# Indices to Determine the Reliability of Rocks under Fatigue Load Based on Strain Energy Method
## 1 Introduction  
Rock is a kind of complicated material composed of mineral grains and cracks [1; 2; 3; 4]. The deformation and failure process of rock is mainly controlled and affected by cracks [5; 6; 7; 8; 9; 10; 11]. During earthquakes, mining, and engineering construction, rock suffers deformation, producing fractures or failure as a result of fatigue load, which can impacts human lives and the environment [12; 13; 14]. Studies on the mechanical properties of rocks under fatigue loads are usually conducted using stress-based, strain-based, or strain energy-based approaches [15; 16; 17; 18]. Due to the essence of rock material, deformation and failure is an energy-driven process [19] and energy evolution runs through the whole process of rock deformation and failure. The analyses based on the strain energy based-approach proved to be an effective approach [20; 21; 22; 23]. In rocks, yield point and peak stress point are very important in evaluating the mechanical properties of rocks [9; 24; 25; 26; 27; 28; 29; 30]. During the yield stage, rock deformation is not easily controlled by stress, and the rock structure becomes unstable. When the deformation of rock exceeds the peak stress point, the whole rock is destroyed. Therefore, it is more significant to determine whether the rock enters the yield state than whether it breaks in engineeringpractice. From the point of view of elastic-plastic mechanics, the mechanical energy input from outside is divided into two parts in rock without considering heat exchange. One part is elastic strain energy, which accumulates in the rock. The other part is plastic strain energy, which leads to the cracks growing, structural changes, and unrecoverable deformations. When some kind of strain energy reaches a certain state, the rock also enters a corresponding deformation or failure stage [31]. Therefore, the study of the effects of elastic and plastic strain energy on the rock deformation process can deepen our understanding of the mechanical behavior of rocks.  
Presently there are no studies in literature that described or proposed a method to determine whether rock has entered the yield stage or failure stage from the point of view of strain energy. Some studies have focused on the evolution of elastic and plastic strain energy in rock and the associated damage to rock structures [32; 33; 34]. Some studies also analyzed the external factors that affect the evolution of elastic energy and plastic strain energy [36; 37]. In addition, some studies analyzed the deformation and failure behavior of rocks under different fatigue loads [13; 38; 39; 40]. These studies provide a good baseline for understanding the relationship between rock and strain energy and the mechanical behavior of rocks under fatigue loads. However, most of these studies are based on the traditional elastic and plastic strain energy division method, which cannot reflect the strain energy effect of rock grains and cracks in the process of rock deformation; hence, it is difficult to make a break through. At the same time, almost all of these studies only explain the phenomena, and do not give indices of strain energy when the rock enters the yielding stage or failure. Therefore, with the further research, the new strain energy division method and the index to determine the rock from stable to unstable stage should to be discussed.  
Focusing on the abovementioned questions, this study tries to give indices to determine whether rocks have entered the yielding stage or failure. Through the new classification of strain energy density types, we found that the ratio of some kinds of strain energy densities to corresponding total strain energy densities obtained by the new method could be used as the index to determine whether the rock has entered the yield stage or failed. We subsequently collected experimental data from extensive literature to confirm this finding. We believe that this discovery is of great significance in rock engineering and earthquake prediction for judging or predicting rock stability.