The nature of the fracture and fragmentation process of concrete medium under blast loading is the transformation process of the medium from continuum to discontinuity, coupled with the significant rate correlation of concrete medium, its mechanical behavior presents a high degree of complexity. Problems such as grid distortion and even negative volume are often encountered in the finite element method and others when solving this problem, while the material point method can effectively avoid these problems. In addition, the widely used HJC concrete constitutive model does not take into account the segmented characteristics of the calculation function for dynamic increasing factor. Therefore, in this paper, firstly, the calculation function for dynamic increasing factor in the HJC concrete model is modified by SHPB experiment, and the improved HJC concrete model is proposed; secondly, the material point method simulation program is developed, and the improved HJC concrete model is embedded into the simulation program; finally, the simulation program is verified by numerical examples, and the results show that the developed simulation program can better simulate the fracture and fragmentation process of concrete medium under blast loading, especially the pulverization characteristics of the medium in the near zone of the load.