4.3.2. Results for the *α*-Brass Sample

Similarly to the Ni sample, the same simulations have been performed for the A6 slip system in the crystal I of the *α*-Brass sample. In this case, the out-of-surface component of the Burgers vector is about 0.14 nm. The maximum slip step height is about 10.03 nm at *d* ≈ 2.44 μm and the slip step height at GB due to slip transmission and/or dislocation absorption is about 4.29 nm. Based on Equation (2), the number of dislocations in the pile-up is 72 and the number of transmitted and/or absorbed dislocations is 31. Therefore, the number of dislocations in the pile-up is equal to 41. The applied stress is 127.4 MPa following experimental results, and it corresponds to a resolved shear stress *τ* = 55.4 MPa. The thicknesses of Grains I and II are *H*<sup>I</sup> = 3.97 μm and *H*II = 5.05 μm as shown in Figure 1d.

The effects of free surfaces and critical force for the *α*-Brass sample are the same as for the Ni sample. However, the number of transmitted and/or absorbed dislocations is much larger in the *α*-Brass sample than in the Ni one, i.e., 31 instead of 6. Thus, the repulsive force from these interfacial dislocations is more important if their Burgers vector is considered as the same as the incoming dislocations (see the Simulation 1 in Figure 8b). All dislocations are moved far away from GB into the direction of free surface due to this repulsive force. The transmission mechanism observed in Figure 6a leads us to consider a different residual Burgers vector for interfacial dislocations. After verifying all the possible residual Burgers vectors with the 12 slip systems in the adjacent grain, it is found that the best solution is with A6 slip system in the adjacent grain while considering free surfaces and a reasonable critical force *Fc* = 0.011 N/m (*τ<sup>c</sup>* = 43 MPa) as shown in Figure 8b with Simulation 3. Furthermore, the theoretical analysis of the slip trace of A6 slip system on the upper surface of the adjacent grain agrees well with the experimental observation as shown in Figure 6a compared to other slip systems, such as B2 slip system with the maximum transmission factor [4] or B4 slip system with the maximum Schmid factor in the adjacent grain (see Table 2). As discussed in Section 3.3 for the *α*-Brass sample, the peak valley in the middle part of the measured curve is certainly caused by the crossing with a second non coplanar active slip system. In the present L–E–S two-dimensional anisotropic elastic theory, it is not possible to consider the effect of two dislocations whose dislocation lines are not parallel. Thus, this effect of peak valley part cannot be simulated within the present theory. However, it is important to well identify the part close to GB with *d* < 0.75 μm. Here, the suitable critical stress is *τ<sup>c</sup>* = 43 MPa. It is larger than the critical stress for the Ni sample, which may be due to alloying element hardening. Furthermore, the total resolved shear stresses for the last fixed dislocation are 997.3 MPa for Simulation 1, 258.9 MPa for Simulation 2 and 28.6 MPa for Simulation 3. Thus, Simulation 3 appears to be the most realistic one.

## 4.3.3. Discussion

Except the free surfaces effects, it is found that the Burgers vector of interfacial dislocations has an obvious influence on the dislocation distribution in the pile-up. It should be pointed out that, for the Ni sample, *b*GB has been tried with all the 12 residual Burgers vectors, but the results are never better than in the case with *b*GB = *b*In. Thus, the Burgers vector of interfacial dislocations can be regarded as the same as incoming dislocations in case of a low number (6) of transmitted and/or absorbed dislocations in the Ni sample. However, in case of a high number (31) of transmitted and/or absorbed dislocations in the *α*-Brass sample, one needs to consider the residual Burgers vector of interfacial dislocations.

In addition to the effects of interfacial dislocations, the critical force is also another important parameter to find the correct equilibrated dislocation distribution. Comparing the simulation 2 (*Fc* = 0 N/m) with the Simulation 3 (*Fc* = 0 N/m) presented in Figure 8 for both Ni and *α*-Brass samples, it is found that the critical force does not have an obvious effect on dislocations that are located at a distance less than 0.3 μm from GB. From Equation (1), it is found that this critical force does not have a uniform effect on all the dislocations within the pile-up. For example, when the total force on a dislocation is towards free surface for positions close to the free surface, the critical force will point towards the GB. However, the stress state is more complicated in regions close to the GB. The total force on a dislocation here can point towards the GB due to repulsive forces from other dislocations, attractive misorientation [41], compliant GB [41], attractive force from interfacial dislocations, and applied stress. On the contrary, it can point towards the opposite direction due to repulsive misorientation [41], rigid GB [41], repulsive force from interfacial dislocations, other dislocations, and free surfaces. Hence, the total force depends on many physical parameters. Thus, the critical force could either move the dislocations towards GB or to the opposite direction.

As mentioned in Section 3.3, the ex-situ AFM measurements were carried out after compression and elastic unloading; thus, for the present static dislocation pile-up simulation, the applied stress should be set to zero. However, in the present study, the dislocations in the pile-up are supposed to be unchanged during unloading, which means that the equilibrium positions of dislocations obtained at the onset of plastic loading are consistent before and after unloading process due to a large critical force in the crystal. The found critical forces *Fc* = 0.003 N/m for Ni and *Fc* = 0.011 N/m for *α*-Brass are supposed to be the original critical force before slip activation with a low dislocation density in the crystal.

Furthermore, the effects of incompatibility stresses [37], elastic anisotropy compared to elastic isotropy, number of interfacial dislocations, applied stress, and misorientation can be also investigated with the developed model [44]. As a summary, for the present samples, the main effects come from the external stress, heterogeneous and anisotropic elasticity, and the number of interfacial dislocations which should be equal to the one of the transmitted and/or absorbed dislocations.

In Figure 8b, it was also found that the measured slip step height decreases significantly after its maximum value along the slip direction. This tendency might be the signature of a double-ended dislocation pile-up due to a less penetrable layer than the free surface coming from surface defects produced by FIB polishing. The effect of these surface defects seems to be stronger for the *α*-Brass sample compared to the Ni sample. Hence, the present model which was designed to account for single slip within a single-ended dislocation pile-up configuration is more suitable for the study of the Ni sample than the *α*-brass sample. In general, the present model can be applied to any crystalline structure where dislocation pile-ups can be characterized experimentally. However, it is more suitable for FCC materials where slip is relatively planar.
