# Micromechanical Analysis in Applications of Active Mono-Slip and Continuum Dislocations in the MDCM
## 2 Continuum Dislocation Formulation
### Thermodynamic Principles
The GND energy components imply two important details. First, the network of dislocation energy is directly proportional to the DOD when the dislocation is small. Second, the existing SDD, which indicates the nearby grouping of dislocation, can have identical signs to those of the newly created permissible dislocation in the unit cell. Assuming that the material is isotropic and contains inclusions, for simplicity, the elastic behavior of the unit cell in both the matrix and fiber materials is considered isotropic [18]. The modulus of elasticity of the matrix can be formulated as  
\[\Re=2G_{12}\left(\mathrm{I}^{\prime}-\frac{\
u}{1-2\
u}\mathrm{I}\otimes \mathrm{I}\right) \tag{21b}\]  
where \(G_{12}\) is the shear modulus, \(\
u\) is Poisson's ratio, \(\mathrm{I}\) represents a quadratic-order identity tensor, and \(\mathrm{I}^{\prime}\) denotes a polynomial-order identity tensor. This is also true for the fiber, but the shear modulus changes to \(G_{12}^{*}\). The GND term [41] ensures a direct increase in energy from a low density of dislocation (\(\rho\)). Then, energy approaches infinity as the density of dislocation reaches the saturated level, symbolized as (\(\rho_{\mathrm{s}}\)). Therefore, an energetic barrier is provided to prevent over-saturation. For the dissipation potential, numerous models can be considered. The simplest model assumes that the dissipation is zero. In this case, all tensors (\(\zeta_{ij}\), \(\lambda_{ij}\) and \(\zeta_{ijk}\)) vanish, and the functions \(u_{ij}\) and \(\mathrm{Y}_{ij}\) can be found from energy minimization. From the above point of view, the total energy functional is represented by the following:  
\[\prod[u_{1},u_{2},\mathrm{Y}]=wL\left\langle \begin{array}{c}\frac{1}{2}\Re( u_{2,2})^{2}+\frac{1}{2}G_{12}(u_{1,2}-\mathrm{Y}\cos 2\varphi)^{2}+\frac{1}{4}G_{12 }\mathrm{Y}^{2}\sin^{2}2\varphi\\ +G_{12}\Big{(}u_{2,2}-\frac{1}{2}\mathrm{Y}\sin 2\varphi\Big{)}^{2}+G_{12}k* \ln\left(1-\frac{|\chi_{y}||\sin\varphi|}{b\varphi}\right)^{-1}\end{array}  \right\rangle_{a=g} \tag{21c}\]  
where \(\langle\bullet\rangle=\int\limits_{0}^{a}\bullet dx_{2}\). Based on the continuum dislocations, the main constitutive and balance equation is defined as a basic formula for the active mono-slip equation. The first visco-plastic framework can be translated into a purely elastoplastic context, for which analytical equations can be determined through the shear analysis. The displacement field in a unit cell and the mean shear distortions can be measured by the continuum dislocation approach. To evaluate this displacement, the function of the overall energy is simplified so that it depends on a functional \(\mathrm{Y}(x_{2})\) only. First, the function \(\mathrm{Y}(x_{2})\) is defined, and then variational calculus is applied to Equation (21a) with respect to \(u_{1}\) and \(u_{2}\) to determine the strain in the analysis of shear (shear strain) deformation. \[\left\{ \begin{array}{c}u_{1,22}=\mathrm{Y}_{,2}\cos 2\varphi\\ \Re u_{2,22}+2G_{12}u_{2,22}=G_{12}\mathrm{Y}_{,2}\sin 2\varphi\end{array} \right\} \tag{21d}\]  
As previously stated, using Equation (22a) as a boundary condition, we can apply simple integration to Equation (21a) to formulate the displacement fields and shear strain. Then, we can obtain a general equation for the continuum approach to describe the analogous shear strain component in the defined boundary conditions [42]. \[\mathrm{Y}(0)=\mathrm{Y}(\xi)=0 \tag{22a}\]  
Subsequently, we can apply terms to divide the displacement into \(\mathbf{u_{1}}\) and \(\mathbf{u_{2}}\) components. The effect of the two components of displacement results in shear strain on the unit cell and a mesoscopic internal parametric length created from the overall heterogeneous internal strain component. Thus, in a small-strain calculation, the two displacement components can be directly derived because the small-strain equation is linear. In the unit cell, the mesoscopic internal parametric length of shear strain can be stated in terms of the DOD pattern.