## 1 Introduction  
Rock bursts have become more and more severe and occurred frequently with the increase in mining depth over the years at the rate of 10-25 m per year, and a growing number of mines reach a depth of 1000 m [1, 2]. The deep coal extraction is facing stress concentration with acute occurrence and frequency of rock burst incidents induced by dynamic loads in coal mines. A large number of case studies of rock bursts are documented in America [3, 4], Australia [5], Poland [6, 7], Russia [8], Czech Republic [9, 10] and many other countries. In China, nearly 300 coal mines have reported rock burst disasters so far, and the number is increasing [11]. Table 1 lists the major cases of rock burst disasters that occurred in China in the past five years. This table clearly reveals one common characteristic among the causes of rock bursts: stress concentration and dynamic load disturbance. Stresses in coal mines include initial stress and mining-induced stress [12], and the local concentrated high stress is the internal cause of rock bursts. Dynamic loads are mainly caused by drilling, strata fracturing, fault reactivation, coal pillar collapse and other mining and excavating-related activities [13, 14, 15, 16, 17, 18]. Dynamic load disturbance and loading are the external causes of rock bursts. The superimposed static and dynamic loads are the fundamental reason for rock bursts [19, 20]. The coal mines affected by hard and thick roofs account for more than 50% of the mining areas, and the hard roof is one of the leading causes for rock bursts [21; 22]. Hard roof rocks generally have high strength and low probability of caving. However, with the continuous advance of the mining face and the increasing overhang area, the hanging roof could collapse, causing seismic waves [23; 24; 25] to propagate in the form of dynamic loads. Once the shock wave propagates to the static loading area, this leads to the superimposition of static and dynamic loads. Therefore, the deformation and failure of thick and hard roofs in coal mines are not only affected by the static in situ stress, but also the dynamic loading rate caused by mining activities. Especially, the failure of roof and coal pillars due to dynamic loading rate becomes more critical as the mining depth increases and mining activities intensify. Therefore, when the rock failure cannot be explained solely by static or dynamic theory [26], it is crucial to consider the combination of static in situ stress and dynamic loading rate to understand the mechanical behavior and failure characteristics of roof rocks [27]. To date, no consensus understanding exists regarding the cutoff between static and dynamic loading rates. The concept of static and dynamic loads is not definite, and is closely related to material types and the environment [28]. The relationship between the dynamic strain rate and seismic wave parameters in coal mines was obtained through an in situ test [29]. Unlike explosions, air blasts, dynamic compaction, piling and rapid loading tests, the maximum dynamic strain rate in coal mines ranges from \(10^{-3}\) to \(10^{-1}\) s\({}^{-1}\), which is a medium strain rate range [30]. Therefore, the analysis of the mechanical behaviors of rock at medium strain rates is of great significance to mine earthquakes, rock bursts and coal