removed the influence of subjectivity.

3.4 AE Method

Acoustic emission parameters are an effective means to obtain the segment of rock in a rock compression test. The closure, generation and expansion of microcracks are accompanied by the occurrence of many acoustic emission activities. The ringdown counts are the number of times the signal exceeds the preset threshold datum. The purpose of setting a threshold datum is to distinguish fracture-related acoustic events from background noise, but this threshold must be below enough to detect the beginning of the micro fracturing process. The ringdown count is a valid parameter that can represent the segment of rock compression.

Figure 8 shows the evolution of acoustic emission ringdown counts during the rock compression test. There will also be some acoustic emission in the initial stage of specimen loading, which is caused by the closing of the preexisting cracks in the specimens. Barely ringing counts occur until the specimen is loaded to 110 MPa. However, once the specimen reaches 110 MPa, the ringdown counts begin to occur and continue to increase steadily. This method has good consistency with σci calculated by both the crack volumetric strain method and moving point regression method. Starting from 200 MPa, the ringdown counts began to substantially increase. Many AE ringdown counts reflected the unstable growth of the crack, which is the onset of unstable crack growth (σcd). This result demonstrates that acoustic emission can be used as an effective means to confirm these obtained stress thresholds.

Fig. 8
figure 8

Log acoustic emission response for a siliceous siltstone

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4 Conclusion

These stress thresholds are important parameters for analyzing the deformation and failure of brittle rocks. If the stress level is lower than crack initiation stress (σci), no damage occurred in the rock. If the stress level is higher than crack initiation stress (σci) but lower than crack damage stress (σcd), cracks randomly initiated but they will cease at a certain length without coalescence. It is a stable cracking stage. If the stress level is higher than the crack damage stress (σcd) but lower than peak strength (σf), cracks will propagate continuously and coalescence to reach macroscopic failure. Based on the measured in situ stress and stress thresholds, we can predict the cracking condition and design support/grouting measures before excavation. Hence, it is important to determine the stress thresholds precisely.

In the present study, using strain gauge analysis and acoustic emission monitoring, a variety of methods is used to obtain the stress thresholds. The following observations were made with respect to uniaxial testing performed on specimens of siliceous siltstone and limestone:

  1. 1.

    When the stress thresholds are obtained by the crack volumetric strain method, the stress thresholds of these specimens are 47% (σci/σf), 98% (σcd/σf) of peak stress (siliceous siltstone), and 50% (σci/σf), 98% (σcd/σf) of peak stress (limestone).The crack volumetric strain method is a relatively objective method to obtain these stress thresholds, but its accuracy is easily affected by Poisson’s ratio.

  2. 2.

    When utilizing the moving point regression technique to determine the stress thresholds, the measured cracking stress are approximately 24% (σcc/σf) and 50% (σci/σf) (siliceous siltstone), 30% (σcc/σf) and 50% (σci/σf) (limestone). There still are certain subjective factors, but this method is independent of Poisson’s ratio. Thus, the stress threshold of the specimens can be more accurately obtained.

  3. 3.

    The lateral strain response method for crack initiation stress (σci) can remove errors caused by human subjective factors. The crack initiation stress (σci) is 64% (siliceous siltstone) and 55% (limestone) of the peak stress, which is larger than the crack volumetric strain method and the moving point regression technique. The physical meaning of this method is not yet clear, and its practicality requires further study.

  4. 4.

    Acoustic emission activity is monitored throughout the uniaxial compression test and can reflect the variation of the crack inside the rock to some extent. Therefore, the use of this activity is recommended as a supplementary strain measurement method to obtain the stress thresholds in the compression test.

It should be noted that these conclusions were drawn from only siliceous siltstone and limestone. In the future, more rock types should be tested and compared to see if they obey the same rule.