## 3 Experimental Results and Analysis
### Dynamic Strength Criterion
#### 3.4.2 Dynamic Mohr-Coulomb Strength Criterion  
Figure 14 presents the Mohr stress circles and corresponding strength envelopes under varying strain rates. The results demonstrate that with an increasing strain rate, the inclination angle of the strength envelopes (corresponding to the internal friction angle) remains relatively stable, while the intercept (representing cohesion) exhibits a marked increasing trend. This observation aligns with findings in [37], confirming that rock strength enhancement during quasi-static loading primarily originates from cohesion improvement, whereas the internal friction angle demonstrates rate-insensitive characteristics. Figure 14: Mohr stress circle and strength envelope curves under different strain rates ((**a**)-10\({}^{-6}\) s\({}^{-1}\); (**b**)-10\({}^{-5}\) s\({}^{-1}\); (**c**)-10\({}^{-4}\) s\({}^{-1}\); (**d**)-10\({}^{-3}\) s\({}^{-1}\); (**e**)-10\({}^{-2}\) s\({}^{-1}\)). Figure 15 delineates the evolution of minor principal stress and major principal stress relationships under varying strain rates. Experimental data demonstrate that the compressive strength exhibits significant enhancement with increasing strain rates and confining pressures, where the confining pressure constitutes the predominant contribution to strength improvement. Linear regression analysis of \(\sigma_{1}\)-\(\sigma_{3}\) curves (fitting equations shown in Figure 15) reveals strict linear dependence between principal stresses across the quasi-static loading regime, supported by correlation coefficient values (R\({}^{2}\) > 0.98) at all tested strain rates. Parameter calculations (Table 5) via Equations (9) and (10) show that cohesion increases from 27.76 MPa to 33.85 MPa (\(\Delta\)c = 6.09 MPa) as the strain rate elevates from 10\({}^{-6}\) to 10\({}^{-2}\) s\({}^{-1}\), while the internal friction angle shows minimal variation from 49.93\({}^{\circ}\) to 51.39\({}^{\circ}\) (\(\Delta\)\(\varphi\) = 1.46\({}^{\circ}\)). When normalized to the reference strain rate of 10\({}^{-5}\) s\({}^{-1}\), cohesion demonstrates a 31% increase (\(\Delta\)c = 8.01 MPa), contrasting with the stable internal friction angle, as shown in Figure 16. These quantitative findings confirm that dynamic strength enhancement principally originates from cohesion evolution, with subsequent sections focusing on establishing the relationship between cohesion and the strain rate. [PERSON] et al. [38] derived the relationship between dynamic strength and strain rate under quasi-static loading based on thermal activation theory, expressed as  
\[\sigma_{\rm d}=\rm C+D\cdot\ln\frac{\dot{\epsilon}}{\dot{\epsilon}_{0}} \tag{11}\]  
where \(\sigma_{\rm d}\) is dynamic strength, and C or D is the fitting parameter. Given the established dominance of cohesion enhancement in strain rate effects, we extend this formulation to characterize dynamic cohesion evolution. Defining static strain rate as \(10^{-5}\) s\({}^{-1}\) and dynamic cohesion increase (\(\Delta\rm c_{d}\)) as the difference between dynamic and static cohesion, Figure 17 reveals nonlinear growth patterns: \(\Delta\rm c_{d}\) increases rapidly at \(10^{-5}\)-\(10^{-3}\) s\({}^{-1}\) before plateauing. The fitted relationship is  
\[\Delta\rm c_{d}=0.99+1.19\cdot\ln(\frac{\dot{\epsilon}}{\dot{\epsilon}_{0}}) \tag{12}\]  
where \(\dot{\epsilon}\) is loading strain rate, and \(\dot{\epsilon}_{0}\) is strain rate of \(10^{-5}\) s\({}^{-1}\). Figure 16: The relationship between friction angle and cohesion of rock specimens and strain rates. Figure 17: The relationship between dynamic cohesion increase and strain rates.