In 1963, Rabotnov, a former Soviet scholar, put forward the concept of damage factor. Lemaitre (1984) of France proposed the strain equivalence hypothesis on this basis. According to his hypothesis, the basic relation of rock damage constitutive equation is established by assuming that rock elements obey the generalized Hooke law before failure: σ = σ* (1-D) = Eε (1-D). A large number of research results show that acoustic emission cumulative ringing counts are one of the characteristic parameters that can well reflect the variation of material damage properties 16. Therefore, in this paper, the cumulative acoustic emission ringing counts are taken as the characteristic parameter and the rock specimen PS-0-1 is selected as the analytical object to analyze the damage evolution characteristics of shale.

## 3.3.1 Analysis of shale damage characteristics under uniaxial compression

According to the stress-strain curve of Rock specific PS-0-1 obtained from the test, the elastic modulus E is 5748MPa. According to the formula σ = σ* (1-D) = Eε (1-D), the damage factor D is obtained by using the stress-strain relationship obtained by the test. The variation of damage factor D and cumulative ringing counts N over time is shown in Fig. 5. As can be seen from the picture, damage factor D presents a trend of first decreasing and then increasing with the increase of time. It can be seen that in the crack compaction stage, the micro-cracks in the specimen gradually close and the rock is compacted under the action of axial pressure. The rock strength is enhanced, and the rock damage is reduced. As the axial pressure continues to increase, the cracks inside the specimen expand and connect gradually, and the rock damage increases.

When the cumulative acoustic emission ringing counts and damage factor begin to increase slowly, it indicates that the internal crack of the specimen begins to generate and expand. At this time, the corresponding stress is regarded as the crack initiation strength σci. For example, the σci of the Rock Specimen PS-0-1 is 13.0MPa, about 47.3% of the peak strength. The σci of Rock specimen PS-0-2 is 23.5MPa, about 58.6% of the peak strength.

## 3.3.2 Establishment of shale uniaxial compression damage model based on acoustic emission characteristics

The former Soviet scholar Kachanov (1958) defined the damage variable as

$$D=\frac{{A}_{d}}{A}$$

1

Where, Ad is the fault area where the rock sample is damaged, such as compaction, new crack generation, propagation, convergence, penetration, and macro failure. A is the fracture area at the initial non-destructive injury.

Assuming that the cumulative acoustic emission ringing counts of complete destruction of the whole section A of the nondestructive material is Nc, then the acoustic emission ringing counts Nw when the unit area element is destroyed is

$${N}_{w}=\frac{{N}_{c}}{A}$$

2

When the damage area reaches Ad, the cumulative acoustic emission ringing counts Nd is

$${N}_{w}={N}_{w}{A}_{d}=\frac{{N}_{c}{A}_{d}}{A}$$

3

So

$$D=\frac{{N}_{d}}{{N}_{c}}$$

4

During the test, due to insufficient stiffness of the testing machine or different fracture conditions of rock samples, the testing machine often stops working before the rock is completely destroyed (that is, the damage of rock samples does not reach 1), so the damage variable is corrected as

$$D={D}_{U}\frac{{N}_{d}}{{N}_{c}}$$

5

Where, DU is the critical value of damage.

In formula (5), the value of Nc is the cumulative acoustic emission ringing counts of the whole process of rock sample compression failure, and the value of Nd is the cumulative acoustic emission ringing counts of each stage in the progress of rock sample compression failure. In order to simplify the calculation, Liu et al. (2009) normalized the damage critical value according to the method of linear function conversion, and obtained

$${D}_{U}=1-\frac{{\sigma }_{c}}{{\sigma }_{p}}$$

6

Where: σp is the peak strength, σc is the residual strength.

Therefore, the damage model of coal rock under uniaxial compression based on acoustic emission characteristics can be written as

$$\sigma =E\epsilon \left(1-D\right)=E\epsilon (1-{D}_{U}\frac{{N}_{d}}{{N}_{c}})$$

7

It can be seen from the formula that the damage factor D is proportional to the cumulative acoustic emission ringing counts N when the test piece is undamaged. The damage factor D increases with the increase of acoustic emission signal N.

However, the rock specimen used in the experiment have initial damage to some extent. Existing rock material damage models always ignore the initial damage characteristics of rock materials, and the actual rock materials have initial damage characteristics 18. Similar to this method, the damage constitutive model of shale under uniaxial compression based on acoustic emission characteristics is established as:

Where: σ is principal stress, ε is axial strain, E is elastic modulus, D is damage variable, K1 and K2 is damage evaluation index, Nd is cumulative number of acoustic emission impacts at a certain time, N0 is cumulative number of acoustic emission impacts when damage factor is 0, Nc is cumulative total number of acoustic emission impacts.

With the help of origin software, the damage factor value D and the acoustic emission cumulative ringing counts value N at the same time during the test are marked on the coordinate axis, and then the relationship between the damage factor value D and the acoustic emission cumulative ringing counts value N is obtained by linear fitting. It can be seen from Fig. 5 that the damage factor D decreases first and then increases with the increase of acoustic emission cumulative ringing counts. The relationship between damage factor D and acoustic emission cumulative ringing counts N is obtained by linear fitting in two sections. Alternatively, the cumulative ringing counts value N and damage value D corresponding to the same time obtained from the experiment: (32, 0.54), (244, 0), (1351, 0.053) can be substituted into formula (8). Through this method, the damage constitutive model of rock specific PS-0-1 is obtained as: