# Deformation Analysis of Large-Scale Rock Slopes Considering the Effect of Microseismic Events
## 2 Damage Constitutive Model Considering the Rock Fracture Scale  
MS data obtained by the monitoring systems contain abundant source fracture information. Based on extensive research on MS waveform data, scholars have obtained source damage characteristic parameters related to seismic energy accumulation and release, such as the seismic moment, seismic magnitude, seismic energy, source radius, apparent stress, stress drop and seismic deformation. Cracks, joints and faults in a rock mass are discontinuities that greatly decrease the mechanical properties of the rock mass. In the present study, the source radius is used to represent the rock fracture scale, thus establishing a direct relationship between the rock fracture scale and rock failure. By considering the size and number of fractures, it is possible to determine an efficient method to estimate the compliance matrix of fractured rock mass based on micro-mechanics.  
The constitutive relationship of the fractured rock mass mainly contains the intact rock strain and the fracture induced strain [22]. To study the fractured rock mass more conveniently, it can be divided into many representative volume elements (RVEs) as study objects [22]. [PERSON] and [PERSON] [23] proposed an overall estimate method for the instantaneous modulus based on the loading conditions when studying a linear, elastic and brittle solid with possible fractional sliding.  
\[\overline{\varepsilon}_{i}=\overline{C}_{ij}\overline{\sigma}_{j},\ldots\;i, \;j\;=\;1,\;2,\;3\ldots\ldots 6 \tag{1}\]  
In two dimensions, the flexibility matrix of the rock mass with randomly distributed joints can be written in Equation (2) [24]:  
\[\overline{C}_{ij}=\frac{1}{E}\left[\begin{array}{ccc}1+a_{0}&-\
u&0\\ -\
u&1+a_{0}&0\\ 0&0&2(1+\
u)+2a_{0}\end{array}\right] \tag{2}\]  
where \(E\) is the elastic modulus of the intact rock mass; \(\
u\) is the Poisson's ratio; \(a_{0}\) is the damage variable of the rock mass fractures and can be expressed as follows:  
\[a_{0}=\omega\;(1\;+\;\omega) \tag{3}\]  
where \(\omega\) is the fracture density of the rock mass and can be expressed as follows:  
\[\omega\;=\mathrm{N}\pi\mathrm{r}_{0}^{2}/\mathrm{V} \tag{4}\]  
where N is the number of micro-fractures in the unit volume; V is the volume of the unit; \(\mathrm{r}_{0}\) is the MS source radius.  
In elastic mechanics, the displacement in the length direction can be considered equal to 0 if the length of the subject is much larger than the length in the other two directions, and the size and shape of each part are largely the same. This kind of displacement represents the plane strain problem. In planar problems, the stress, strain and displacement are functions that are independent of the length direction. As the profile of the left bank slope model is selected, it can be approximately simplified to a plane strain problem.  
Based on Equation (2), Equation (5) is transformed to obtain the damage constitutive elastic matrix of the fractured rock mass under three-dimensional plane strain as follows:\[\overline{C}_{ij}=\frac{1}{E}\left[\begin{array}{cccccc}1+a_{0}&-\
u&-\
u&0&0&0\\ -\
u&1&-\
u&0&0&0\\ -\
u&-\
u&1+a_{0}&0&0&0\\ 0&0&0&2(1+\
u)&0&0\\ 0&0&0&0&2(1+\
u)+2a_{0}&0\\ 0&0&0&0&0&2(1+\
u)\end{array}\right] \tag{5}\]