# A Simulation of the Densification Process of NdFeB Bulks by a Modified Drucker-Prager Cap Model
###### Abstract  
During the sintering process of NdFeB bulks, temperature changes and significant temperature differences between the bulk interior and the surface region will produce high residual stress. Temperature field and stress field prediction during the sintering process is one of the key techniques for analyzing residual stress. Therefore, the sintering process simulation and residual stress prediction of NdFeB bulks under different sintering temperatures were conducted based on the modified Drucker-Prager cap (DPC) model in ABAQUS (ABAQUS 2024). The calculated field cloud charts were analyzed against the microstructure of the bulks observed by scanning electron microscope (SEM). The finite element analysis (FEA) results of the sintering process and the residual stress show good agreement with SEM morphologies, which validates the accuracy and predictability of the model. The results indicate that cracks predominantly formed in edge regions. As the sintering temperature increased, longitudinal compressive stress at the edge of the cross-section transitioned into tensile stress. These results indicate that the developed simulation framework effectively identifies crack-prone areas, enabling data-driven optimization to reduce experimental trial-and-error costs in engineering applications. technique is a new energy-saving, efficient, and environment-friendly material thermal processing technology [9]. However, in the NdFeB preparation process, the contraction inconsistency among various parts leads to cracks during the sintering process [10]. In addition, the preparation of magnets has issues like uneven density, stress concentration, etc., which are detrimental to their shaping and performance. As a result, additional adjustment is required to reduce the cracks and generate better performance, causing wasted time and prolonged test cycles. To address this, finite element analysis (FEA) is used to transform the project into a visual model. The real-time changes in the microstructure of the green body (unsintered NdFeB bulk) can be assessed by observing the distribution of density and stress during the shaping of NdFeB powder under different sintering processes. This approach can reduce raw material usage, save trial costs, and further help improve sample quality. Finite element software has a superior pre-post-processing program and a high-strength nonlinear solution system. Several yield models have been implemented into FEA to simulate the characteristics of metal powder [11]. [PERSON] et al. [12] simulated the pressing process of metal powder in ABAQUS software and used the DPC model to describe the properties of metal powders. [PERSON] et al. [13] simulated three powder molding experiments based on the modified DPC model, and the results highly correspond to the density field, which shows that the cap model can reduce the error to make the modified model more accurate. [PERSON] et al. [14] used tungsten carbide powder as the research object to obtain cap model parameters by inverse modeling, showing the flexibility of the cap model. In current studies, the modified DPC model is more commonly applied in powder pressing and density analysis than many constitutive models, and a model with high simulation accuracy can be obtained through appropriate design. This is why researchers have focused more on the study of this type of model in recent years [12; 13]. It was found that the modified DPC model is directly related to material density and yield strength, accurately simulating powder densification. Compared with traditional models, it conforms better to the mechanisms of material hardening. Moreover, a curvature-continuous transition function can be used between the cap and the shear yield surface to avoid stress jumps and improve the convergence of the calculation. The DPC constitutive model is based on the framework of elastic-plastic mechanics and describes the densification and shear failure behavior of powders under external pressure. It is widely used to simulate material behavior in powder metallurgy.