## 3 Incubation Time Criterion  
When the effects of the stress or strain rate on the dynamic strength of brittle materials are investigated, the strength is usually defined as the transient stress at fracture moment, which is extended from quasi-static to dynamic cases. Then, the relationship between dynamic strength and stress or stain rate is empirically modeled [35]. Although this approach is concise and intuitive, the phenomenological law obtained by the simple fitting of experimental data lacks relevant physical meaning. The structure-temporal criterion reflects the fracture mechanism by operating with spatial and temporal scales [36]. \[\frac{1}{\tau}\int_{t-\tau}^{t}\frac{1}{d}\int_{x-d}^{x}\sigma\big{(}x^{ \prime},t^{\prime}\big{)}dx^{\prime}dt^{\prime}\leq\sigma_{0}. \tag{4}\]  
Equation (4) can be used to determine a macroscopic prefabricated crack growing in brittle material at a point \(x\) at time \(t\) under dynamic loading. Where \(\tau\) is the incubation time for a specific material, it is the characteristic constant response to the load on the time scale, and \(d\) is the other material constant reflecting the sensitive size of the fracture process zone, which characterizes the fracture on the spatial scale. The integrand \(\sigma\) (\(x^{\prime}\), \(t^{\prime}\)) is the continuous stress field, and \(\sigma\) is the corresponding quasi-static strength of materials. Intact rock specimens used in laboratory tests generally have dimensions that range from millimeters to several centimeters, and the assumption of uniformity must be satisfied. These naturally correspond to the operations of the structure-temporal criterion on the spatial scale [18]. Thus, Equation (4) can be simplified and rewritten as:  
\[\frac{1}{\tau}\int_{t-\tau}^{t}\sigma\big{(}t^{\prime}\big{)}dt^{\prime}\leq \sigma_{0}. \tag{5}\]  
Equation (5) is recalled as the incubation time criterion. As the criterion reveals, the fracture is considered as a time-dependent process. The upper limit of integration (\(t\)) is  
Figure 2: Schematic view of the dynamic Brazilian test under SHPB loading. the formed moment of fracture, and the integral boundary implies an incubation period preceding the fracture-induced macrostructural transformation. Microscopic scanning of the dynamic fracture surfaces of brittle materials reveals that the fracture surface becomes rougher with the increasing loading rate and is accompanied by additional microcracks that are not parallel to the main fracture surface [37]. [PERSON] et al. [37] suggested that this is a result of the additional microdefects in the vicinity of the crack tip being excited by the stress field with high rates that extend and interact with the main crack. This microscopic incubation process preceding the macroscopic fracture can be synthetically considered as a characteristic time associated with microcracking relaxations. The temporal hysteresis between dynamic fracture and stress level is reflected in the approach of Equation (5). If the rate dependence of rock dynamic strength under SHPB loading is studied by Equation (5), we can assume that the stress increases in the rock at a constant rate before fracture occurs [17; 20]. That is described mathematically as:  
\[\sigma(t)=\dot{\sigma}tH(t) \tag{6}\]  
where \(\dot{\sigma}\) is the stress rate, and \(H(t)\) is the Heaviside step function. Substituting Equation (6) into the incubation time criterion (Equation (5)), the transient stress at fracture is obtained:  
\[\sigma^{*}=\sigma(t^{*})=\left\{ \begin{array}{ll}\sigma_{0}+\dot{\sigma}\tau /2&(t^{*}\geq\tau)\\ \sqrt{2\dot{\sigma}\sigma_{0}\tau}&(t^{*}\leq\tau)\end{array} \right.. \tag{7}\]  
Obviously, for the low loading rate, the quasi-static strength satisfies the state of \(t^{*}\geq\tau\) in Equation (7). Especially under the quasi-static loads, \(\dot{\sigma}\approx 0\), the transient stress at the fracture is equal to the quasi-static strength.