FRH CRACKING MODEL
Material grains of different sizes, phases and mechanical properties are assumed randomly arranged along the specimen minimum circumference (SMC). The specimen surface roughness resembles multi coplanar micro cracks with different sizes existing at the center of each grain along the SMC. The front of a crack is assumed having the geometry of a circular arc 40 - 46. The crack depth is assumed equal to half of the surface crack length as manipulated in some relevant works found in the literature 42, 44. The crack length determines the corresponding crack area. A specimen with an initial configuration as described is cyclically loaded.
Consequently, some grains experience plastic deformation. The tip of a crack in an elastically deformed grain has a plastically deformed zone given by the crack size, the applied stresses and the grain yield stresses. Thus, the stress-strain regime existing at the two opposing tips of two adjacent cracks in two neighboring grains enables the estimation of the extents of the plastic zones at both tips. The rate of advance of a crack tip is estimated by making use of the surface crack length, the extents of the monotonic and cyclic plastic zones ahead of the tip and the average shear strain within the plastically deformed zones.
After N load cycles, (1) some surface cracks become longer, (2) the total cracked area increases (3) the area of the supporting ligament decreases, (4) the tip of a crack has the possibility of either advancing or stop propagating and (5) the two opposing tips of some adjacent cracks approach and possibly stop at a distance or merge to form one crack. A linear coordinate system is devised along the SMC to continuously trace the location of the tips of the existing cracks.
As a result of that cracking activity the following two scenarios are possible, i.e. (1) when the applied stress is greater than the material’s fatigue limit \(\sigma_{F}\), the supporting ligament becomes unable to withstand the applied load after the application of a finite N load cycles and, hence, the specimen breaks, (2) In case of the applied stress be less than \(\sigma_{F}\), all existing cracks become non-propagating at some load cycles and, thus, the specimen remains unbroken after the application of a relatively large N load cycles. The details of the original model can be found elsewhere52.
The model was applied to predict the fatigue lifetime of un-notched round specimens made of a ferrite-pearlite 0.4C-70/30 carbon steel in push-pull axial loading. This cracking simulation enabled an assessment of the specimen’s fatigue lifetime and the endurance \(S/N\) curve of the material with its fatigue limit could, thus, be assessed. Figure 1 presents the prediction of that simulation in the HCF regime compared with published experimental results 3. Further, the model recognized the effect of surface roughness, specimen size and mean stress on fatigue lifetimes 52.