Deformation Interactions in Polycrystalline Alloys

Abstract

Specificities of the theory of deformation interactions in polycrystalline alloys are discussed in this work. It is shown that, in the continuous approximation, these interactions are completely determined by the dilatation components of elastic and inelastic strain fields created by the systems of grain boundaries and the system of impurity atoms. Formulas and estimates for various types of deformation interactions are presented. Problems concerning the effect of the boundaries on the lattice parameters, the concentration expansion of the lattices of inhomogeneous alloys, and the interaction of grain boundaries are discussed.

APPENDIX

EXCESS VOLUME OF BOUNDARIES

Let us consider the excess volume density distribution ${{\varepsilon }_{{0V}}}({\mathbf{r}})$ for a single unrelaxed plane special boundary with the tilt {n10}, $\left| n \right| \geqslant 2$ (Fig. 1а) [13]. The boundary connects crystallites K₁ and K₂ along the planes A₁ and A₂. In Fig. 1b, the shaded square abcd shows the volume occupied by an atom in an ideal lattice. Therefore, the broken line ${{y}_{1}}(x)$ is the upper boundary of crystallite K₁ and the line ${{y}_{2}}(x)$ is the lower boundary of crystallite K₂ (Fig. 1c). Hence, the distribution of excess volume per 1 m² of the boundary is determined by the difference

$$\Delta V(x) = {{y}_{2}}(x) - {{y}_{1}}(x).$$

(55)

From definition (16), we find

$${{\varepsilon }_{{0V}}}(x) = {{\Delta V(x)} \mathord{\left/ {\vphantom {{\Delta V(x)} H}} \right. \kern-0em} H}.$$

(56)

Here, H is the width of the zone of irreversible volume changes. The form of functions ${{y}_{1}}(x)$ and ${{y}_{2}}(x)$ on the length of the period l for boundaries with an inclination angle θ and a relative displacement of crystallites by an arbitrary vector ${\mathbf{d}} = \{ {{x}_{0}},{{y}_{0}}\}$ (y₀ is the distance between planes A₁ and A₂, Fig. 1a) is determined by the formulas (see Figs. 1b and 1c):

$$\begin{gathered} {{y}_{2}}(x) = h + {{y}_{1}}({{x}_{0}} - x),\,\,{{h}_{1}} = 2{{m}_{1}} = 2{{m}_{2}} = a{\text{sin}}(\theta {\text{/}}2), \\ {{x}_{1}} = a{\text{/}}2\sin (\theta {\text{/}}2),\,\,\theta = 2\arctan (1{\text{/}}n), \\ \end{gathered} $$

(57)

$${{y}_{1}}(x) = \left\{ \begin{gathered} (x + {{x}_{1}})\cot (\theta {\text{/}}2),\,\,\,\,0 \leqslant x < {{x}_{1}}; \hfill \\ (l - {{x}_{1}} - x)\tan (\theta {\text{/}}2),\,\,\,\,{{x}_{1}} \leqslant x < l - {{x}_{1}}; \hfill \\ (x - l)\cot (\theta {\text{/}}2),\,\,\,\,l - {{x}_{1}} \leqslant x \leqslant l. \hfill \\ \end{gathered} \right\}$$

(58)

Here, ${{\varepsilon }_{0}} = h$ is the mean excess volume per 1 m² of the unrelaxed boundary (see Fig. 1c) [13].