## 4 Crack Propagation DSIF in Complete Process
### Calculation Method of Dynamic SIFs  
According to the theory of linear elastic fracture mechanics, the SIF can be expressed by the crack tip displacement with Equation (9):  
\[u(r,\theta,t)=\frac{(1+v_{d})K_{\mathrm{I}}(t)}{2E}\sqrt{\frac{r}{2\pi}}\Big{[}( 2k+1)\sin\frac{\theta}{2}-\sin\frac{3\theta}{2}\Big{]} \tag{9}\]  
where \(E\) is the elastic modulus, \(v_{d}\) is the Poisson's ratio, \(k=3\)-\(4v_{d}\), \(u\) is the displacement, and \(K_{\mathrm{I}}(t)\) is SIF. According to the displacement extrapolation method, SIF can be expressed by Equation (10):  
\[K_{\mathrm{I}}(t)=\frac{E}{24(1-v_{d}^{2})}\sqrt{\frac{2\pi}{r_{OA}}}\big{[}8 u(r_{OB},\pi,t)-u(r_{OA},\pi,t)\big{]} \tag{10}\]  
where \(r_{OA}\) and \(r_{OB}\) are the distances from the A and B points to the origin O in the singular element of the crack tip.  
[PERSON] [42] proposed that DSIF in the dynamic crack propagation is related to the crack propagation speed and DSIF of the stationary crack with the corresponding crack length. The relation can be expressed by Equation (11):  
\[K_{\mathrm{I}}^{\mathrm{d}}(t)=k(v)K_{\mathrm{I}}^{0}(t) \tag{11}\]  
where \(K_{\mathrm{I}}^{\mathrm{d}}(t)\) is the DSIF of crack with a certain speed, \(v\) is crack propagation speed, \(K_{\mathrm{I}}^{0}(t)\) is the DSIF of the stationary crack with corresponding crack length, \(k(v)\) is the universal function, as shown by Equation (12):  
\[k(v)\approx\frac{1-v/c_{\mathrm{R}}}{\sqrt{1-v/c_{\mathrm{D}}}} \tag{12}\]  
where \(c_{\mathrm{R}}\) is Rayleigh wave velocity, \(c_{\mathrm{D}}\) is longitudinal wave velocity.  
Figure 14 demonstrates the determination method of the propagation of DSIF. As the curve of dynamic SIF versus time at the crack tip has been obtained by calculation results from ABAQUS, the critical DSIF is determined by the fracture time recorded from the crack propagation gauge in the  
Figure 13: Mesh of finite element model established in ABAQUS code.  
experiment. Then, the determined \(K_{1}{}^{0}(t)\) and the corresponding velocity is substituted into Equations (11) and (12), and the propagation DSIF, \(K_{1}{}^{4}(t)\), can be acquired.  
Therefore, according to the method discussed above, the crack propagation DSIF in the complete process with three loading rates were calculated, as shown in Figure 15. It is clear that the crack initiation DSIF is the highest among the whole crack propagation process. During the crack propagation process, the DSIF has a decreasing trend after initiation. However, when running crack is arrested with very low velocity, the arrest DSIF, which is a particular kind of propagation DSIF, increased significantly.  
Taking the effect of the loading rate into consideration, although the three DSIF curves have a consistent trend, the crack propagation DSIF has an entire enlargement as the loading rate rises. Especially when the loading rate rises from 1500 GPa-s\({}^{-1}\) to 2000 GPa-s\({}^{-1}\), the amplification of DSIF is more pronounced than the one from 1000 GPa-s\({}^{-1}\) to 1500 GPa-s\({}^{-1}\). Under loading rate 1000 GPa-s\({}^{-1}\), the initiation DSIF is 12.21 MPa-m\({}^{1/2}\), the minimum propagation DSIF is 5.25 MPa-m\({}^{1/2}\), and arrest DSIF is 8.39 MPa-m\({}^{1/2}\). Under loading rate 1500 GPa-s\({}^{-1}\), the initiation DSIF is 16.41 MPa-m\({}^{1/2}\), the minimum propagation DSIF is 8.00 MPa-m\({}^{1/2}\), and arrest DSIF is 12.57 MPa-m\({}^{1/2}\). Under loading rate 2000 GPa-s\({}^{-1}\), the initiation DSIF is 19.42 MPa-m\({}^{1/2}\), the minimum propagation DSIF is 11.3 MPa-m\({}^{1/2}\), and arrest DSIF is 16.20 MPa-m\({}^{1/2}\). All the above shows that loading rate does have a significant effect on crack propagation DSIF.  
Figure 14: The curve of dynamic stress intensity factor (SIF) versus time and determination method of propagation dynamic stress intensity factor (DSIF).  
Figure 15: Crack propagation dynamic stress intensity factor in the complete process.