13th International Conference on Fracture June 16–21, 2013, Beijing, China -4- where αg is an irreversibility coefficient and gr d is a length scale of the current grain calculated as: = , n i i gr st nd i d D D (6) in which ωi is the disorientation factor for grain i, Dst is related to the length of the band considered and Di nd relates to the length of all n intersecting bands in adjacent grains. The values of Dst and Di nd are calculated for each averaging band as the square root of the area of the band. The disorientation factor is computed as o = 1 20 dis (7) Here, θdis is the disorientation angle between two grains, and the Macaulay brackets satisfy that <a>=a if >0 a , <a>=0 if 0a . Thus, = 1 when there is no disorientation (i.e., the grain and the neighbor have exactly the same orientation and should be a single grain) and = 0 if the disorientation is 20° or larger. The disorientation factor for randomly oriented grains results in non-zero values for fewer than 10% of the grain boundaries. The MSC crack growth rate is assumed to be controlled by the mechanical irreversibility of dislocations emitted from the crack tip and proportional to the crack tip displacement range, i.e., th = FIP CTD , msc da A dN (8) where (A ~2) is a scaling constant that depends on microstructure attributes, ∆CTDth is a threshold that has a value close to the magnitude of the Burgers vector. The factor ф=0.077 measures the mechanical irreversibility at the crack tip process zone and depends on environment. The number of cycles to extend the crack front through the ith grain along slip system α in the MSC regime, | iG MSC N , is determined by integrating Equation (8) with respect to the crack length. The crack growth rate Figure 1: Schematic representation of elements, bands and grains in which FIPs are averaged to estimate transgranular fatigue crack growth. The implementation in a FEM model with unstructured, voxellated meshing is shown, with bands color coded and numbered for a single spherical grain.
RkJQdWJsaXNoZXIy MjM0NDE=