New Die-Compaction Equations for Powders as a Result of Known Equations Correction: Part 2—Modernization of M Yu Balshin’s Equations

Based on the generalization of M. Yu. Balshin’s well-known equations in the framework of a discrete model of powder compaction process (PCP), two new die-compaction equations for powders have been derived that show the dependence of the compaction pressure p on the relative density ρ of the powder sample. The first equation, p=w(1−ρ0)(n−m)·(ρ−ρ0)n(1−ρ)m, contains, in addition to the initial density ρ0 of the powder in die, three constant parameters—w, n and m. The second equation in the form p=H1−ρ0b−c·ρ−ρ0b1−ρ0c−aρ−ρ0c also takes into account the initial density of the powder and contains four constant parameters H, a, b, and c. The values of the constant parameters in both equations are determined by fitting the theoretical curve according to these equations to the experimental powder compaction curve. The adequacy of the PCP description with these equations has been verified by approximating experimental data on the compaction of various powders, including usual metal powders such as iron, copper, and nickel, highly plastic powders such as tin and lead, a mixture of plastic powder (Ni) with non-plastic powder (Al2O3), nickel-plated alumina powder, as well as powder of a brittle compound, in particular titanium carbide TiC. The proposed equations make it possible to describe PCP with high accuracy, at which the coefficient of determination R2 reaches values from 0.9900 to 0.9999. The four-constant equation provides a very accurate description of PCP from start to finish when the density of the samples stops increasing once the pressure increases to an extremely high level, despite the presence of porosity.