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Equations of plasticity of a porous body proposed by different authors and obtained under the condition that the yield surface of a porous body has the shape of an ellipsoid of revolution are considered in this paper. Such equations have two independent parameters which are the functions of relative density. Various theoretical dependences of these parameters on the relative density and, as a result, various equations for describing the die-compaction of powders are presented. It is shown that the correction of two density-dependent parameters, taking into account the initial density, makes it possible to significantly increase the accuracy of approximation of experimental data on the powder compaction process (PCP) of various powders. Among the considered “continuum” equations of powder die-compaction, the PCP to a density of 0.95 is the most accurately described by the equation in which the corrected Skorokhod’s theoretical density functions are used and which contains one constant as a result. Another equation which contains four constants allows one to accurately (R2 > 0.9990–0.9999) describe the PCP to a density of >0.95. This equation is obtained by replacing one of two independent parameters in the traditional continuum equation with the lateral pressure coefficient followed by substituting, instead of those parameters, their dependencies on the density in the form of power function. The adequacy of the PCP description by this equation was verified by approximating experimental data on the die-compaction of iron powders to a relative density greater than 0.95, as well as highly plastic powders with a final density of ~1.0.

The well-known equations for the powder compaction process (PCP) in a rigid die published from the beginning of the last century until today were considered in this review. Most of the considered equations are converted into the dependences of densification pressure on the powder’s relative density. The equations were analyzed and their ability to describe PCP was assessed by defining the coefficient of determination when approximating experimental data on the compaction of various powders. It was shown that most of the equations contain two constants the values of which are determined by fitting the mathematical dependence to the experimental curve. Such equations are able to describe PCP with high accuracy for the compaction of powders up to a relative density of 0.9–0.95. It was also shown that different equations can describe PCP in the density range from the initial density to 0.9 with the same high accuracy, but when the process of compaction is extrapolated to higher values of density, the curves diverge. This indicates the importance of equations that can unambiguously describe PCP to a relative density equal to or close to 1.0. For an adequate description of PCP for relative density greater than 0.95, equations containing three or four constants have proven useful.

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.