# Micromechanical Analysis in Applications of Active Mono-Slip and Continuum Dislocations in the MDCM
## 2 Continuum Dislocation Formulation
### Mono-Slip Analysis in the Continuum Formulation  
Here, we show that the lateral boundary condition in the free form can be investigated analytically. Additionally, clamped micro-BCs of the top and bottom sections of the unit cell can be formulated using the standard analytical model, where, based on the principle of plasticity, the overall displacement field in the gradient \(\mathbf{u}\) signifies the total displacement field in a compatible formulation. The dislocation loop is singular and positioned on the single-slip plane, and the Burgers vector points in the right direction toward the path of the  
Figure 1: 2D simplest representation of the MDCS unit cell. Figure 2: Active mono-slip system with a unit cell (UC) of \(\xi\) thickness. slip. Based on the above concept, the inverse plastic distortion on the loop is formulated as Equation (2). \[-\mathrm{Y}_{ij}=b_{i}n_{j}\delta(s) \tag{2}\]  
where \(b_{i}\) denotes the Burgers vector, \(\delta\)S is the cut surface lying on the slip plane, and \(n_{j}\) is the vector perpendicular to the plane of the slip. This inverse plastic distortion describes the creation of dislocation by cutting a perfect crystal along the surface \(S\) and shifting the structure beneath this surface by one Burgers vector [37]. In general, the summation of the reversible and irreversible distortion fields contributes to the total distortion result:  
\[\mathrm{Y}=\mathrm{Y}^{e}+\mathrm{Y}^{p} \tag{3}\]  
where \(\mathrm{Y}^{e}\) and \(\mathrm{Y}^{p}\) describe the reversible and irreversible distortions in the material system. Due to the existence of boundary lines between dislocations, the incompatibly of the two phases are represented by the GND [38]. A huge number of loops pass through the slip planes, and the average distance between them is considerably smaller than the material characteristic scale size of the standard piece; we go one step further and propose a unique formulation for the plastic distortions created by this slip system, expressed as  
\[\mathrm{Y}_{ij}=\mathrm{Y}(x)s_{i}m_{j} \tag{4}\]  
where \(m\) represents a normal vector that points toward the slip direction, and \(s\) denotes the slip path. The analytical analysis fundamentally depends on the continuum dislocation principle of unit cell plasticity owing to particular situations in which shape variations are insignificant and the response of the material is rate independent [39]. The governing equations can be abridged by employing tensor analysis in a coordinate system. The plastic strain, represented by \(\varepsilon_{ij}^{p}\), and plastic rotations, designated by \(\Omega_{ij}\), are the symmetric and anti-symmetric parts of the irreversible deformations:  
\[\varepsilon_{ij}^{p}=\left(\mathrm{Y}_{ij}+\mathrm{Y}_{ji}\right)/2,\quad \Omega_{ij}=\left(\mathrm{Y}_{ij}-\mathrm{Y}_{ji}\right)/2 \tag{5}\]  
The strains in the reversible phase can be estimated as the differences between the overall incompatible strains and the irreversible strain state. Then, the reversible strain tensors are  
\[\varepsilon_{ij}^{e}=1/2\left(u_{ij}+u_{ji}\right)-\varepsilon_{ij}^{p} \tag{6}\]  
The fundamentals of the continuum dislocation principle of unit cell plasticity applied in this work are supported by [PERSON], whose described the concept of lattice incompatibility [40]. The plastic distortion field \(\mathrm{Y}^{e}\) can be obtained by the sum of its symmetric strain and anti-symmetric rotation fields,  
\[\mathrm{Y}^{e}=\varepsilon^{e}+\Omega^{e} \tag{7}\]  
where \(\varepsilon^{e}\) denotes elastic strain and \(\Omega^{e}\) is the reversible torsional strain. We assumed that the elasticity is linear, disregarding the'shape nonlinearity' portion. The fundamental concept of the elastic deformation of the UC is not harmonious with regular distortion, which is obtained through a continuously differentiable displacement component. However, reversible distortion is sufficient to represent geometrically necessary dislocations [8].