## 4.1 Establishment of A Damage Constitutive Model

The damage of rock mass results in changes in micro-structures and fracture of bearing surfaces under the loading of rock materials. According to the concept of damage factor and effective stress proposed by Kachanov and Krajcinovic (1986), the continuous damage variable D of rock can be defined as

$$D=\frac{A-\tilde{A}}{A}$$

1

Where \(\tilde{A }\)is the effective bearing area of the material after damage, \(A\) is the cross-sectional area of the material without damage.

The loading force \(F\) on rock specimens can be expressed as

$$F=\sigma \bullet A=\tilde{\sigma }\bullet \tilde{A}$$

2

Where \(\sigma\) is the initial stress without damage, \(\tilde{\sigma }\) is the effective stress corresponding to the effective area \(\tilde{A}\) after damage.

Combine Eq. (1) and Eq. (2), get Eq. (3).

$$\tilde{\sigma }=\frac{\sigma }{1-D}$$

3

According to the relation between deformation component and stress component of ideal elastomer in theory of elasticity

$${\epsilon }_{1}=\frac{\sigma }{E}=\frac{1}{E}\left[{\sigma }_{1}-\mu \left({\sigma }_{2}+{\sigma }_{3}\right)\right]$$

4

Where \(E\) is the elastic modulus of the material, \(\epsilon\) is the strain.

According to the equivalent strain hypothesis, get Eq. (5)

$${\epsilon }_{1}=\frac{\tilde{\sigma }}{E}=\frac{\sigma }{(1-D)E}=\frac{\left[{\sigma }_{1}-\mu ({\sigma }_{2}+{\sigma }_{3})\right]}{(1-D)E}$$

5

The constitutive model of rock mass damage is obtained after deformation as shown in Eq. (6).

$${\sigma }_{1}=E{\epsilon }_{1}\left(1-D\right)+\mu \left({\sigma }_{2}+{\sigma }_{3}\right)$$

6

During the process of applying external load to a rock mass, various randomly distributed micro-damages occur inside the rock mass, so that the damage under load can be studied from a statistical point of view and the corresponding statistical damage constitutive model can be established. In this paper, the Weibull distribution model is used for constitutive analysis of AE damage in jointed rock mass.

The probability density function is

$$\phi \left(\epsilon \right)=\frac{m}{\alpha }{\left(\frac{\epsilon }{\alpha }\right)}^{m-1}exp\left[{-\left(\frac{\epsilon }{\alpha }\right)}^{m}\right]$$

7

Where \(\phi \left(\epsilon \right)\) is the distribution function of rock micro element strength, \(\epsilon\) is rock micro-element strain, \(m\) and \(\alpha\) are distribution parameters.

It is assumed that under certain loading conditions the material strain is \(\epsilon\), the probability of failure of the micro-elements in the material section is obtained as

$$\left\{\begin{array}{c}F\left(\epsilon \right)=P\left(\epsilon \ge 0\right) \\ P\left(\epsilon \ge 0\right)={\int }_{0}^{\epsilon }\phi \left(x\right)dx\end{array}\right.$$

8

The upper and lower expressions in Eq. (8) are derived from \(\epsilon\).

$$\frac{dF\left(\epsilon \right)}{d\epsilon }=\phi \left(\epsilon \right)$$

9

Where \(\phi \left(\epsilon \right)\)is the probability density of failure of the material element.

The effective area of the specimen can be expressed as

$$\tilde{A}=A\left[1-F\left(\epsilon \right)\right]$$

10

The damage variable D based on Weibull distribution statistical model is Eq. (11).

$$D=F\left(\epsilon \right)={\int }_{0}^{\epsilon }\phi \left(x\right)dx$$

11

When rock mass is loaded to the strain level\(\epsilon\), its damage variable can be expressed as

$$D={\int }_{0}^{\epsilon }\phi \left(x\right)dx=1-exp\left[-{\left(\frac{\epsilon }{\alpha }\right)}^{m}\right]$$

12

In the process of damage, AE cumulative relationship is

$$N=\frac{{N}_{f}}{A}\bullet \varDelta A$$

13

Where \(A\) is the cross-sectional area of the whole specimen. \({N}_{f}\) is the accumulation of AE when the whole section is completely destroyed.\(\varDelta A\) is a micro area element. \(N\) is the accumulation of AE when the compressive strain of the rock specimen increases to ε.

Based on the strength distribution of the element, it is assumed that when the strain of the specimen increases \(\varDelta \epsilon\), the increment of the section area resulting in failure is

$$\varDelta A=A\bullet \phi \left(\epsilon \right)\bullet \varDelta \epsilon$$

14

By simultaneous Eqs. (13) and (14), get Eq. (15). it can be obtained that the AE accumulation of rock specimen when the compressive strain increases to \(\epsilon\).

$$N={N}_{f}{\int }_{0}^{\epsilon }\phi \left(x\right)dx$$

15

Substituting Eq. (12) into Eq. (15), the load rock damage variable based on AE characteristic parameters is obtained as

$${D}_{s}=\frac{N}{{N}_{f}}$$

16

The damage of the jointed rock mass under load can be equivalent to the coupling of two damage states, one is the initial damage caused by the initial defect of the joint surface, and the other is the damage caused by the loading of the rock mass, then the jointed rock mass The internal damage constitutive relationship (Kachanov and Krajcinovic 1986) can be expressed as

$$\sigma =(1-{D}_{s}){E}_{\phi }\epsilon$$

17

Where \({E}_{\phi }\) is the elastic modulus of the rock mass with the joint dip angle \(\phi\), \({D}_{s}\) is the damage variable of the rock mass under load.

The initial damage inside the rock caused by joints is more complicated, and the response of the macro-physical properties of the rock can represent the degree of internal deterioration of the material. Since the elastic modulus of jointed rock masses with different dip angles is easier to analyze and measure, the degree of deterioration of the elastic modulus of the jointed rock mass can be used to characterize the initial joint damage value, so the initial damage variable \({D}_{\phi }\) of jointed rock mass can be expressed as (Wang et al. 2018).

$${D}_{\phi }=1-\frac{{E}_{\phi }}{{E}_{0}}$$

18

Where \({E}_{0}\) is the initial elastic modulus of the complete rock mass.

From equations (17) and (18), the stress-strain relationship of the jointed rock mass expressed by the initial joint damage variable and the loaded damage variable is

$${\sigma }=(1-{D}_{\phi })(1-{D}_{s}){E}_{0}\epsilon$$

19

From equations (16), (18) and (19), the total damage variable of rock mass joints and load coupling based on the characteristic parameters of AE is

$$D=1-\frac{{E}_{\phi }}{{E}_{0}}(1-\frac{N}{{N}_{f}})$$

20

By substituting Eq. (20) into Eq. (6), the statistical constitutive relation of jointed rock damage based on AE characteristic parameters is

$${\sigma }_{1}=\frac{{{E}_{\phi }}^{2}}{{E}_{0}}{\epsilon }_{1}\left(1-\frac{N}{{N}_{f}}\right)+\mu \left({\sigma }_{2}+{\sigma }_{3}\right)$$

21

Where \({\sigma }_{1}\)、\({\sigma }_{2}\)and \({\sigma }_{3 }\)is the stress component in three directions of the specimen, \({\epsilon }_{1 }\)is the strain in the direction \({\sigma }_{1 }\)of the specimen, \(E\) is the elastic modulus of the material, \(\mu\)is Poisson's ratio of material, \(N\) is the AE accumulation when the strain reaches \({\epsilon }_{1}\), \({N}_{f}\) is the AE accumulation when the specimen is completely destroyed.

## 4.2 Verification of The Damage Constitutive Model

Based on the AE damage constitutive Eq. (21) and AE ringing count, the AE constitutive curve is plotted. As shown in Fig. 7.

When comparing the damage constitutive curves of different joint dip angles with the complete specimens in the Fig. 7, the evolution law is basically the same, which indicates that the joint dip angle does not affect the damage evolution law of rock mass. There is a certain difference between the theoretical curve and the experimental curve, and there is an obvious compaction stage in the initial loading stage of the experimental curve, but it is not obvious in the theoretical curve. This may be because the acoustic emission signal at this stage is difficult to detect, causing the theoretical damage value to be smaller than the actual damage value. Therefore, there is a great difference between the theoretical curve and the test curve in the compaction stage, which changes the subsequent trend of the theoretical curve, resulting in the low fitting degree of the two curves as a whole. In the line plastic and failure stages, the AE activity is obvious and easy to detect. So, the theoretical damage value is gradually getting close to the actual damage value, and the theoretical curve is in good agreement with the experimental curve. The stress peak value is also very close. The above analysis shows that the AE damage constitutive curve is basically consistent with the experimental curve, which indicates that the AE ringing count has good consistency with the rock damage failure. The damage constitutive model is appropriate and reasonable for the damage evolution of rocks.