The equilibrium positions of the N dislocations can be found out by minimizing the component of the Peach–Koehler (P–K) force along the slip direction for each dislocation to a material’s critical force

${F}_{c}$ as follows:

where

$({X}_{1}\left(\gamma \right),{X}_{2}\left(\gamma \right))$ denotes the position of the

$\gamma $th dislocation.

$\mathit{v}$ is a unit vector that belongs to the slip plane and is normal to the dislocation line. It indicates the glide direction of all the dislocations. It is considered to be directed towards the GB so that a pile-up can form.

${\mathit{\sigma}}_{\mathrm{ext}}$ is a homogeneous applied stress tensor to the bi-crystal and

${\mathit{\sigma}}_{\mathrm{int}}={\mathit{\sigma}}_{\mathrm{im}}+{\mathit{\sigma}}_{\mathrm{dis}}$ is the internal stress tensor produced by all the other dislocations,

${\mathit{\sigma}}_{\mathrm{dis}}$, and the image stress tensor

${\mathit{\sigma}}_{\mathrm{im}}$ on this particular dislocation coming from all elastic heterogeneities, see the detailed calculations in [

41] using the Leknitskii–Eshelby–Stroh (L–E–S) formalism for two-dimensional anisotropic elasticity. Here, the elastic heterogeneities include the GB seen as an interphase [

41], both grains related by a misorientation and two free surfaces (denoted as

${\Lambda}_{1}$ and

${\Lambda}_{2}$ in

Figure 7a). The interphase is supposed to be the region between two planar interfaces

${\Gamma}_{1}$ and

${\Gamma}_{2}$ with a thickness

${H}^{\mathrm{GB}}$ as shown in

Figure 7a. The image stress produced by these heterogeneities can be calculated using the image decomposition method for anisotropic multilayers [

42]. In this method, all the interfaces and the free surfaces are regarded as a distribution of image dislocation densities which can be resolved through boundary conditions [

42]. The interphase model allows considering a non-zero thickness in the nanometer range and a specific elastic stiffness tensor for the GB region. While the configuration with two free surfaces can be used to study size effects [

43], more discussions about the interphase model and the effect of free surfaces can be found in [

44]. At the end, the critical force

${F}_{c}$ may include the lattice friction force (primary) and other forces due to obstacles to dislocation motion (solute atoms, precipitates, etc.). Meanwhile, this critical force can be converted into a shear stress on a dislocation by dividing by the corresponding Burgers vector magnitude as

${\tau}_{c}={F}_{c}/\left|\mathit{b}\right|$. Most past studies assume

${F}_{c}=0$ N/m in Equation (

1) because of the low value of the lattice friction stress in pure FCC crystals, which is around

$1\sim 2$ MPa [

45]. However, in the present study, it is found that a non-zero critical force has a crucial effect on the discrete distribution of dislocations in the presence of GB and free surfaces. The reason is that there are already lattice defects in the material after sample preparation, such as the

${\mathrm{Ga}}^{+}$ ions from FIB [

46]. Furthermore, for

$\alpha $-Brass, the theoretical value of the lattice friction force should be higher than in pure FCC crystals, like pure Ni. Thus, the critical force cannot be ignored in Equation (

1) for realistic discrete pile-up calculations. The calculations of the discrete dislocation pile-up equilibrium positions are thus obtained by following an iterative relaxation scheme that minimizes all the

${F}^{\left(\gamma \right)}$ after an initial configuration is specified [

41,

47].