6.2 Effect of Dislocation Sources on Flow Stress Behaviour
6.2.1 Number of sources
The effect of dislocation source concentration within a grain was investigated by varying the num- ber of grain-boundary sources on the domain wall. Fig. 6.2.1 shows slip band formation after 3% applied strain for the source spacings ranging between 200nm and 1 µm. Regions of plastic defor- mation are clearly defined and seperated by interstitial elastic regions with no dislocation activity which have F12p values of 0.
Figure 6.2.1: Plasticity maps (showing the shear component of the plastic deformation gradient tensor) for simulations with a differing number of sources placed across the left-hand grain wall (λjog = 25nm)
The average magnitude of the F12p values within the slip bands increases as the number of sources is reduced. This is expected as the same strain is applied to all grains, but the number of slip bands available to share the plastic deformation is reduced. In the simulation with only 5 available sources the plastic deformation in each band is the largest, with the individual sources generating at the highest observed rate and the 5 slip bands sharing the plastic shear evenly. As the number of sources is increased the plastic shear is spread over more bands and the deformation sharing becomes less even. In the 10 source simulation the slip bands near the top and bottom of the domain contain more slip activity than those in the middle.
Fig. 6.2.2 shows the variance of the local shear stress (π12∞+ πd
12) along the left-hand domain
boundary. It can be seen that the stress undulates with changing height such that the minima cor- respond with the locations of the boundary sources, since the plastic shear reduces the elastic strain in this vicinity, dropping the local stress. The maxima fall within the elastic domains between the slip bands. Both the minima and maxima follow a trend that they are lowest towards the middle of the grain, aligning precisely with the regions of low plasticity in Fig. 6.2.1. The minima vary over a small stress range of 110-140 MPa, occurring just below the threshold at which a dislocation will accelerate through the jog-drag and cause yield (see Section 4.2.2). The maxima, however, in the regions without the mitigating effect of plasticity, vary over a much wider stress range and the absolute stress values increase as source number is reduced. For the 10 and 5 source case, where there is wide enough spacing to resolve stress changes between slip bands, it can be seen that the stress is highest directly adjacent to the slip band and forms a local minima in the centre of the elastic region.
Figure 6.2.2: Shear stress fields for a verticle cross section of the domain, taken level with the dislocation sources, for increasing applied strain, when different numbers of sources are present.
The accompanying flow stress behaviour and source generation count for these simulations is shown in Fig. 6.2.3. The simulations all yield at the same point, but softening post-yield is shown to increase with source concentration. When fewer sources are present the domain must still ac- commodate the same amount of deformation, associated with the fixed strain rate, but with less plasticity available due to reduced source activity this leads to an increase in the strain hardening rate. The 23 source simulation showed the most ideally plastic behaviour and so was used as the control setup for all simulations in later sections. It has been shown in Fig. 6.2.1 that the amount
of sources present does not necessarily equate to the total active slip bands at a given time, but the use of 23 evenly spaced sources allows the maximum opportunity for slip to occur during deforma- tion without sources occupying adjacent elements and negatively impacting source operation. This choice of maximum source density is justified by the fact that the 2D model deals with a single slip system, where in reality further out-of-plane slip systems would offer more opportunity for plastic flow.
The total dislocation generation within the grain is shown in Fig. 6.2.4 where it can be seen that the greater dislocation source activity corresponds with softer flow stress curve. It is also evi- dent that the amount of dislocation generation does not scale linearly with the number of sources available. For the 5 source simulation the total generation after 3% applied strain (620 events) is approximately half of the generation for the simulation with 23 sources (1248 events). This result is consistent with the higher flow stress for the 5 source grain, but implies that the individual sources are generating at a greater rate than those in the 23 source grain. Averaging to find the number of generations by an individual source in each of these simulations gives 124 for the 5 source case, 87 for the 10 source, 69 for the 15 source, 59 for the 20 source and 54 for the 23 source. This implies that when fewer dislocation sources are available within a grain, those sources will generate more often than in a grain with higher source density.
The data in Fig. 6.2.2 is consistent with this observation, with the average stress at the dislo- cation sources being lower for the simulations with higher source density: 137.9 MPa for the 23 source case and 145.9 MPa for the 5 source case, at 0.5% strain.
Figure 6.2.3: Flow stress response for the a grain with a differing number of sources placed across the left-hand grain wall (λjog= 25nm).
Figure 6.2.4: Number of dislocation generation events for the a grain with a differing number of sources placed across the left-hand grain wall (λjog= 25nm).