## 2 Methodology
### Elastic-Plastic Model and Yield Criteria  
Stress-strain curves of rock under dynamic load show obvious strain rate-related properties. The yield limit strength of rock obviously increases as the increase in strain rate. In this study, the Cowper-Symonds plasticity with dynamic strengthening model [22] was used to describe the stress-strain relation of rock under blasting load.  
On the basis of a large number of experiments, [PERSON] (1958) presented the following empirical formula concerning the dynamic limit yield stress and strain rate [22].  
The Cowper-Symonds plasticity with the dynamic strengthening model may be denoted by Equation (6).  
\[\sigma_{\rm y}=\left[1+(\frac{\dot{\varepsilon}}{\rm C})^{1/{\rm P}}\right]( \sigma_{\rm y0}+\beta E_{\rm p}\varepsilon_{\rm p}^{\rm eff}) \tag{6}\]  
where \(\sigma_{\rm s}\) is static limit yield stress; \(\sigma_{\rm y}\) is dynamic limit yield stress; \(\sigma_{\rm y0}\) is initial yield stress; C, P are constants of the Cowper-Symonds bilinear elasto-plastic model; \(\dot{\varepsilon}\) is strain rate; \(\beta\) is rock hardening parameter; \(E_{\rm p}\) is plastic hardening modulus; \(\varepsilon_{\rm p}^{\rm eff}\) is equivalent plastic strain. \(\beta=0\) when rock shows the plasticity with dynamic strengthening, and \(\beta=1\) when rock shows isotropic strengthening.  
\[E_{\rm p}=\frac{E_{0}E_{\rm tan}}{E_{0}-E_{\rm tan}} \tag{7}\]  
\[\varepsilon_{\rm p}^{\rm eff}=\int_{0}^{t}\big{(}\frac{2}{3}\dot{\varepsilon} _{ij}^{\rm p}\dot{\varepsilon}_{ij}^{\rm p}\big{)}^{1/2}\mathrm{d}t \tag{8}\]  
\[\dot{\varepsilon}_{ij}^{\rm p}=\dot{\varepsilon}_{ij}-\dot{\varepsilon}_{ij}^ {\rm e} \tag{9}\]  
where \(E_{\rm tan}\) is tangent modulus; \(E_{0}\) is elasticity modulus; \(\dot{\varepsilon}_{ij}\) is total strain rate; \(\dot{\varepsilon}_{ij}^{\rm e}\) is elastic strain rate; \(\dot{\varepsilon}_{ij}^{\rm p}\) is plastic strain rate; \(t\) is time.  
The yield surface of rock varies with the states of tension and compression under the blasting load. Rock damage and plastic flow generate at the same time. Rock residual strength decrease with the development of damage. In this study, the dynamic characteristics of rock were denoted by considering the effect of damage on the yield function. According to the damage mechanics theory, the plastic problem may be considered in the effective stress space. Therefore, the yield function \(\phi\) is denoted by Equation (10).  
\[\phi=\overline{\sigma}_{i}^{2}-\overline{\sigma}_{y}^{2}=0 \tag{10}\]where \(\overline{\sigma}_{i}\) is effective stress strength; \(\overline{\sigma}_{y}\), \(\overline{\sigma}_{y}^{n}\) are current yield stress.  
\[\begin{split}\overline{\sigma}_{i}=\sqrt{\frac{3}{2}(\overline{ \xi}_{ij}-\overline{\alpha}_{ij})(\overline{S}_{ij}-\overline{\alpha}_{ij})}\\ \overline{S}_{ij}=\overline{\sigma}_{ij}-\frac{1}{3}\overline{ \sigma}_{\rm kk}\\ \Delta\overline{\alpha}_{ij}=\frac{2}{3}(1-\beta)E_{\rm P} \dot{\overline{\epsilon}}_{ij}^{\rm P}\Delta t\end{split} \tag{11}\]  
where \(\overline{S}_{ij}\) is a component of effective deviatoric stress tensor; \(\overline{\sigma}_{ij}\) is effective stress tensor \(\overline{\sigma}_{\rm kk}\) is the first invariant of stress tensor; \(\overline{\alpha}_{ij}\) is movement tensor at the center of yield surface; \(\Delta\overline{\alpha}_{ij}\) is movement tensor increment at the center of yield surface; \(\dot{\overline{\epsilon}}_{ij}^{\rm P}\) is effective plastic strain rate; \(\dot{\overline{\epsilon}}_{ij}^{\rm P}\) is effective plastic strain; \(\Delta t\) is time increment.