# Material Point Simulation Method for Concrete Medium Fracture and Fragmentation under Blast Loading
## 3 An Improved HJC Model Considering the Segmentation of the DIF Function
### The Improved Method of the HJC Model
#### 3.2.2 Calculation Expression of the \(DIF_{t}\)  
At the end of the Brazilian dynamic splitting experiments, the dynamic tensile strength under different strain rate conditions was plotted as a scatter plot by the experimental results, as shown in Figure 7a. Several functional forms, including the linear function, the Boxlucas2 function, the Explp1 function, the parabola function, and the PWL2 function, were fitted to the experimental results, and the expressions for each function form are shown in Table 1. Figure 7b-f shows the results of fitting the tensile dynamic increasing factor for different functional forms.  
It can be seen that the Boxlucas2 function has the highest goodness of fit compared to the other functional forms when analyzing and comparing the goodness of fit of the different functional forms. The Boxlucas2 function is therefore a more accurate description of the tensile dynamic increasing factor. In addition, analysis of the dynamic tension plot and various data fitting plots shows that the dynamic tension of concrete, like dynamic compression, also exhibits the \"strain rate segmentation property\".  
Based on the consideration of the \"strain rate segmentation characteristics\", the expression of the tensile dynamic increasing factor is proposed as follows:  
\[DIF_{t}=\begin{cases}DIF&\dot{\epsilon}\leq 10^{2}s^{-1}\\ \frac{0.11132}{0.11132+1.45258}\left(e^{1.45258\log{(\dot{\epsilon}^{*})}}-e^ {-0.11132\log{(\dot{\epsilon}^{*})}}\right)&\dot{\epsilon}>10^{2}s^{-1}\end{cases} \tag{8}\]  
In the equation, \(DIF=1+C\ln{\dot{\epsilon}^{*}}\) is the formula for calculating the dynamic increasing factor in the HJC model, \(\dot{\epsilon}\) is the strain rate, and \(\dot{\epsilon}^{*}=\dot{\epsilon}/\dot{\epsilon}_{0}\) is the dimensionless equivalent strain rate (\(\dot{\epsilon}_{0}=1s^{-1}\) is reference strain rate).  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Function Form** & **Curve Expression** \\ \hline Linear curve & \(y=Ax+B\) \\ Boxlucas1 curve & \(y=A(1-e^{Bx})\) \\ Boxlucas2 curve & \(y=\frac{A}{A+B}\left(e^{Bx}-e^{-Ax}\right)\) \\ Exp1p1 curve & \(y=e^{x-A}\) \\ Parabola curve & \(y=A+Bx+Cx^{2}\) \\ PWL2 curve & \(y=\left\{\begin{array}{cc}A+Bx&x<E\\ C+D(x-E)&x\geq E\end{array}\right.\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: The expressions of each function curve.  
Based on this research, the HJC model has been improved, and an improved expression for yield stress has been obtained as follows:  
\[\sigma_{y}^{*}=\left\{\begin{array}{ll}[A(1-D)+Bp^{*n}]&DIF_{c}&p^{*}\geq 0\\ [A(1-D+p^{*}/T^{*})]&DIF_{t}&p^{*}<0\end{array}\right. \tag{9}\]  
Figure 7: Fitting results of the tensile dynamic increasing factor: (**a**) the scatter plot of the experiment result, (**b**) Linear curve fit result, (**c**) Boxlucas2 curve fit result, (**d**) Exp1p1 curve fit result, (**e**) Parabola curve fit result, (**f**) PWL2 curve fit result.  
In the equation, \(T^{*}\) is the normalized maximum static tensile stress, and the other symbols are explained in the previous text.