The stochastic response of fatigue crack growth in scaled components

In fatigue testing the number of cycles and crack length to failure are recognised to be stochastic quantities necessitating repetitive testing to draw out statistically meaningful measures. The origin of this phenomena is uncertainty in the detailed microstructure, where (from a microstructural perspective) there exists no two identical test pieces. The situation is further compounded under scaled testing of polycrystalline specimens due to a form of coarsening with scaled samples containing less grains than the full-size test piece with samples drawn from the same material stock. Although microstructural uncertainly impedes predictability, implying the need for a probabilistic framework, it is nonetheless important to capture deterministic aspects that feature in any stochastic model. The theory advanced in this paper builds on the finite similitude scaling theory and the two-experiment first-order rule that features length as an invariant. Length invariance is shown to be critically important for fatigue as it caters for the many aspects of the microstructure that are not scaled under scaled testing. A diffusional stochastic model is introduced in the paper that is constrained by the first-order finite similitude rule. The approach is shown to enforce necessary determinism despite the uncertainties present and unlike many of the rules that are currently applied in fatigue analysis, it is applicable over a large range of length scales. Popular growth laws for long cracks are examined under the new framework, which transpires to be remarkably straightforward to apply. • A new stochastic diffusion model is established for the analysis of fatigue. • Scaling requirements for stochastic diffusion equations are established. • The first-order finite-similitude scaling theory is shown applicable to stochastic fatigue. • Paris empirical law is confirmed to relate both deterministic and averaged data.