# Material Point Simulation Method for Concrete Medium Fracture and Fragmentation under Blast Loading
## 1 Introduction  
The mechanical performance characteristics of concrete medium are: when external forces reach a certain limit, the concrete medium will undergo abrupt failure without prior warning; it has significant strength asymmetry, with compressive strength far greater than tensile strength; the failure is random and is a typical disordered medium, with an extremely small ultimate strain value during failure; and it exhibits significant rate sensitivity under blast loading. In addition, the process of concrete medium failure under blast loading exhibits significant progressive (dynamic evolution process of failure), grading (scale of fragments), and zoning (mechanical response differences at different distances from the load source) characteristics: the medium in the near zone undergoes violent deformation and shows characteristics of crushing failure; the medium in the middle zone undergoes elastic-plastic deformation, and the degree of fragmentation is relatively larger compared to the medium in the near zone; the medium in the far zone undergoes elastic vibration. Therefore, the mechanical behavior of concrete medium under blast loading is highly complex. Numerical simulation studies of the concrete medium under blast loading mainly include: the finite element method (FEM), the discontinuous deformation analysis method (DDA), the discrete element method (DEM), the material point method (MPM), etc. As a relatively mature numerical simulation method, FEM is widely used. [PERSON] et al. discussed the failure modes of concrete under different working conditions by reviewing numerical/analytical studies on air blast loading [1]. [PERSON] et al. studied the response of tubular concrete under blast loading [2]. [PERSON] et al. simulated the dynamic fracturing and crack propagation processes of brittle materials under blast loading [3]. [PERSON] et al. simulated the explosion characteristics of a hydrogen-air mixture near a concrete barrier [4]. [PERSON] et al. proposed an Arbitrary Lagrangian Eulerian-FEM-smoothed particle hydrodynamics (ALE-FEM-SPH) coupling method and predicted the damage process of reinforced concrete slabs under explosive loads [5]. [PERSON] et al. predicted the fracture mode of reinforced concrete slabs under explosive loads [6]. [PERSON] et al. conducted numerical research on the response of reinforced concrete slabs under explosive loads through a hybrid discretization combining FEM and smoothed particle hydrodynamics (SPH) [7]. [PERSON] et al. predicted the dynamic response of blast-loaded reinforced concrete beams and derive the Iso-Damage curve of a reinforced concrete beam exposed to a blast wave [8; 9]. To solve discontinuous mechanical problems and overcome the limitations of the finite element method, the extended finite element method has been developed in recent years. [PERSON] et al. analyzed the two-dimensional and three-dimensional contact-type crack expansion based on contact fatigue tests and empirical formulas for friction-type cracks [10; 11]. However, the FEM often encounters grid distortion or even negative volume problems when solving this problem, resulting in abnormal termination of calculations. The DDA is a numerical model proposed by [PERSON] for analyzing the motion and deformation of block systems [12]. DDA is capable of solving problems such as large displacements, large deformations, and discontinuous movements such as slipping, cracking, and rotating. In recent years, scholars have started using this method to simulate the failure process of reinforced concrete structures and have achieved good results [13; 14]. However, this method is currently limited to simulating two-dimensional problems and is still a certain distance away from simulating three-dimensional problems. The DEM was first proposed by [PERSON] in 1971 in the United States and is suitable for solving large displacement and nonlinear problems [15]. [PERSON] et al. used a two-dimensional discrete element model to study the failure process of concrete [16]. [PERSON] et al.