Apparent Hall-Petch effects in polycrystalline lamellar TiAl

Abstract

The mechanisms responsible for the extremely high Hall-Petch slope of K_y ≈ 5MPa m^½, for grain sizes d in the range 3000–200 μm, in fully lamellar (FL) and nearly lamellar (NL) TiAl microstructures are explored. It is shown that the constraints exerted by the neighbouring crystals, particularly those oriented in the hard mode, are the dominant factors controlling the yield stress. At large grain sizes (and small sample dimensions), these constraints are largely relaxed, because of free boundary conditions, thereby severely underestimating the true yield stress. Numerically computed flow stress curves illustrate that the sample gauge diameter D, for approximating bulk behaviour, for Hall-Petch measurements, scales with the hard-mode (g°₂) against soft-mode (g°₁) critical resolved shear stress anisotropy. Thus, while D/d ≈ 10 meets the bulk requirement for nearly isotropic, fully γ alloys, the minimum number of crystals required to approach bulk behaviour is 400 or more (i.e. D/d ≥ 20) for nominal plastic anisotropy (g°₂/(g°₁ ≈ 12) of the (FL, NL) lamellar alloys. This requirement is expected to be further exacerbated as the ratio increases, with the limit (g°₂/(g°₁ → ∞, as (g°₂ → ∞) of having only one slip system as input. A similar high Hall-Petch slope (K_y ≈ 4·42 MPa m^½) is derived from computed results by simply considering the geometrical factors pertaining to material constraint for D/d in the range 1–20 for lamellar microstructures. Thus, increasing constrained volume fraction, as the grain size decreases, is concomitant with the increase in yield stress, and the ensuing Hall-Petch slope.