13th International Conference on Fracture June 16–21, 2013, Beijing, China -5- Figure 3. Cross section of a crack in primary alpha. 4. Short crack model Several models for stage I-cracks have been presented, which consider the abnormal propagation behaviour of short cracks described above. Very promising one-dimensional analytical models for stage I-crack growth have been developed by Taira [8] and Navarro and de los Rios [9]. These models account for the barrier effect of grain and phase boundaries by blocking the extension of the plastic zone at the next grain boundary. Only if the stress on a dislocation source behind the grain boundary reaches a critical value, slip is transferred over the barrier resulting in an increased crack propagation rate. In order to take real microstructures with arbitrary grain shapes and orientations into account the analytical approaches have been extended to a two-dimensional model, which has been applied successfully to simulate crack growth in a duplex stainless steel [6,10]. This model is taken as the basis for the simulation of short crack growth in forged Ti6Al4V. The crack problem is solved numerically by means of a discretisation with dislocation dipole boundary elements (see Fig. 4). Such a dipole element consists of a negative and a positive dislocation and represents a constant relative displacement over the element. Within a crack element two dipoles with a Burgers vector normal and tangential to the crack are combined to allow for an opening and a tangential displacement. On a slip band only a slide deformation is possible, so the respective boundary elements only consist of a dipole representing a tangential displacement. To monitor the stress state behind the grain boundary sensor elements with a constant length are positioned at the intersection point between active slip band and grain boundary. The sensor orientations match those of possible slip planes so that it is possible to calculate, in which direction a slip band is activated. As grain boundaries act as obstacles to plastic deformation, the extension of the plastic zone is blocked by the next grain boundary. Thus, the crack propagation rate, which is controlled by the plastic deformation at the crack tip, decreases. Only if a critical stress intensity is reached on a dislocation source beyond the barrier in the neighbouring grain, the respective slip band is activated and the plastic zone extends into the new grain. The dislocation pile up in front of the grain
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