# Analysis of Optimal Loading Angle in Dynamic Flattened Brazilian Disc Splitting Test for Concrete
## 2 Numerical Simulations  
The mechanics of Brazilian splitting of disc self-compacting concrete under dynamic loads is first investigated by means of numerical simulations. The set speeds of the projectile with a length of 400 mm are 4.9 m/s, 9.6 m/s, and 12.5 m/s, respectively, where the projectile was defined as kinetic energy penetrators, and the impact compression of the disc concrete specimen is obtained to obtain the whole process of dynamic fracture evolution. The numerical simulation of Brazilian splitting uses a plane strain assumption, the diameter of the numerical specimen is 50 mm, the model mesh adopts a mapping mesh,the element size is 0.25 mm, and the whole model contains a total of 10,000 cells and 10,201 nodes. The loading process is controlled by speed, and the boundary conditions are deformation in the z-direction of the fixed model, the speed boundary condition in the x direction of the projectile is applied to the model, and the transmission rod is fixed and constrained. A total of 61 measuring points are arranged along the central profile of the length of the Brazilian splitting disc specimen, and the spacing between each measuring point is 5 mm, of which the No. 6 measurement point is arranged in the center of the circle, as shown in Figure 1. Using the JHC (Johnson Holmquist Concrete) constitutive model [35; 36], the numerical model parameters are shown in Table 1. When the model element is yielded, the cohesion and tensile strength of the unit gradually soften linearly with the accumulation of plastic shear strain and plastic tensile strain. When the plastic shear strain and plastic tensile strain accumulate to the critical value, the cohesion and tensile strength of the unit weaken to the residual strength and remain unchanged. In the plane of the Brazilian disc, any pair of symmetry points, as shown in Figure 2, the compressive stress on the symmetry point are symmetrical, indicating that the specimen can be maintained in a dynamic force equilibrium state during the dynamic load process. As shown in Figure 3(a\({}_{0}\))-(c\({}_{0}\)), at a velocity of 4.9 m/s, the stress wave first forms a compression crack at the end of the incident rod due to the dynamic load. From Figure 4a, it can be seen that the compressive stress peak at 32.08 \(\upmu\)s formed a compressive stress of 205.6 MPa at measurement point 1, and then the compressive stress decayed rapidly. As the tensile stress increases along the radial loading direction, as shown in Figure 4d, before \(t=40\)\(\upmu\)s, the tensile stress is small, and the mean tensile stress is about 2.5 MPa, which is not enough to produce damage. In \(t=50\)-\(80\)\(\upmu\)s, the tensile stress from measurement point 3 to measurement point 9 is generally increased, and the average value is about 7.5 MPa. At \(t=50\)\(\upmu\)s, the compressive stress propagates to the transmission bar end and begins to reflect, and the incident wave and the reflected wave are the same sign, showing the loading process. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline
**Sign** & \(\rho\)**/kg.m\({}^{-3}\)** & \(f_{\text{c}}\)**/MPa** & \(\mathbf{G}\)**/GPa** & \(\mathbf{A}\) & \(\mathbf{B}\) & \(\mathbf{C}\) & \(\mathbf{N}\) & \(\mathbf{T}\)**/MPa** & \(\mathbf{\varepsilon}_{\text{fmin}}\) \\ \hline
**Value** & 2425 & 52 & 13.9 & 0.80 & 1.60 & 0.007 & 0.61 & 3 & 0.007 \\ \hline
**Sign** & \(\mathbf{P}_{\text{e}}\)**/MPa** & \(\mathbf{\mu}_{\text{c}}\) & \(\mathbf{P}_{\text{J}}\)**/MPa** & \(\mathbf{\mu}_{\text{n}}\) & \(\mathbf{D_{1}}\) & \(\mathbf{D_{2}}\) & \(\mathbf{K_{3}}\)**/MPa** & \(\mathbf{K_{2}}\)**/MPa** & \(\mathbf{K_{3}}\)**/MPa** \\ \hline
**Value** & 11.7 & \(6.26\times 10^{-4}\) & 800 & 0.06 & 0.0452 & 10 & 85 & \(-\)171 & 208 \\ \hline \end{tabular}
\end{table} 
Table 1: Material parameters of JHC model. Figure 1: Schematic diagram of Brazilian cleavage (The numbers in the figure indicate the monitoring points).