*3.2. Results of AE Analysis*

As the number of oscillations exceeds the AE threshold, the AE ring count can further characterize filling specimens' internal micro-fracture and damage evolution. The AE cumulative ringing count refers to the cumulative value of the ringing count of AE in an AE process. This parameter can describe the total strength of AE and the accumulation of internal damage to materials, which is the external manifestation of the accumulation effect of internal structural changes in materials. Through AE monitoring of ASPM specimens under uniaxial compression test at different loading rates, the internal defects and damage can be reflected, and the failure of the filling body can be predicted [55].

### AE Ringing Count Analysis

The AE ringing counts, stress, and time relationships of the filling specimens under uniaxial compression at different loading rates are shown in Figure 7. It can be seen from Figure 7 that the number of AE ringing reaches the maximum when the peak stress approaches. The reason is that the micro-cracks initiate, develop and expand after the micro-pore compaction in the early stage of the specimen. The macroscopic penetrating cracks are formed near the peak stress. The accumulated energy is released rapidly so that the AE ringing count signal value is suddenly increased to the peak value, which can be used as a precursor signal to determine the failure of the filling specimen [50].

According to the change rule of AE ringing count, the whole loading process can be divided into four stages: initial active stage, pre-peak rise stage, active stage, and post-peak stability stage.


decrease of axial stress after the peak. This stage accounts for about 23.6%, 24.2%, 28.0%, and 32.1% of the process, as shown in Figure 8. The proportion of the active stage is positively correlated with the loading rate, indicating that the increase in the loading rate that makes the internal energy in the compression process cannot be released until close to the peak. The higher the loading rate is, the more energy is released, so it can be seen that the proportion of this stage is gradually increasing.

(4) The post-peak stability stage *T*<sup>4</sup> (A11: 714~779 s, A21: 292~342 s, A32: 177~223 s, A43: 104~117 s): This stage corresponds to the post-peak failure stage of the stress–strain curve. After the sudden increase and decrease of the ringing count in the previous stage, the ringing count in this stage is at a low level as a whole. Because there is some friction and slipping between the fracture surfaces after the failure of the specimen, there is a small range of growth at some time, but it does not affect the overall development trend. This stage accounts for 8.3%, 14.9%, 20.9%, and 11.0% of the process, as shown in Figure 8.

**Figure 7.** AE ringing counts, stress and time curves of ASPM specimens. (**a**) 0.002; (**b**) 0.005; (**c**) 0.0075; (**d**) 0.01.

**Figure 8.** The proportion of each stage.

As shown in Figure 9, it can be seen that with the increase in loading rate, the AE cumulative ringing count negatively correlates with the loading rate. It is concluded that when the loading rate is less than 0.002 mm/s, the total number of AE cumulative ringing will be at a high level. When the loading rate exceeds 0.01 mm/s, the total number of AE cumulative ringing will be further reduced.

**Figure 9.** Relationship between cumulative ringing count and loading rate.

#### **4. Establishment of Damage Constitutive Model of the ASPM**

*4.1. Fitting of AE Cumulative Ringing Count and Strain*

The analysis of the above experimental results shows that the AE characteristics are closely related to the development of microcracks in ASPM specimens. Cracks and defects inside the filling specimen are essential factors affecting its mechanical properties. Therefore, there is an inevitable connection between the AE cumulative ringing count and the mechanical properties of the filling specimen.

According to the experimental data, the relationship between strain and time is fitted. That is the following relationship between *ε* and time:

$$
\varepsilon = kt + \varepsilon\_0 \tag{2}
$$

The formula: *ε* is the strain of the filling body specimen; *k* is the strain rate; *t* is time; *ε*<sup>0</sup> is the initial strain of the filling body, obtained by linear fitting experimental data.

The Boltzmann function can express the relationship between the measured cumulative ringing count and time [56,57], that is:

$$N = \frac{A - B}{1 + \exp\left(\frac{t - \zeta}{G}\right)} + B \tag{3}$$

The formula: *N* is the AE cumulative ringing count in the loading stage; *t* is time; *A*, *B*, *C*, and *G* are all fitting parameters.

According to the experimental results, the total number of AE cumulative ringing decreases with the increase of loading rate, the loading rate *v* is introduced, and the fitting relationship is further modified as follows:

$$N = \frac{A - B}{1 + \exp\left[\frac{v(t - C)}{G}\right]} + B \tag{4}$$

The formula: *v* is the uniaxial loading rate of the filling specimen.

Using the Formula (4) to fit, as shown in Figure 10, the correlation coefficients corresponding to different loading rates are 0.9992 (0.002 mm/s), 0.9988 (0.005 mm/s), 0.9972 (0.0075 mm/s) and 0.9987 (0.01 mm/s). Therefore, the function can represent AE cumulative ringing count and time variation.

Formulas (2) and (4) can be obtained:

$$N = \frac{A - B}{1 + \exp\left[\frac{v(x - x\_0 - kC)}{kG}\right]} + B \tag{5}$$

$$\varepsilon = k \left[ \frac{G}{v} \ln \left( \frac{A - N}{N - B} + \mathcal{C} \right) \right] + \varepsilon\_0 \tag{6}$$

Formulas (5) and (6) establish the coupling relationship between cumulative ringing counts and strain of ASPM specimens under different loading rates.

#### *4.2. Establishment of the ASPM Damage Model*

In this paper, AE ringing count and AE cumulative ringing count are selected as characteristic parameters to characterize the damage characteristics of filling specimens during compression.

Kachanov [58] proposed the concept of damage variable *D* and defined it as:

$$D = \frac{A'}{A} \tag{7}$$

The formula *A* is the total area of micro defects on the bearing section and *A'* is the fracture area when there is no initial damage.

Considering that it is difficult to determine the effective bearing area of damaged materials, Lemaitre [59] proposed the strain equivalence hypothesis, that is, to indirectly measure the damage through effective stress:

$$
\sigma = \sigma \ast (1 - D) = E \varepsilon (1 - D) \tag{8}
$$

**Figure 10.** Fitting of the AE cumulative ringing count with time. (**a**) 0.002; (**b**) 0.005; (**c**) 0.0075; (**d**) 0.01.

Formula: *σ* is nominal stress, *σ*\* is effective stress, *E* is the elastic modulus of the filling body, and *ε* is strain.

Assuming that the AE cumulative ringing count is *Nf* when the whole section A of the non-destructive material is completely damaged, the AE ringing count *Nw* when the unit area is damaged is:

$$N\_w = \frac{N\_f}{A} \tag{9}$$

When the cross-section damage reaches *A'*, the AE cumulative ringing count is:

$$N\_d = N\_{\text{iv}} A' = \frac{N\_f}{A} A' \tag{10}$$

Formulas (7) and (10) show that the relationship between the damage variable and AE cumulative ringing count is:

$$D = \frac{A'}{A} = \frac{N\_d}{N\_f} \tag{11}$$

During the experiment, due to the insufficient stiffness of the testing machine or the different failure conditions set, the testing machine was stopped when the specimen was not completely damaged (the damage variable *D* does not reach 1). There is still a specific residual strength of the specimen. Therefore, the modified damage variable is:

$$D = D\_{\nu} \frac{N\_d}{N\_f} \tag{12}$$

Formula: *Du* is the critical value of the damage.

For the convenience of calculation, the critical damage value *Du* is:

$$D\_{\rm u} = 1 - \frac{\sigma\_{\rm c}}{\sigma\_{pk}} \tag{13}$$

Formula: *σpk* is peak strength, *σ<sup>c</sup>* is residual strength.

The coupling relationship between cumulative ringing count *N*, damage variable *D*, and stress *σ* of ASPM at different loading rates can be obtained by combining Formulas (5), (8), (12), and (13) as follows:

$$D = (1 - \frac{\sigma\_{\mathcal{E}}}{\sigma\_{pk}}) \frac{N\_{\mathcal{E}}}{N\_f} \tag{14}$$

$$
\sigma = E\varepsilon (1 - D) = E\varepsilon \left[ 1 - (1 - \frac{\sigma\_c}{\sigma\_{pk}}) \frac{N\_d}{N\_f} \right] \tag{15}
$$

Formula: *Nf* is the cumulative ringing count produced at the end of the experiment,

$$N\_d = \frac{A - B}{1 + \exp\left[\frac{v\left(x - x\_0 - kC\right)}{kC}\right]} + B.$$

#### *4.3. Model Validation and Discussion*

To verify the rationality and effectiveness of the model, combined with experimental data, the strain-time curve and the cumulative count-time curve of AE ringing are fitted, respectively. The statistical fitting parameters are shown in Table 5. They are substituting the fitting parameters into formulas (6), (4) and (15), the comparison between AE cumulative ringing count and strain, damage variable and strain, stress–strain fitting based on AE ringing count, and measured stress–strain under ASPM uniaxial compression can be determined.

**Table 5.** Fitting parameters of different loading rates.


As shown in Figure 11, the comparison between the experimental and theoretical results shows that the theoretical and experimental curves of strain and AE cumulative ringing count have a high matching, which indicates that the coupling relationship between AE cumulative ringing count and strain considering loading rate is appropriate. The model can provide a reference for the deformation prediction of the same type of filling materials.

**Figure 11.** Comparison of Experiment value and Theoretical value of *N* and *ε.* (**a**) 0.002; (**b**) 0.005; (**c**) 0.0075; (**d**) 0.01.

Figure 12 compares the stress–strain curves of ASPM specimens under different loading rates and the experimental results. At a low loading rate, the theoretical stress peak is smaller than the measured stress peak, and the peak strain is ahead of the measured value. With the increased loading rate, the theoretical peak stress and peak strain are close to the experimental results, which can better reflect the failure process of ASPM specimens from linear elastic transition to plastic deformation. Compared with the experimental results, the established model cannot effectively reflect the compaction stage of ASPM specimens under different loading rates, and the theoretical residual strength after the peak is greater than the experimental value.

**Figure 12.** Comparison of Experiment value and Theoretical value of *σ* and *ε.* (**a**) 0.002; (**b**) 0.005; (**c**) 0.0075; (**d**) 0.01.

Figure 13 shows the relationship between AE cumulative ringing counts and damage variables at different loading rates. It can be seen that the higher the loading rate is, the smaller the final AE cumulative ringing counts are. The final damage values at different loading rates are about 0.7, which again shows that ASPM specimens still have a specific residual strength at the end of the experiment. In the Figure, the higher the loading rate is, the greater the slope of the curve is, indicating that the higher the loading rate per unit time is, the greater the damage to the ASPM specimen is.

Overall, the strain, stress, and damage variables of ASPM specimens agree with the measured and simulated cumulative counts of AE ringing, which proves the rationality and effectiveness of the coupling model.

**Figure 13.** Relationship between *N* and *D*.
