# Indices to Determine the Reliability of Rocks under Fatigue Load Based on Strain Energy Method
## 2 Classification and Calculation Methods of Rock Strain Energy by Tests Results  
### Classifications and Methods of CalculatingStrain Energy  
It is of great importance to divide suitable strain energy types to explain the elastic-plastic transformation and the mechanical properties of rocks. Although many methods in rock researches are derived from metal, the unique properties of properties of rocks clearly makes the two materials to be treated differently in some instances. In this study, we use the strain energy density \(u_{i}\) to represent the strain energy in rocks. In conventional rock cyclic loading and unloading studies, the researchers only divided the strain energy density into elastic strain energy density \(u_{i}^{c}\) and plastic strain energy density \(u_{i}^{p}\), which as shown in Figure 1a. The deformation of rock consists of two parts: deformation of rock grains and deformation of cracks. [PERSON] et al. [3] pointed out that during the unloading process, due to the previously compacted cracks and the opening of newly formed crack as a result of decrease in load, the slope of the unloading curve would gradually slow down. The deformation rebound of rock, at the beginning of unloading, was mainly due to the elastic deformation of grains because the crack in the rock cannot immediately change from compression to opening. Therefore, in the pre-peak stage, in order to obtain the elastic strain energy density of rock grains when unloading. We draw a line section with the slope of unloading modulus obtained at the initial stage of unloading from the unloading point \(A\), and then cross the strain axis at point \(C\), as the line \(AC\) shown in Figure 1b. So in this paper, the elastic strain energy density \(u_{i}^{c}\) is divided into two parts. One part is the elastic strain energy density of rock grains \(u_{i}^{g}\), the second part is the crack strain energy density \(u_{i}^{c}\). The represents the strain energy density generated by the recoverable deformation of cracks. The method of dividing strain energy in post-peak stage is the same as that in pre-peak stage which are shown in Figure 1c,d. Thus, the relationship between these kinds of strain energy density can be written as  
\[u_{i}=u_{i}^{p}+u_{i}^{\varepsilon}=u_{i}^{p}+\left(u_{i}^{g}+u_{i}^{\varepsilon }\right), \tag{1}\]  
According to Figure 1b,d, the calculation method of strain energy density in a loading and unloading process can be written as follows:  
\[u_{i}=\int_{0}^{\varepsilon_{i1}}\sigma_{i1}d\varepsilon_{i1}+\int_{0}^{ \varepsilon_{i2}}\sigma_{i2}d\varepsilon_{i2}+\int_{0}^{\varepsilon_{i3}} \sigma_{i3}d\varepsilon_{i3}, \tag{2}\]  
\[u_{i}^{\varepsilon}=\int_{a}^{b}\sigma_{i1}^{\prime}d\varepsilon_{i1}^{ \varepsilon}+\int_{a}^{b}\sigma_{i2}^{\prime}d\varepsilon_{i2}^{\varepsilon}+ \int_{a}^{b}\sigma_{i3}^{\prime}d\varepsilon_{i3}^{\varepsilon}, \tag{3}\]  
\[u_{i}^{p}=u_{i}-u_{i}^{\varepsilon}, \tag{4}\]  
\[u_{i}^{g}=u_{i1}^{g}+u_{i2}^{g}+u_{i3}^{g}=\frac{1}{2}\sigma_{i1}\varepsilon_ {i1}^{g}+\frac{1}{2}\sigma_{i2}\varepsilon_{i2}^{g}+\frac{1}{2}\sigma_{i3} \varepsilon_{i3}^{g}, \tag{5}\]  
Figure 1: Comparison of two methods for dividing strain energy in pre-peak stage and post-peak stage: (**a**) The pattern of strain energy division in pre-peak stage which has been used in the former studies; (**b**) the new pattern of strain energy division in pre-peak stage which has been used in this study; (**c**) the pattern of strain energy division in post-peak stage which has been used in the former studies; (**d**) the new pattern of strain energy division in post-peak stage which has been used in this study.