A Deterministic Limit for High-Cycle Fatigue Nucleation: Deriving the Critical Slip Radius via Elastic Restitution

The prediction of high-cycle fatigue (HCF) in metallic components relies heavily on empirical Wöhler (S-N) curves and linear damage accumulation models such as Miner's Rule. While these statistical frameworks effectively estimate aggregate fatigue life based on cyclic probability, they fail to deterministically define the exact spatial boundary where localized dislocation pile-up triggers the irreversible nucleation of a micro-crack. This paper introduces a strict continuum framework for micro-mechanical crystal plasticity. By modeling the metallic lattice as a dynamic kinematic balance between the spatial capacity for elastic strain recovery (elastic restitution) and the localized rate of plastic dislocation slip, we derive a universal critical fatigue radius (Rfatigue). We demonstrate that fatigue crack initiation is not a probabilistic function of time or cycle count, but an exact deterministic limit where localized plastic ratcheting strictly overpowers the advective elastic-recovery capacity of the surrounding crystal lattice. We propose a blueprint for Active Pre-Crack Telemetry (APT) using Nonlinear Elastic Wave Spectroscopy (NEWS) to provide real-time structural prediction prior to the formation of a physical flaw.