# Effect of Holes on Dynamic Crack Propagation under Impact Loading
## 3 Numerical Method
### Introduction of the Algorithm in AUTODYN  
To illustrate the algorithm of AUTODYN [16; 17], a group of elements undergoing dynamic loading and their potential deformation are presented in Figure 8. The material parameters were ascribed to all elements with the same value. When subjected to dynamic loading, the nodes of the elements move with independent velocity, causing deformation. \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline
**Specimen** & \begin{tabular}{c} **AT** \\ **\(\mu\)s** \\ \end{tabular} & \begin{tabular}{c} **V\({}_{\text{min}}\)** \\ **\(m\)/s** \\ \end{tabular} & \begin{tabular}{c} **V\({}_{\text{avg}}\)** \\ **\(m\)/s** \\ \end{tabular} & \begin{tabular}{c} **specimen** \\ **\(\mu\)s** \\ \end{tabular} & \begin{tabular}{c} **AT** \\ **\(m\)/s** \\ \end{tabular} & \begin{tabular}{c} **V\({}_{\text{avg}}\)** \\ **\(m\)/s** \\ \end{tabular} & \begin{tabular}{c} **Specimen** \\ **\(\mu\)s** \\ \end{tabular} & \
\begin{tabular}{c} **AT** \\ **\(m\)/s** \\ \end{tabular} &
\begin{tabular}{c} **V\({}_{\text{min}}\)** \\ **\(m\)/s** \\ \end{tabular} \\ \hline B12-D15 & 85.87 & 23.29 & 205.95 & B12-D30 & 51.77 & 38.63 & 164.46 & B12-D45 & 338.98 & 5.90 & 152.82 \\ B16-D15 & 56.60 & 35.33 & 186.69 & B16-D30 & 99.90 & 20.02 & 208.30 & B16-D45 & 309.11 & 6.47 & 183.93 \\ B20-D15 & 14.40 & 138.88 & 212.32 & B20-D30 & 24.20 & 82.64 & 205.18 & B20-D45 & 50.80 & 39.37 & 225.88 \\ \hline \hline \end{tabular}  
In this table, AT is short for arrest time, V\({}_{\text{min}}\) is minimum velocity, and V\({}_{\text{avg}}\) is average velocity.  
\end{table} 
Table 4: Crucial data of crack propagation. Figure 8: Elements and potential deformation under dynamic loading. The acceleration of node A in Figure 8 can be expressed by Equation (5), according to [PERSON]'s law. \[ \begin{array}{rcl}F_{x}&=&m\cdot\overset{\bullet\bullet}{u}\\ F_{y}&=&m\cdot\overset{\bullet\bullet}{v}\end{array} , \tag{5}\]  
where \(\overset{\bullet\bullet}{u}\) and \(\overset{\bullet\bullet}{v}\) are the accelerations in \(x\)- and \(y\)-directions, respectively, \(m\) is mass, and \(F_{x}\) and \(F_{y}\) are the forces in \(x\)- and \(y\)-directions. respectively. Then, the velocity \(\overset{\bullet}{u}\) and \(\overset{\bullet}{v}\) of node A with time in \(x\)- and \(y\)-directions can be determined using Equation (6). \[ \begin{array}{rcl}\overset{\bullet}{u}_{n+1}&=&\overset{\bullet}{u}_{n}+ \overset{\bullet\bullet}{u}_{n}\Delta t\\ \overset{\bullet}{v}_{n+1}&=&\overset{\bullet}{v}_{n}+\overset{\bullet \bullet}{v}_{n}\Delta t\end{array} , \tag{6}\]  
where \(n\) is the time step and \(\Delta t\) is the time increment. The strain rates of elements (surrounded by A, B, C, and D nodes) are calculated using Equation (7). \[ \begin{array}{rcl}\overset{\bullet}{\varepsilon}_{x}&=&\frac{\partial \overset{\bullet}{u}}{\partial x}&=&\frac{1}{2\Delta^{n+\frac{1}{2}}}[( \overset{\bullet}{u}_{A}-\overset{\bullet}{u}_{c})(y_{B}-y_{D})-(\overset{ \bullet}{u}_{B}-\overset{\bullet}{u}_{D})(y_{A}-y_{C})]\\ \overset{\bullet}{\varepsilon}_{y}&=&\frac{\partial\overset{\bullet}{v}}{ \partial y}&=&\frac{1}{2\Delta^{n+\frac{1}{2}}}[(\overset{\bullet}{v}_{A}- \overset{\bullet}{v}_{C})(x_{B}-x_{D})-(\overset{\bullet}{v}_{B}-\overset{ \bullet}{v}_{D})(x_{A}-x_{C})]\\ \overset{\bullet}{\gamma}_{xy}&=&\frac{\partial\overset{\bullet}{v}}{ \partial x}+\frac{\partial\overset{\bullet}{v}}{\partial y}&=&\frac{1}{2 \Delta^{n+\frac{1}{2}}}[(\overset{\bullet}{v}_{A}-\overset{\bullet}{v}_{C}) (y_{B}-y_{D})-(\overset{\bullet}{v}_{B}-\overset{\bullet}{v}_{D})(y_{A}-y_{ C})\\ &&+(\overset{\bullet}{u}_{A}-\overset{\bullet}{u}_{c})(x_{B}-x_{D})-( \overset{\bullet}{u}_{B}-\overset{\bullet}{u}_{D})(x_{A}-x_{C})]\end{array} , \tag{7}\]  
where \(\overset{\bullet}{\varepsilon}_{x}\) and \(\overset{\bullet}{\varepsilon}_{y}\) are the strain rates of the element ABCD in \(x\)- and \(y\)-directions, respectively, whereas \(\overset{\bullet}{\gamma}_{xy}\) is the strain rate of element ABCD in the \(xy\) plane.