Introduction

Rock is typically characterized by heterogeneity, anisotropy and discontinuity, there are a large number of pores, fractures, schistosity, bedding and other defects. These defective structures determine the non-uniform damage and deformation localization of rock under loading (Erarslan and Williams 2012; Zhu et al. 2019). The deformation localization is the mechanism of microcrack initiation, propagation, coalescence and macrocrack development (Yuan and Harrison 2006; Ghasemi et al. 2021). Revealing the localized deformation process is of great significance to the stability assessment of rock slopes and underground chambers in active tectonic areas (Shen et al. 2014; Gischig et al. 2016; Zhu et al. 2021).

Many uniaxial and triaxial tests have been carried out to study the deformation evolution of rock under loading (Liu et al. 2014; Cerfontaine and Collin 2018; Fu et al. 2020; Yang et al. 2020; Zheng et al. 2020; Liu and Dai 2021). The rock damage evaluation methods based on residual strain, elastic modulus, energy dissipation and acoustic emission (AE) signals were proposed (Antonaci et al. 2012; Liu and He 2012; Kim et al. 2015; Peng et al. 2019; Li et al. 2020). To deeply understand the deformation development process of rock under loading, various advanced monitoring technologies have been introduced, including holographic interference, AE, thermal infrared, computed tomography (CT), nuclear magnetic resonance imaging (NMRI) so on. Holographic interference technique could reveal the formation of permanent deformation under loading from the microscopic viewpoint of rock particle deformation, slip and loosening. The propagation process of microcracks in rocks during loading is revealed by obtaining AE parameters, including AE counts, dissipation energy, risetime/amplitude (RF) and average frequency (AR), and spectral features (Meng et al. 2018; Zhao et al. 2020). Monitoring the change of rock temperature under loading by thermal infrared can reveal the properties of initiated cracks and obtain the precursory information of rock failure (Cai et al. 2020; Cao et al. 2020). In addition, CT and NMR techniques were widely used to quantitatively evaluate the changes of pores and cracks in rocks (Feng et al. 2004; Xie et al. 2018; Wang et al. 2021). Although the above monitoring technologies have been widely used, monitoring and quantifying non-uniform damage and deformation localization during loading is difficult.

Digital image correlation (DIC) is an effective technology for full-field and real-time measurement of surface displacement and localization band of rock. It has been successfully applied to the observation of surface displacement of different rocks and test types. Huang et al. (2020) studied the crack initiation and propagation of tunnel sidewall under external loading and revealed that faults had an important impact on the stability of rock tunnels. Cheng et al. (2017) studied the deformation localization and crack propagation process of a series of composite rock samples with different dip angles. Liu et al. (2021) revealed the crack initiation process of granite with a single defect under uniaxial compression and identified three crack types with different initiation mechanisms by the DIC technique. Song et al. (2013) reported the whole process of crack propagation and coalescence in the sandstone through surface strain. Song et al. (2013) investigated the completed process from crack propagation to failure of sandstone through surface displacement. Zhang et al. (2020a) determined the crack formation and development process of rock samples with different geometric defects by deformation localization.

DIC technique has been widely used in rock mechanics tests. However, few studies involve the quantitative evaluation of deformation localization under monotonic and cyclic loading. In this paper, the variance of the surface deformation field is introduced, the characteristics of stress–strain and stress-variance curves are compared. The development process of the stress-variance hysteresis curve during cyclic loading was investigated. Based on the AE counts and variance, the damage evolution process of the sample under cyclic loading was revealed. The differentiation rate of the deformation field is defined, and the evolution law of differentiation rate and the precursory failure information is analyzed. A new method for evaluating the deformation evolution of rock under loading is proposed in this paper.

Test method

Test scheme and instrument

The sample was taken from the same rock block in the slate of the Zagunao formation of the Triassic and cut into a cylinder with a diameter of 50 mm and a length of 100 mm. The preparation process strictly followed the standards of the international society of rock mechanics (ISRM). The slate was slightly weathered, and the mineral composition was analyzed by X-ray diffraction, and it was found that the slate was mainly composed of 48% quartz, 38% plagioclase, 5% chlorite and 10% clay minerals. Some fine and unevenly distributed quartz, sericite and other minerals could be seen from the microscope. The quartz was unevenly extinction due to stress. The minerals were mostly in the form of aggregates with obvious orientation, the quartz grain size was mainly distributed in 0.1– 0.6 mm with coarse crystals, in addition to more cryptocrystals with obvious plastic deformation (Fig. 1). From the regional geological background of the sampling site, the slate was the product of low-level regional metamorphism, formed in a regional low-temperature power metamorphism with lower temperature and stronger stress. Monotonic and multi-level & multi-cycle loading uniaxial tests were carried out. The axial displacement rate of the monotonic loading uniaxial test was 0.1 mm/min. For the multi-level & multi-cycle loading uniaxial tests, the sine wave with a frequency of 0.5 Hz was used, and the minimum axial force was set to 1kN. The upper limit stress difference ratio between two adjacent stress levels was 0.05 and 30 cycles were performed at each stress level. The upper limit stress ratio of the first cyclic stress level was 30%. The upper limit stress applied in each one of the eight sets of 30 cycles was shown in Table 1. The monotonic loading was imposed at a loading rate of 1 kN/s after cyclic loading. The axial loading was set according to the stress path (Fig. 2) until the failure of the rock samples. Of which: σmax and σmin were the upper and lower limit stress respectively. σamp was the cyclic stress amplitude. σst was the stress difference between two adjacent stress levels.

Fig. 1
figure 1

Microstructure of slate, a single polarized light, b perpendicular polarized light. ① cryptocrystalline texture, ② quartz, ③ sericite

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Table 1 Multi-level and multi-cycle loading test scheme
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Fig. 2
figure 2

The stress path of the multi-level and multi-cycle loading test

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The test was conducted on a MTS815 servo-controlled testing machine, which could apply a maximum axial force of 4600kN and a maximum axial strain of 5 mm. Micro-II AE instrument was used to monitor the fracture process of the sample in real-time (Fig. 3). The sampling threshold was 40 dB and the sampling interval was 1 μs. The monitoring frequency range of the AE sensor was 75–750 kHz.

Fig. 3
figure 3

MTS 815 rock mechanics test system, AE system and DIC system

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DIC and variance of the deformation field

DIC is an optical measurement technique widely used in surface deformation measurement in rock mechanics (Munoz et al. 2016; Cao et al. 2018). As shown in Fig. 4, the principle of DIC is to match the square subset centered on a pixel based on the undeformed image and deformed image. The displacement is obtained by calculating the coordinate difference of the same pixel before and after deformation. The displacement field is determined by repeating this process. Song et al. (2013) reported that the correlation coefficient between the reference subset and the target subset was defined as follows:

$$C(X) = \frac{{\sum\limits_{i = 1}^{m} {\sum\limits_{j = 1}^{m} {\left[ {f(x_{i} ,y_{i} ) - \overline{f} } \right] \cdot \left[ {g(x_{i} *,y_{i} *) - \overline{g} } \right]} } }}{{\sqrt {\sum\limits_{i = 1}^{m} {\sum\limits_{j = 1}^{m} {\left[ {f(x_{i} ,y_{j} ) - \overline{f} } \right]^{2} \cdot \sum\limits_{i = 1}^{m} {\sum\limits_{j = 1}^{m} {\left[ {g(x_{i} *,y_{j} *) - \overline{g} } \right]^{2} } } } } } }}$$
(1)

where f(x, y) is the gray value of the reference image at coordinates (x, y), g(x*, y*) is the gray value of the target image at coordinates (x*,y*), and f and g are the average values of functions f(x,y) and g(x*,y*) respectively.

Fig. 4
figure 4

Schematic diagram of digital image correlation (Song et al. 2013)

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The surface deformation measurement reveals that the rock sample has significant non-uniform deformation characteristics under loading. The non-uniform deformation will result in a localized deformation zone, which is different from the uniform deformation field. First, the deformation of a few pixel points is much greater than that of other pixel points. Second, these pixel points with large deformation are concentrated in one or a few connected bands (Yang et al. 2018). In mathematical statistics, variance is a measure of the dispersion degree in a group of data. The formation and development of deformation localization lead to a significant dispersion in a deformation field. Therefore, the variance is defined to quantify the dispersion degree of the deformation field. The calculation formula is as follows:

$$S_{{\varepsilon {\text{,i}}}}^{2} = \frac{1}{N - 1}\sum\limits_{i = 1}^{N} {(\varepsilon_{ij} - \overline{{\varepsilon_{i} }} )^{2} }$$
(2)

where Sε,i2 is the variance of the deformation field in the ith time step, and N is the number of subsets. εij is the ith time step deformation corresponding to the jth subset. \(\overline{{\varepsilon_{i} }}\) is the average value of the ith time step deformation field, which is defined as:

$$\overline{{\varepsilon_{i} }} = \frac{1}{N}\sum\limits_{j = 1}^{N} {\varepsilon_{ij} }$$
(3)

Test result

Monotonic loading test result

According to the stress–strain curve characteristics under monotonic loading, the rock deformation is divided into five stages in sequence, i.e., crack closure (stage I), elastic region (stage II), stable crack growth (stage III), initiation of macro-scale failure (stage IV) and global post-peak failure (stage V) (Huang et al. 2016) (Fig. 5a). The strength decreases rapidly after failure (σc: 43.34 MPa), showing apparent brittle failure characteristics. The axial strain corresponding to the peak strength is 0.72%. However, the variance under monotonic loading has no apparent increase in the crack closure stage, elastic region stage, stable crack growth stage, as shown in Fig. 5b. The evolution process of the variance can be divided into three stages. (i) Initial deformation stage, the variance increases slowly, There are no or few new microcracks, and the distribution of the deformation field is relatively uniform. (ii) Initiation of macro-scale failure stage, the variance begins to increase significantly. The shear crack initiates and propagates, and the angle between the loading direction and the shear crack is 45°. The shear crack propagates to the middle of the sample, and the tensile crack appears along the loading direction. Gradually, the crack coalescence occurs. (iii) Global post-peak failure stage, the stress falls instantaneously, after which the variance increases rapidly with decreasing stress, indicating the development of macroscopic cracks and the occurrence of failure to the sample. To reveal the development difference of strain and variance during loading, we proposed the strain/variance ratio (defined as the ratio of the strain/variance generated in each stage to the failure strain/variance). As shown in Table 2, the strain is mainly generated in the initiation of macro-scale failure stage (49.65%) and crack closure stage (27.22%). However, the variance is minimal in the crack closure stage, elastic region stage, stable crack growth stage, accounting for 13.23%. In the initiation of the macro-scale failure stage, the variance increases rapidly, accounting for 86.77%.

Fig. 5
figure 5

Stress–strain and stress-variance curves

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Table 2 Strain/variance ratio at different stages
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Figure 6 shows the time series data of the stress, variance, AE count rate and cumulative AE counts. From Fig. 6, the variance and cumulative AE counts increase barely in the crack closure stage, elastic region stage, stable crack growth stage, while the variance and cumulative AE counts increase significantly in the initiation of macro-scale failure stage. It is noted that the curves of the variance and cumulative AE counts show similar growth characteristics.

Fig. 6
figure 6

Time series data of the stress, variance, AE count rate and cumulative AE counts

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Cyclic loading test result

Figure 7 shows the stress–strain and stress-variance curves under multi-level & multi-cycle loading. The two curves show similarities and differences. The residual strain of the sample continuously develops with increasing upper limit stress, representing the damage accumulation. Similarly, the variance gradually accumulates under cyclic loading. At the eighth cyclic stress level, both the strain and variance increase rapidly. The sample was failed after applying the eighth cyclic stress level for five cycles. However, at the end of the seventh cyclic stress level, the total strain is 0.511%, about 47% of the failure strain (1.08%). In comparison, the total variance is 0.00445, which is only 4.13% of the failure variance (0.1078). This result reveals that the increase of variance mainly occurs at the eighth cycle stress level.

Fig. 7
figure 7

Stress–strain and stress-variance curves

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Pearson correlation coefficient, a statistical index describing the degree of correlation between different variables, is introduced to determine the correlation between axial strain and variance during cyclic loading. The absolute value of the Pearson correlation coefficient indicates the degree of correlation, as shown in Table 3. Table 4 shows the correlation coefficient of axial strain and variance at different stress levels. From Table 4, the correlation coefficient ranges from 0.26 to 0.97. In addition, there is a highly positive correlation between axial strain and variance in the first three cyclic stress levels. The correlation coefficient decreases gradually with increasing upper limit stress. It is considered that the axial strain indicates the overall deformation, but the variance indicates the deformation localization. Before the obvious deformation localization, the axial strain is highly correlated with the variance. With the occurrence and progressive development of the localized deformation zone, the correlation decreases gradually, indicating that the axial strain can not show the process of localized deformation.

Table 3 Relationship between the correlation coefficient and correlation degree
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Table 4 Pearson correlation coefficient of axial strain and variance
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Figure 8 shows the contour of the deformation field and time-series data of stress, variance, AE count rate and cumulative AE counts under multi-level & multi-cycle loading. The contour shows that the deformation localization is progressively induced with raising upper limit cyclic stress. Comparing the deformation field contour under different cyclic stress levels, it is found that no obvious localized deformation zone appears in the first three cyclic stress levels. Afterwards, a significant localized deformation zone is formed. With increasing upper limit stress, the band expands and the deformation difference between the band and the adjacent area increases. Finally, the sample is failed along with the band at the eighth cyclic stress level. The variance increases significantly before sample failure, which is also the period of intensive AE signals. However, it is noted that the cumulative AE counts curve shows different development stages at the first seventh cyclic stress levels, which will be later discussed. The monotonic and cyclic loading test results also show similar characteristics. From the viewpoint of microscopic, the sample deformation in the initial period is attributed to pore and fissure closure, particle compression and a small number of microcracks. However, there is no obvious deformation localization band on the sample surface. Thus the variation of variance is tiny. After the deformation localization band is generated, the variance will sharply increase. The variance can accurately show the development process of the crack propagation and deformation localization band.

Fig. 8
figure 8

Contour of the deformation field and time-series data of stress, variance, AE count rate and cumulative AE counts

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The amplitude distribution of AE can be described by the b value, which is a parameter representing the magnitude-frequency characteristic. The variation characteristic of the b value is helpful to understand the damage evolution process of samples. The b value can be obtained by (Gutenberg and Richter 1944):

$$\lg (N(A)) = C - b\lg (A)$$
(4)

where A is the amplitude of AE events, N(A) is the total number of AE events greater than amplitude A. The increase of b value indicates that the small-scale microcracks are mainly generated in the sample, and the constant b value suggests that the scale distribution of the microcracks is relatively stable, and the decrease of b value indicates the increase of large-scale microcracks (Lockner et al. 1991). The evolution process of the b value is shown in Fig. 9, the b value is large and stable at the first seven cyclic stress levels, indicating that the small-scale microcracks are mainly generated. The small change of the b value reveals a steady development of microcracks and a progressive damage process. Afterwards, the rapid decrease of the b value indicates that the large-amplitude AE events increase significantly and the macrocracks appear. This result shows that the microcracks propagation mainly occurs in the first seven cyclic stress levels, and macrocracks appear in the eighth cyclic stress level, which interprets the significant increase of variance at the eighth cyclic stress level.

Fig. 9
figure 9

Time series data of axial stress and b-value

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Result analysis

Variance response characteristics

Figure 10 shows the time series data of axial stress and variance. The variance varies with the cyclic stress, and the variance increases in the loading stage and decreases in the unloading stage. In addition, the total variance and residual variance rise gradually with growing cycles. As a natural geological material, rock has a large number of microcracks, pores and other inner defects. The microstructure characteristic of rock results in its nonlinear elastic behavior and significant hysteresis behavior (Chen et al. 2003; Li et al. 2019). The hysteresis curves of the stress–strain and stress-variance at different stress levels are shown in Fig. 11 and Fig. 12. At the first seven cyclic stress levels, the strain phase continually advances the stress phase at the loading stage, and the stress–strain hysteresis curves are crescent. However, the variance phase changes from advancing to lagging stress phase at the first three stress levels, the stress-variance hysteresis curves are long eggplant. Gradually, the variance phase lags behind the stress phase, and the hysteresis curves become oval. At the eighth stress level, the shape of the stress–strain and stress-variance hysteresis curves is similar. The hysteresis curves are not closed, indicating significant residual strain/variance is induced in the sample.

Fig. 10
figure 10

Time series data of axial stress and variance

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Fig. 11
figure 11

Stress- strain hysteresis curves, a 15th cycle (first stress level), b 45th cycle (second stress level), c 75th cycle (third stress level), d 105th cycle (fourth stress level), e 135th cycle (fifth stress level), f 165th cycle (sixth stress level), g 195th cycle (seventh stress level), h 212 cycle (eighth stress level), i 215th cycle (eighth stress level)

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Fig. 12
figure 12

Stress- variance hysteresis curves, a 15th cycle (first stress level), b 45th cycle (second stress level), c 75th cycle (third stress level), d 105th cycle (fourth stress level), e 135th cycle (fifth stress level), f 165th cycle (sixth stress level), g 195th cycle (seventh stress level), h 212 cycle (eighth stress level), i 215th cycle (eighth stress level)

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Based on the above findings, the types of stress-variance hysteresis curves under multi-level & multi-cycle loading were summarized. As shown in Fig. 13. Point A and point B are the maximum stress point and the maximum variance point respectively. According to the characteristic of the hysteresis curve, the evolution process of the hysteresis curve is revealed. If the variance phase changes from advancing to lagging stress phase, point B lags behind point A and the hysteresis curve is long eggplant, it indicates that the deformation induced by crack closure is dominant. When point A and point B appear at the same time or point B lags behind briefly, the hysteresis curves are oval. The sample shows good elastic behavior and there are no extensive cracks in the sample. The lag time of point B increases, indicating the gradual damage accumulation. A longer lag time means a higher failure potential (Tutuncu et al. 1998; Song et al. 2018, 2021). Figure 14 shows the relationship between the lag time of point B and cyclic stress level. It can be seen that the variation characteristic of the average lag time of strain and variance is the same. The lag time increases suddenly at the eighth cyclic stress level, which reveals that the sample's damage degree and failure potential increases sharply.

Fig. 13
figure 13

Different types of stress-variance hysteresis during cyclic loading (modified from Zhang et al. 2020c)

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Fig. 14
figure 14

Average lag time versus stress level

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The “lag ratio” is defined as the ratio of the number of cycles with “strain/variance lag” to the total number of cycles at each cycle stress level. The total number of cycles for each stress level in this test is 30. Song et al. (2021) reported that the increase of the “lag ratio” presented the gradual increase of the failure potential. Figure 15 shows the evolution characteristics of the "lag ratio" of axial strain and variance. The "lag ratio" of variance increases with increasing upper limit stress, and there is a linear relationship between the "lag ratio" and the stress level. However, the "lag ratio" of axial strain increases significantly only at the eighth stress level. It is revealed that the "lag ratio" of axial strain presents unsignificant response of the damage accumulation and the sudden increase of the failure potential. However, the "lag ratio" of variance reveals the progressive increase of failure potential.

Fig. 15
figure 15

Evolution of axial strain “lag ratio” versus stress level (a), the evolution of variance “lag ratio” versus stress level (b)

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Damage evolution characteristics

Figure 16 shows the relationship between residual variance, residual strain and cycle at each stress level. From Fig. 16, the residual variance increases linearly with increasing cycles at the first seven cyclic stress levels. However, the rising rate of the residual strain decreases gradually at the first three cyclic stress levels. The scatters of residual strain and cycle are fitted by a logarithmic function. Afterwards, the residual strain increases linearly with increasing cycles. At the eighth cyclic stress level, the residual variance and residual strain increase exponentially with increasing cycles, and the sample is failed. The development of residual variance and residual strain under cyclic loading shows the same law except in the first three cyclic stress levels, which is attributed to the crack closure. The test time was very short when the sample was close to failure, so only a small amount of test data was collected, however, we conducted tests on three samples and all demonstrated the same pattern.

Fig. 16
figure 16

Residual stain and residual variance versus cycle curves, a first stress level, b second stress level, c third stress level, d fourth stress level, e fifth stress level, f sixth stress level, g seventh stress level, h eighth stress level, i eighth stress level

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AE and deformation field variance data were used to quantitatively evaluate the rock damage evolution. Damage variables are respectively defined from internal and external damage by cumulative AE counts and variance. Liu et al. (2009) have derived the analytical expression of damage variable based on cumulative AE counts, but there are few studies on the expression of damage variable based on variance.

The damage variable based on cumulative AE counts can be defined as:

$$D_{AE} { = }D_{u} \frac{{C_{d} }}{{C_{0} }} = \left( {1 - \frac{{\sigma_{c} }}{{\sigma_{p} }}} \right)\frac{{C_{d} }}{{C_{0} }}$$
(5)

where Du is the critical damage; Cd is the cumulative AE counts at any time in the process of rock deformation; C0 is the cumulative AE counts when the damage variable is Du; σc is the residual strength; σp is the peak strength.

The deformation localization is induced by the propagation and coalescence of microcracks and the formation of macrocracks. The variance can be used as a parameter to quantify the damage evolution of rock.

Damage variable D can be obtained by:

$$D = A_{d} /A$$
(6)

where Ad is the damaged area, A is the cross-section area of the undamaged sample.

V0 is defined as the variance when the whole section of the sample is failed. The variance Vw induced by micro-unit damage in the unit area, Vw can be written as:

$$V_{w} = \frac{{V_{0} }}{A}$$
(7)

When the damaged area reaches Ad, the variance, Vd, can be expressed as:

$$V_{d} = V_{w} A_{d} = \frac{{V_{0} }}{A}A_{d}$$
(8)

Therefore, damage variable DVA can be written as:

$$D_{VA} = \frac{{V_{d} }}{{V_{0} }}$$
(9)

In the rock mechanics test, the loading process will be terminated when the sample is not completed failed. The damage variable DVA cannot reach 1, the DVA is revised as (Cao et al. 2019):

$$D_{VA} = D_{u} \frac{{V_{d} }}{{V_{0} }}$$
(10)

where Du is the critical damage variable.

Du can be described as:

$$D_{u} = 1 - \frac{{\sigma_{c} }}{{\sigma_{p} }}$$
(11)

Therefore, the damage variable DVA can be re-written as:

$$D_{VA} { = }\left( {1 - \frac{{\sigma_{c} }}{{\sigma_{p} }}} \right)\frac{{V_{d} }}{{V_{0} }}$$
(12)

Based on the characteristics of the DAE and DVA, the evolution process can be divided into five stages (Fig. 17): (i) Crack closure stage, the DVA does not increase, and DAE increases obviously, which is attributed to a lot AE counts induced by crack and pore closure. (ii) Initial damage stage, the slopes of DAE and DVA approach 0, which means that few new microcracks are generated in this stage. (iii) Stable development stage, the DAE and DVA increase slowly with increasing cycles, indicating the generation and continuous propagation of the microcracks. (iv) Accelerated development stage, the increase of DAE and DVA is gradually accelerated, indicating the initiation, propagation and coalescence of a large number of microcracks. (v) Failure stage, the DAE and DVA increase significantly and the sample is failed in this stage. In addition, it is noted that the sudden increase of DAE is earlier than that of DVA, indicating that the significant damage first appears in the interior of the sample. Afterwards, the gradual development of internal cracks induced the surface deformation localization, macrocracks and the obvious increase of DVA.

Fig. 17
figure 17

Axial stress and damage variable versus time curves

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Failure precursory characteristic

The development characteristic of AE parameters and waveform under loading conditions can indicate the damage evolution process of rock. Generally, the spectrum characteristics obtained from the AE waveform can comprehensively describe the development process of microcracks (Fan et al. 2020). Therefore, the evolution feature of the AE spectrum in the process of rock fracture is investigated, and the failure mechanism and capture the precursor points of rock failure is revealed. In this paper, the peak frequency of the recorded AE waveform is obtained by fast Fourier transform. As shown in Fig. 18. Based on the distribution feature of peak frequency, the ranges of low, medium, high peak frequencies are divided into 40–128 kHz, 128–256 kHz and 256–500 kHz respectively.

Fig. 18
figure 18

Waveform and spectrum of AE signal

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The variance has undergone a variation process from slow growth to rapid growth. To quantitatively describe the variation rate of the deformation field, the differentiation rate is obtained by the derivative of the variance with respect to the axial strain, the formula is as follows (Zhang et al. 2020b),

$$\nu_{\varepsilon } { = }\frac{{\partial S_{\varepsilon ,i}^{2} }}{\partial \varepsilon } \approx \frac{{S_{\varepsilon ,i + 1}^{2} - S_{\varepsilon ,i - 1}^{2} }}{2\Delta \varepsilon }$$
(13)

Figure 19 shows the time series data of the stress, peak frequency under monotonic loading. It is found that the distribution range of peak frequency depends on the deformation stage. In the crack closure stage, elastic region stage and stable crack growth stage, the peak frequency ranges from 228 to 278 kHz. However, in the initiation of the macro-scale failure stage, the peak frequency ranges from 95 to 445 kHz, and the distribution range of peak frequency is expanded. Figure 20 shows the horizontal and vertical differentiation rate (εxx and εyy) under monotonic loading. The horizontal and vertical differentiation rates approach 0 in the crack closure, elastic region, and stable crack growth stages. However, the obvious precursor points appear in the initiation of the macro-scale failure stage. The horizontal and vertical differentiation rate curves show similar development characteristics.

Fig. 19
figure 19

Time series data of the stress, peak frequency under monotonic loading

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Fig. 20
figure 20

Evolution of the differentiation rate under static loading, a εxx, b εyy

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Figure 21 shows the time series data of the stress, peak frequency under cyclic loading. From Fig. 21, the wide distribution range of peak frequency is presented in the initial loading stage due to the fast loading rate. In addition, the distribution range of peak frequency is relatively stable compared with the static uniaxial test, and there is no obvious expansion of the distribution range of peak frequency with increasing upper limit stress. We believe that the multi-level and multi-cycle loading results in progressive damage and crack propagation process, thus no obvious precursor formation based on AE spectrum is obtained under cyclic loading. In addition, the variation curves of the horizontal and vertical differentiation rate (εxx and εyy) under cyclic loading is obtained. As shown in Fig. 22, the horizontal and vertical differentiation rates approach 0 in the first seven cyclic stress levels. However, the obvious precursor points appear in the eighth cyclic stress level. Therefore, it is clear that the failure precursor point under cyclic loading can be obtained by differentiation rate.

Fig. 21
figure 21

Time series data of the stress, peak frequency under cyclic loading

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Fig. 22
figure 22

Evolution of the differentiation rate for the deformation field under cyclic loading, a εxx, b εyy

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At present, the failure precursory point is often obtained by analyzing AE parameters and spectrum. The peak frequency and differentiation rate under static loading show the apparent failure precursory point. However, there is always progressive crack propagation and energy dissipation during cyclic loading, thus it is poor to reveal the failure precursor point based on the peak frequency. However, the variance has a high susceptibility after the local fracture band appears, and the differentiation rate of variance can obtain the precursory point.

Conclusion

  1. 1.

    The variance begins to increase significantly in the initiation of macro-scale failure stage under monotonic loading, and it increases sharply under cyclic loading before sample failure.

  2. 2.

    In the initial stage, the stress-variance hysteresis curve is long eggplant, which indicates that the deformation induced by crack closure is dominant. Gradually, the hysteresis curves are oval. The sample shows good elastic behavior and there are no extensive cracks in the sample. The lag time of variance increases, indicating the gradual damage accumulation. A longer lag time indicates a higher failure potential

  3. 3.

    According to the evolution characteristics of the damage variable, the evolution process of damage variable can be divided into the crack closure stage, initial damage stage, stable development stage, accelerated development stage and failure stage.

  4. 4.

    The differentiation rate is proposed to reveal the failure precursory point under monotonic and cyclic loading. Compared with the characteristic of the AE spectrum, the curves of differentiation rate show obvious precursor points under cyclic loading.