Statistical approach to flow stress and generalized Hall-Petch law for equilibrium polycrystalline materials

A theory of flow stress, including the yield strength is proposed for the class of PC materials with equilibrium defect structure (EDS), which is established in the PC material after series of $N_0$ similar treatments of severe plastic deformation at fixed temperature T and characterized by stabilized scalar dislocation density (SDD) and average grain size d. We calculate both the stationary SDD $\rho(b,d,T)$ and suggest a way to calculate $\varepsilon$-evolution of an equilibrium SDD $\rho_{\varepsilon}$ in PC sample under quasy-static loading depending on the average size $d$ of a grain in the range of $10^{-8}- 10^{-2}$ m, on grain boundaries orientation. The analytical dependence is realized within a disclination-dislocation mechanism in approximation of single dislocation ensemble for given phase and T. It is based on a statistical model of Boltzmann-like distribution (smoothly dependent on a strain ${\varepsilon}$) for discrete energy spectrum in each grain of a single-mode one-phase PC material with respect to quasi-stationary levels under plastic loading with the highest level equal to the energy of dislocation with maximal length. The difference of equilibrium SDD, $\rho_{\varepsilon}- \rho$, leads to a flow stress from the Taylor strain hardening mechanism containing (for $\varepsilon$ = $0.002$) the normal and anomalous Hall-Petch relations for coarse and nanocrystalline grains, respectively, and gains a maximum at floe stress values for an extreme size containing $d_0$ of order $10^{-8}- 10^{-7}$ m. The maximum undergoes a shift to the region of larger grains for decreasing temperatures, revealing temperature-dimension effect. Coincidence is well established between the theoretical and experimental data on ${\sigma}_y$ for the materials with EDS with BCC (${\alpha}$-Fe), FCC (Cu, Al, Ni) and HCP (${\alpha}$-Ti, Zr) crystal lattices with closely packed grains at T=300K.