Integrating Simulation and Experimentation for Optimized Compaction Powder Forming: A Study on RVE Size, Friction, and Particle Size Distribution

This work comprehensively represents multi-particle finite element simulations for powder compaction, including examining representative volume element (RVE) size, friction effects, and particle distribution. The analysis provides valuable insights into the correlation between RVE size and relative density, facilitating a comprehensive understanding of the sensitivity of process parameters to relative density. Furthermore, the research investigates the impact of particle size distribution. Moreover, the study investigates the influence of friction between powder particles and the die wall. Results demonstrate that increased friction leads to a significant reduction in relative density.


Introduction
Powder compaction is fundamental in various industries, such as pharmaceuticals, ceramics, and metallurgy.The process involves transforming powder into a solid form using mechanical forces.The quality of the compacted material can be influenced by a myriad of factors, namely representative volume element (RVE) size, friction, and particle size distribution.Michrafy et al. [1] investigated the impact of friction between powder and die walls during the compaction of pharmaceutical binders across three lubrication techniques: lubricated die, non-lubricated die, and powder mixed with lubricant.Using the principles of Coulomb friction, the wall friction coefficient was assessed based on transmission, transfer, and aspect ratios.Frictional behavior was found to vary depending on the powder and lubrication method.While adding a lubricant to the powder reduced friction compared to a lubricated die, the friction coefficient changed during densification.A model combining the Janssen-Walker analysis with the Heckel equation was used to assess the effect of wall friction on the axial density distribution of tablets.However, this model was deemed unsuitable for powder mixed with lubricant due to its lack of relevant physical parameters.Korim et al. [2] developed a Cu-15Fe alloy using powder metallurgy, incorporating solid lubricants paraffin wax (PW), stearic acid (SA), and their blend.The compacts were shaped under varying pressures and sintered at 1050 •C for an hour.Assessments included densification, porosity, hardness, and morphological changes.Tribological properties were analyzed through pin-on-disk tests.Lubrication type and compaction pressure significantly affected the alloy attributes.PW achieved the best results at 50 MPa, decreasing with pressure, while SA performed best at 350 MPa, improving with increased pressure.PW samples demonstrated reduced friction and wear, but SA enhanced wear resistance, reducing the wear rate by roughly 50% compared to other samples.Guner et al. [3] examined metal powder compaction, focusing on 200 μm copper particles using the multi-particle finite element method (MPFEM).The von Mises model analyzed friction and contact dynamics in the compaction process.Three friction models, Amonton-Coulomb, Wanheim-Bay, and Levanov, were evaluated.Validation involved compaction experiments and microscopy comparisons.Friction coefficients for Wanheim-Bay and Levanov were between 0.04-0.07.The Wanheim-Bay model was suited for high-density compaction, while Levanov was preferable for lowdensity cases.Kahhal et al. [4] discuss how loading conditions, geometry, and wall friction impact punch force and stress distribution.They found that Increased wall friction (from 0.0 to 0.2) raises compaction force by 14.70%, and differences between upper and lower face forces increase.Changing the loading path from compression to hydrostatic compaction reduces compaction force by 57.7% (upper) and 65.75% (lower), decreases axial stress by 27.70%, and increases transverse stress by 40.41%.Compaction volume strain is 35.33% smaller than hydrostatic compression.This study performs multi-particle finite element simulations for powder compaction, emphasizing the size of the RVE, friction, and particle distribution.The influence of RVE size is analyzed, aiming to discern its relationship with relative density.The impact of particle size distribution and the effects of friction between powder particles and the die wall are studied.

Material and Method
The modeling framework was devised by integrating both DEM and traditional FEM.In the initial phase, particle center coordinates from the DEM experiments, informed by empirical data, were utilized as the foundational configuration for the MPFEM.This led to the MPFEM-based execution of powder compaction simulations as the final step.The construction and execution of both DEM and MPFEM models were carried out using the commercial FEM software Abaqus (Version 2020).Due to the intensive computational nature of the model, the Abaqus explicit algorithm was employed.A detailed breakdown of this modeling approach is presented in the following sections.

Material Properties
The Fe-Si-Al-P alloy powder compaction was evaluated utilizing the elastic-plastic material framework.Table 1 illustrates the material characteristics of this model.Compression testing was executed using the Instron 8501 Servo Hydraulic Machine, which utilizes a servo-regulated hydraulic mechanism to exert compressive forces on specimens.The peak pressure sustained during the study was set to 1.2 GPa.Additionally, compressive assessments were conducted on samples sintered at 950 °C.By employing the power law swift hardening technique, extending the experimental compression results and identifying the stress-strain trajectory became feasible.The equation below represents the flow stress-plastic strain correlation for the given material: σ = 1380 * (ε + 0.0354) .

Details of MPFEM Models
The DEM simulation, employed to define particle properties and initialize configurations for MPFEM simulations, utilized the Edinburgh-Elasto-Plastic-Adhesion (EEPA) model with a model size of 1 mm 3 .
The EEPA model governed the interaction forces between particles, encompassing elastic, plastic deformation, and adhesive forces.After completing the DEM simulation, coordinate data of particle centers were extracted and transferred to the corresponding MPFEM model through Python scripts.
The MPFEM model incorporates both a punch and a die in addition to the meshed particles.Fig. 1 illustrates the generated particles and dies.Each particle is depicted as a solid component and is meshed utilizing C3D4 elements.These elements are determined by a seeding size of around 20 micrometers.
The other parts are depicted as discrete 3D rigid structures.Initial relative density was considered 40 percent.Interparticle friction and particle-die friction coefficient were chosen at 0.2.

Fig. 1 MPFEM model
A customized MATLAB code, complemented by a Python script, was crafted specifically for Generating Representative Volume Elements (RVEs).This dual-software approach facilitated the import of particle data directly from the discrete element method results, particularly focusing on the positions and radii of each particle.The process for integrating particles into the RVE was multifaceted.
If the designated position of a particle was within the die boundaries, it was included in the RVE.Conversely, any particle outside these confines was excluded, and a new particle was imported.This method ensured that all particles populated the RVE without overcrowding.Furthermore, a particle in the RVE had to meet three stringent criteria.Firstly, it had to be fully encapsulated by the die.Secondly, the initial relative density of the RVE should be maintained at 40 percent.Lastly, the RVE composition is needed to reflect the target distribution accurately.This rigorous approach to RVE generation ensures that the subsequent analyses are grounded in realistic and representative models of the material under investigation.

Influence of Friction Coefficient on Density
A thorough examination was conducted to understand how the friction coefficient between particles affects the compaction process.In our study, the friction between particles was varied, with values of 0.1 and 0.2 adopted.Fig. 2 presents the results demonstrating the significant impact of changing the friction coefficient between particles.A clear pattern can be observed: as the friction decreases, the density under load increases, and the spring back reduces, leading to a higher final density.This indicates that the friction coefficient of powder should be reduced to increase the density difference between various mixtures.

Fig. 2 Effect of friction between particles on relative density
In the subsequent analysis, the influence of inter-particle friction is investigated.For the kinetic friction evaluation, a coefficient of 0.2 was applied.In the case of static friction, two distinct coefficients, 0.2 and 1, were considered.Referring to Fig. 3, an evident trend was observed: with an increase in static friction, there was a decline in density under applied load.
Fig. 3 Influence of static and kinetic friction on the density

Influence of RVE Size on Stress-Relative Density Curve
This section investigates the impact of the RVE size on relative density.For our study, RVE sizes ranging from 350 to 800 micrometers were chosen.Notably, smaller RVE sizes can introduce significant fraction errors, underscoring the importance of selecting an appropriate RVE size.On the other hand, with larger RVE sizes, there is an increase in the particle count, leading to a rise in computational demands.Fig. 4 illustrates the impact of RVE size on relative density.Increasing the RVE size results in a corresponding increase in initial density.However, it is important to note that while this variation becomes more significant at lower pressures, the difference decreases and becomes relatively insignificant in areas where higher pressures are applied.This subtle interplay between RVE size and density, particularly at varying pressure levels, highlights the complex dynamics of powder compaction.Understanding the relationship between pressure, deformation, and relative density is crucial when compacting powder mixtures.The comparison between the three cases provides valuable insights into how fine and coarse particles respond during compaction, with higher pressure consistently leading to increased relative density.

Conclusion
Throughout our extensive study of powder compaction, various factors that affect the final relative density of powder mixtures under various conditions were examined.The MPFEM results have revealed several critical insights into the behavior of these mixtures.Notably, as pressure increased, there was a consistent rise in relative density.Fine powders exhibited a higher density compared to their coarser counterparts.As the friction decreases, the density under load increases, and the spring back reduces, leading to a higher final density.

Fig. 4 3 .
Fig. 4 Effect of RVE size on the relative density

Fig. 5
Fig.5The pressure-density relationship as predicted by MPFEM at the loading stage

Fig. 6
Fig. 6 Relative density comparison within EXP and MPFEM at 1600 MPa at the unloading stage

Table 1 .
Material parameters of Fe-Si-Al-P alloy powder.