br The connection between the yield point of a

The connection between the yield point of a material and the surface tension melk inhibitor of its particles As follows from Fig. 1, the surface tension coefficient rapidly decreases with a decrease in the particle size R/δ of the material, for particles with sizes less than 20δ (about 20nm). Its values for most metals with larger-sized structural elements are about σ0=1–2J/m2. The elastic energy stored in a metal through uniaxial tension up to the yield point is about 90kJ/m3 for steels. Let us compare this energy with the bulk density of the surface tension energy ρσ. Let us assume, to simplify the estimates, that the particles have a spherical form, then the energy is going to equal ρσ=3σ0/R. Already for a typical crystallite size of about 2R=120 – 200µm, the ρσ value becomes equal to the maximum elastic strain energy. Consequently, the surface energy needs to be taken into account when constructing plastic deformation models for ultrafine materials. We can speculate that it is this quantity that determines the plastic deformation mechanisms in this case. To test this hypothesis, let us use the Hall–Petch law [16,7] which was discovered empirically. To explain this law, Petch suggested examining the formation and the growth of a microcrack in a grain with a characteristic dimension d (this diameter d = 2R for our assumptions). Currently, the law itself, and the deviations from it for nanometer-sized crystallites are interpreted solely on the basis of the dislocation mechanism of plastic deformation. According to the Hall–Petch law, the yield point τ depends on the grain dimension d=2R of a polycrystalline material and is expressed as where τ0 is some friction stress required for the dislocation glide; K is the material constant, often referred to as the Hall–Petch coefficient (both values are assessed from the experimental data). Let us consider the beginning of the plastic straining process. For most metals, the strain grows essentially without any additional increase in stress. In the simplest case of uniaxial tension, this stress can be taken as constant. Therefore, the plastic strain energy should be proportional to the yield point. Let us assume that plastic strains lead to a growth of the internal crystallite surface without substantially changing their volume; then, when the plastic flow starts, the mechanical stresses should compensate both the elastic strains of the crystallites and the surface tension forces. It follows, then, that the yield point is linearly related to the bulk density of the surface energy. The behavior of this relationship makes it possible to approximate the Hall–Petch law using the dependence of the specific surface energy of the material on the particle size. With the exact solution of the Gibbs–Tolman–König–Buff Eq. (4), we can obtain an expression for the bulk density of the surface tension energy in a material containing spherical particles. The obtained relationship is expressed in the following way:
The normalized bulk density of the surface energy /σ0 is shown in Fig. 3 versus the value. The curve has several quasilinear parts. The figure shows linear approximations of the curve parts corresponding to the Hall–Petch dependence. We should note that for grain sizes (2R) ranging from 4δ to 2δ, there is a strong change in the approximating dependence parameters. The obtained dependence can be approximated using the Hall–Petch law; however, when the grain size is less than 2δ, the bulk density of the surface energy reaches its limit value (about 0.913 σ0/δ). The plots shown in Fig. 3 give reason for interpreting the plasticity as a variation in the surface energy. In other words, if the material started flowing, it means that the allowed value of the surface energy has been exceeded. The estimates we have presented show that the exact solutions of the Gibbs–Tolman–König–Buff equation allow to calculate the melting point with an accuracy that is acceptable for practical use. The curve of the normalized bulk density of the surface tension versus the particle diameter is in agreement with the Hall–Petch law in the d >4δ range and differs from it for lower diameter values, which is consistent with the experimental data. In the second case the exponent at the particle size 2R (that was equal to –1/2) changes in the Hall–Petch law. For many metals this size is just (4–8) δ.