Abstract
Abstract
Whether a crystal fails by cleavage or by dislocation emission is decided by a competition between two load thresholds, and that competition is classically evaluated at an atomically sharp crack. Real flaws have finite tip radii, so the force balance that decides the outcome is itself a function of notch acuity. The present work resolves that competition at a blunt notch in closed form. The crack is replaced by a traction-free elliptic void, and the elastic image force on a straight mixed dislocation near it is obtained exactly: the three infinite series that the perturbation construction generates for the edge component are summed in closed form, giving a finite rational expression that is free of truncation error for every exterior dislocation position and every elliptic aspect ratio and is evaluated at O(1) cost, while the screw component follows independently in closed form. Combining the resulting mixed image force with the exact remote-tension field of the same void, and retaining a core-cutoff emission criterion together with a blunt-notch decohesion criterion, the emission threshold K Id (ρ) and the cleavage threshold K Ic (ρ) are computed for six crystals (Cu, Na, LiF, Si, Be, Fe). Their crossing defines a critical blunting radius ρ ⋆ , expressed in units of the Burgers-vector magnitude b, at which the competition reverses. Iron (ρ ⋆ = 2.03 b) cleaves below its crossing and emits above it. The formal copper (0.68 b) and sodium (0.98 b) crossings lie below the nominal continuum-resolution scale ρ ∼ b, so both are predicted ductile for ρ ≳ b, while LiF, Si and Be remain brittle across the whole range ρ/b ∈ [0.05, 100]. The results assemble a closed-form notch-acuity phase diagram in (material, ρ/b) space, within which the classical sharp-crack classification is recovered as one limiting (atomically sharp) edge and the emission–cleavage competition is a joint property of the crystal and the acuity of the flaw it contains.