Yielding transitions and grain-size effects in dislocation theory

The statistical-thermodynamic dislocation theory developed in previous papers is used here in an analysis of yielding transitions and grain-size effects in polycrystalline solids. Calculations are based on the 1995 experimental results of Meyers, Andrade, and Chokshi [Metall. Mater. Trans. A 26, 2881 (1995)] for polycrystalline copper under strain-hardening conditions. The main assertion is that the well-known Hall-Petch effects are caused by enhanced strengths of dislocation sources at the edges of grains instead of the commonly assumed resistance to dislocation flow across grain boundaries. The theory describes rapid transitions between elastic and plastic deformation at yield points; thus it can be used to predict grain-size dependence of both yield stresses and flow stresses.

Figure 1

Theoretical stress-strain curves for polycrystalline copper at the small strain rate $\dot{\epsilon }={10}^{-3}{\text{s}}^{-1}$, for grain diameters $d=9.5,25,117$, and $315\mu \text{m}$, shown from top to bottom. The experimental points are taken from [8].

Figure 2

Theoretical stress-strain curves for polycrystalline copper at the large strain rate $\dot{\epsilon }=3\times {10}^3{\text{s}}^{-1}$, for grain diameters $d=9.5,25,117$, and $315\mu \text{m}$, shown from top to bottom. The experimental points are taken from [8].

Figure 3

Hall-Petch plots for the conversion factor ${\kappa }_1$ as a function of grain diameter $d$. The lower solid curve is for the small strain rate, $\dot{\epsilon }={10}^{-3}{\text{s}}^{-1}$, and the upper dashed curve is for $\dot{\epsilon }=3\times {10}^3{\text{s}}^{-1}$. These curves are fit by the formulas shown in Eq. (3.1).

Figure 4

Stress-strain curves for prehardened isothermal samples. The grain diameters are $d=9.5,25,117$, and $315\mu \text{m}$ from top to bottom.

Figure 5

Enlarged graphs of the elastic-to-plastic transitions shown in Fig. 4.

Figure 6

Hall-Petch plots for the yield stresses shown in Figs. 4 and 5 (upper solid curve), and for the flow stresses at $\epsilon =0.1$ on the hardening curves shown in Fig. 1 (lower dashed curve).

Figure 7

Stress-strain curves analogous to those shown in Fig. 4, but with thermal softening and an initial perturbation added to induce shear-banding failure. Going from left to right, these failures occur for grain diameters $d=9.5,25,117$, and $315\mu \text{m}$.

Figure 8

Stress-strain curves analogous to those shown in Fig. 7 for grain diameters $d=9.5$ and $315\mu \text{m}$, but with initial values of $\tilde{\chi }$ slightly reduced to simulate mild annealing between the slow prehardening and rapid reloading stages.