Emergence of fatigue crack-initiation life statistics from dislocation dynamics: the surviving open core of Mott's 1955 theory of fracture and fatigue

This task targets the open core of N. F. Mott's 1955 dislocation-based theory of fracture and fatigue (J. Phys. Soc. Jpn. 10, 650). Later literature closed most of the 1955 gaps: Stroh's published pile-up analyses supplied the pending numerical constants; direct dislocation-velocity measurements (LiF, Si-Fe) replaced the near-sonic 'dynamic hypothesis' with drag-limited, strongly stress-dependent motion; detailed work on polycrystalline zinc showed that the single-cleavage-plane pile-up picture must be extended there (multiple cleavage systems and twinning dominate); stress-assisted vacancy processes were established for creep cavitation and quantified for fatigue surface roughening, while the fatigue crack nucleus itself is a dislocation-structure intrusion; and the requested flow-stress temperature-dependence and grain-size/fracture-stress experiments were carried out. What survives is the item the source itself identified as central to fatigue prediction: how fatigue-life distributions and rate effects emerge from underlying dislocation dynamics. Current models either give deterministic accumulation laws (Coffin-Manson-type mean-life and Petch-type grain-size relations) or fit the scatter phenomenologically (Weibull statistics with calibrated parameters). The problem posed here is this surviving open core: derive the probability distribution of cycles to fatigue crack initiation - including its scatter and its dependence on amplitude, loading rate/frequency, temperature, and microstructure - from dislocation-level dynamics, and validate it against measured initiation statistics.

Background

In a crystalline metal, plastic deformation is carried by dislocations - line defects whose motion produces slip on crystallographic planes; the elementary step of slip is quantified by the Burgers vector $b$, a lattice-translation vector of interatomic length. A Frank-Read source is a pinned dislocation segment that, under a shear stress exceeding $Gb/l$ (with $G$ the shear modulus and $l$ the segment length), repeatedly emits dislocations on one slip plane. Dislocations of like sign blocked by an obstacle (a grain boundary or a sessile, i.e. immobile, dislocation) accumulate into a pile-up, a wedge-like group that concentrates stress at its head. Pre-1950 fracture theory rested on Griffith's energy criterion - a pre-existing crack propagates when the elastic energy released exceeds the surface energy spent, giving a propagation stress of order $G\sqrt{b/r}$ for a crack of length $r$ (Griffith 1921) - but it offered no microscopic origin for cracks in ductile metals containing no macroscopic pre-existing flaws. Mott (1955) proposed a unified dislocation-based picture of fracture and fatigue initiation. Using the pile-up stress law attributed to a then-unpublished analysis by Stroh, $\sigma_{\text{local}}(r)=\sigma A\sqrt{L/r}$, where $\sigma$ is the applied shear stress, $L$ the slip-line length, and $A$ a numerical factor depending on the slip-plane orientation angle $\theta$ (largest near $\theta=45^\circ$), and comparing it with the Griffith-type crack-propagation stress, Mott obtained the scale-free initiation criterion $\sigma/G>C\sqrt{b/L}$ for annealed metal and $\sigma/G>l/L$ for cold-worked (work-hardened) metal. For fatigue (failure under repeated cyclic loading well below the static strength), he posited that a source operates only during short-lived upward fluctuations of the local internal stress $\sigma_i$ - the fluctuating stress field produced by other dislocations - so the effective threshold is $\sigma_1=Gb/l+\sigma_i$; each resulting slip line can nucleate a micron-scale cracklet at its end, and rare stochastic events cumulatively crack a grain. The paper explicitly acknowledged six gaps: (i) the validity of the 'dynamic hypothesis' that dislocations can move at nearly the speed of sound; (ii) the unspecified numerical constants $A$, $A'$, $C$ and their orientation dependence; (iii) the unsettled applicability to materials such as zinc where fine slip dominates and slip lines are not seen at small deformations; (iv) the undetermined microscopic nature and statistics of the fluctuations $\sigma_i$; (v) the unresolved role of vacancy formation and coagulation (vacancies are empty lattice sites; their clustering was a proposed crack nucleus) in coupling to crack nucleation; and (vi) needed experiments on the temperature dependence of the flow stress (the stress needed to sustain plastic flow) in work-hardened metals and on the link between slip-line lengths, grain size, and fracture stresses.

Most of this program was subsequently completed or superseded. In the modern picture, fatigue crack initiation in ductile metals occurs at persistent slip bands (PSBs) - ladder-like dislocation structures that localize cyclic strain and produce persistent slip markings on the surface, consisting of extrusions (ridges) and intrusions (grooves); the deep intrusions act as crack-like defects from which stage I cracks (the early, shear-mode growth of a microcrack along slip planes, typically of order the grain size) develop. Dislocation motion is now known from direct measurement to be drag-limited (velocity set by dissipative drag, not inertia) and strongly stress-dependent, far below sound speed at ordinary stresses, and dislocation activity is intermittent and avalanche-like. Long-range internal stresses in cyclic and monotonic dislocation structures have been measured and modeled. Quantitative initiation-life prediction, however, remains either deterministic dislocation-accumulation modeling - which reproduces mean Coffin-Manson-type amplitude laws (an empirical low-cycle-fatigue law relating plastic strain amplitude to cycles-to-failure) and Petch-type grain-size laws - or phenomenological statistics, in which the scatter of life is fitted by a Weibull distribution (a flexible extreme-value-type distribution with adjustable shape and scale) rather than derived. What was never achieved, and what the source left open, is the derivation of the statistics itself: the probability distribution of cycles-to-initiation, its scatter, and its rate effects, as an emergent consequence of dislocation dynamics.

Problem Statement

Mott (1955) left as an explicit further-work item: how fatigue-life distributions and rate effects emerge from underlying dislocation dynamics - in his framing, the statistics of the internal-stress fluctuations that trigger rare slip-line nucleation events during cyclic loading. Restated on the corrected mechanistic basis established by later literature - initiation at persistent slip bands (the ladder-like dislocation structures that localize cyclic slip) and their extrusion/intrusion surface markings (the ridges and grooves those bands leave on the surface), rather than Mott's random-fluctuation mechanism, which was superseded - the surviving open problem is:

Derive, from dislocation-level dynamics - source-operation thresholds, internal-stress statistics, cyclic dislocation-structure (persistent slip band) evolution, and point-defect production and migration - the probability distribution of the number of cycles required to initiate a stage I fatigue crack (an early shear-mode microcrack growing along a slip plane) of order the grain size in a crystalline metal of specified microstructure, together with the dependence of that distribution on cyclic stress/strain amplitude, loading rate/frequency, temperature, and microstructural lengths (grain size, slip-band spacing); and validate the derived statistics against measured initiation statistics.

The sought result is the probability distribution of cycles to initiation together with its scatter - the shape and width of the life distribution, not only its mean - its dependence on cyclic stress/strain amplitude, loading rate/frequency, temperature, and microstructural lengths, and its validation against measured initiation statistics. The statistical parameters of the life distribution may not be fitted to the target life data: they must follow from the dislocation dynamics, with any calibration done on independently measured dislocation-scale quantities. The derivation concerns crystalline metals generally, with face-centred cubic metals as the paradigm case.

References

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