# Research Advances in Large Deformation Analysis and Applications of the Material Point Method
## 5 Application in Dynamic Cracking  
Crack propagation introduces discontinuities and local gradients. The MPM, not being constrained by the background mesh when simulating crack propagation, can conveniently  
Figure 8: The computational region for the CFDMP [51]. Figure 9: Schematics of MPM–CDEM coupled analysis: (**a**) contact detection; (**b**) contact force calculation [53]. simulate the dynamic fracturing process of objects and handle the significant topological changes that occur during the fracturing process. [PERSON] et al. [54], based on the traditional MPM, solved the problem of the energy release rate in dynamic fracture mechanics, using adaptive meshing and a full mass matrix to improve accuracy, and were the first to validate the value of the MPM in studying dynamic crack propagation. [PERSON] et al. [55] further demonstrated the potential of the MPM in simulating dynamic fracture by comparing the FEM solutions and experimental results that simulate dynamic brittle failure. [PERSON] et al. [56] introduced the characteristic erosion method into the MPM to solve the dynamic fracture problems of brittle materials. [PERSON] et al. [57] simulated dynamic fracture under large elastoplastic deformation. [PERSON] et al. [58] analyzed the effects of frictional heating on crack propagation, crack branching, and the fracturing process. To avoid non-physical oscillations and numerical fragmentation, [PERSON] et al. [59] utilized a cohesive zone model in the GIMP method to simulate dynamic crack propagation and successfully applied it to the problems of crack propagation in both ductile and brittle materials. [PERSON] [60] developed an improved MPM scheme (CRAMP) for simulating cracks with discontinuities in displacement and velocity. In this method, background grid nodes can have multiple velocity fields, allowing adjacent particles to move and deform independently to simulate discontinuities. However, this CRAMP scheme does not have the capability to simulate concurrent crack initiation and propagation nor the ability to propagate cracks to the material's edge. To address this, [PERSON] et al. [61] made improvements by introducing additional grids to simulate any number of crack paths (see Figure 10). They defined a ghost tip point for each active crack tip, located on the trajectory of the crack tip and at a distance equal to twice the defined propagation length, thus enabling the simulation of multiple crack initiations, extensions, and coalescences. Due to the great flexibility of the peridynamics (PD) in simulating dynamic fractures [62; 63], and the fact that both PD and the MPM are particle methods that simulate the dynamic processes of particles through explicit time integration, the combination of PD and the MPM is very natural. [PERSON] et al. [64] simulated plastic fracture. [PERSON] et al. [65] made improvements by modifying the cutoff radius \(\delta\) to \(\delta+l_{e}/2\), thereby eliminating the restrictions of the original PD due to volume integration, and proposed a volume-corrected PD combined with the MPM/GIMP scheme to simulate the evolution of fracturing under  
Figure 10: Arbitrary crack paths in parallel grids [61]. transient loading. The PD combined with the MPM scheme requires the pre-segmentation of the simulation subdomains. To overcome this disadvantage, [PERSON] et al. [56] proposed an adaptive peridynamics MPM, where the simulation region is initially discretized by MPM particles, and during crack growth, the MPM subregions will adaptively switch to PD subregions (see Figure 11), successfully simulating the spallation of a one-dimensional rod, mode I fracture propagation in a pre-cracked plate, mixed-mode crack initiation problems, and the Kalthoff-Winkler experiment. In the process of dynamic fracture, the thin geometric shapes of small debris and new FSI interfaces are produced, making the reconstruction of FSI interfaces challenging. The immersed boundary method (IBM) has shown significant advantages in simulating FSI with complex boundary geometries [66; 67; 68]. [PERSON] et al.