The Hall–Petch Relation and High-Temperature Subgrains

Abstract

Although originally proposed for high-angle boundaries, a relation of the Hall–Petch type has been applied by a number of authors to low-angle boundaries produced by high-temperature deformation. Investigations published to date include data on commercially pure Al and Fe and on silicon steel; in these studies room-temperature testing showed that the following relation applies: S = S ₀′ + K′d^−1/2. Here S represents the mechanical property being measured, d is the mean sub grain size, and S ₀′ and K′ are empirical constants. However, this relation has previously been applied only to small ranges of sub grain size, and neither S ₀′ nor K′ has any fundamental significance. A rationalization is proposed in which the published observations are reinterpreted in terms of sub-boundary strengths that depend on sub-boundary misorientation and perfection and therefore, indirectly, on sub grain size. According to this view, the relatively perfect sub-boundaries formed at creep strain rates have low strengths; by contrast, the relatively tangled sub-boundaries formed at hot-working strain rate have strengths approaching those of grain boundaries. When experimental results for sub grains covering a wide range of sizes are considered, an empirical correlation of the following type is obtained: σ = σ₀ + K ₄ d^−m. Here σ₀ is the strength of a single crystal, K ₄ is a constant, and m is a constant ≃ 3/4. Its anomalously high value is attributed to the variation of the Hall–Petch “constant” K ₁ with (1/d), as given approximately by K ₁ ∝ (1/d)^p, where p = m − 1/2 ≃ 1/4. The correlation is attributed to an increase in both the sub-boundary misorientation and the density of redundant dislocations in the sub-boundaries with increasing (1/d).