# Material Point Simulation Method for Concrete Medium Fracture and Fragmentation under Blast Loading
## 4 Numerical Example  
The improved HJC model was embedded into the MPM simulation program, and its effectiveness was investigated through a numerical example. The numerical example is a contact explosion experiment on a plain concrete slab carried out by [PERSON] et al. [32]. The dimension of the concrete slab is 750 mm \(\times\) 750 mm \(\times\) 75 mm. The concrete slab was reinforced with 12-mm-diameter rebar mesh spaced at 100 mmc/c each way at the top and bottom with a 30-mm cover. The concrete slab was supported on a 900-mm-high steel frame without edge fixity. Experimental tests were performed to investigate both failure modes by subjecting the concrete slab to the contact explosion effects of 500 g of TNT. The experiment result is shown in Figure 8.  
The numerical model was established first before using the program to solve this problem. The material points were used to discretize the charge and concrete slab, and a background grid was generated. The spacing between material points of the charge was set to 0.005 m, resulting in a total of 2184 material points. The spacing between material points of the concrete slab was also set to 0.005 m, producing a total of 339,684 material points. The range of the background grid was set to X: 0.0 m-0.75 m, Y: 0.0 m\(\sim\)0.75 m, and Z: \(-\)0.8 m-0.3 m, with a spacing of 0.01 m. Boundary conditions were applied to the concrete slab using the background grid, with the length and width being fixed and the thickness being free. The explosive calculation model in contact with the concrete slab is shown in Figure 9.  
Figure 8: Response of the concrete slab to the contact explosion effect [32].  
Figure 9: Calculation model for the contact explosion of the concrete slab. (**a**) Geometric model; (**b**) Numerical model.  
Figure 10: The equivalent plastic strain of the slab.  
When the damage value of a certain material point inside the slab exceeds 1.0, it is considered that the material point has failed, as shown in Figure 13. From the figure, it can be seen that after the detonation of the charge, the first damage caused is the crushing damage to the front explosive surface, followed by the formation of a through-hole at the center of the slab, and finally the tensile damage to the back explosive surface, with the damaged area on the back of the slab being significantly larger than that on the front. The diameter of the damaged area on the front of the slab calculated by this method is 315 mm, with an error rate of 10% compared to the experimental results, indicating that this method can accurately simulate the process of the concrete medium under blast loading.  
Figure 11: Blast pressure time curve.  
Figure 12: The ultimate damage distribution of the slab. (**a**) The front explosive surface; (**b**) The back explosive surface.  
Figure 14 depicts the process of changing system energy over time. When the charge explodes, a massive amount of energy is released in an instant. Compression and tensile failure occur in the concrete slab. As the explosive burns completely, its internal energy progressively changes into the system's kinetic energy. However, the entire energy of the system is conserved both before and after the explosive detonation, which indirectly reflects the method's accuracy.  
This numerical example demonstrates that, compared to other continuum mechanics methods such as the FEM, the MPM can accurately simulate material fracture problems, whereas the FEM frequently generates grid distortion and distorted calculation results when solving those problems. However, the MPM is less accurate than the FEM when calculating tiny deformation problems.  
Figure 13: Failed area of the slab. (**a**) The XY plane of the slab; (**b**) The YZ plane of the slab; (**c**) The whole slab.