# Micromechanical Analysis in Applications of Active Mono-Slip and Continuum Dislocations in the MDCM
## 2 Continuum Dislocation Formulation
### Thermodynamic Principles
#### 2.3.2 Non-Zero Energy Dissipation Case  
Dislocation appears in crystals to reduce their energy. Energy dissipation analysis is more accurate than any other method to define the material features in the plastic phase. The displacement associated with reversible-to-irreversible reactions of the materials is commonly linked to the accumulation of continuum dislocations. The methods can be implemented to analyze the hysteresis energy triggered by the inclusions that create resistance in the unit cell. If the dislocation motion resistance cannot be avoided, the irreversible deformation advances as long as the yield condition is satisfied, that is, \(\aleph=\aleph\) K [37]. The evolution of dislocation in MDCS can also be highly sensitive to the interaction between fibers and the matrix interface. If \(|\aleph|<\aleph\), then Y is fixed, the DOD is assumed to be constant, and the unit cell is reversibly distorted. Analyzing the variational derivative of Equation (21) yields the condition. The first case considered is \(\varphi<\pi/4\). \[\left|-\mathrm{Y}\sin^{2}2\varphi+\kappa\mathrm{Y}\sin^{2}2\varphi-\cos^{2}2 \varphi\mathrm{Y}^{\alpha\
u}+k\mathrm{Y}^{\alpha\
u}\sin^{2}2\varphi+k \mathrm{Y}_{,22}\sin^{2}\varphi\left(b^{2}\rho_{s}^{2}\right)^{-1}+\Gamma \cos 2\varphi\right|=K/G_{12}\cos 2\varphi \tag{37}\]  
The distribution of dislocation also leads to an escalation in entropy related to distortion. A strain energy state limits individual dislocation, which absorbs some of the plastic flow as the DOD increases [19; 32]. According to \(\Gamma\), the definite time function can be used to estimate \(\mathrm{Y}(t,x_{2})\). The key is to determine the distortion as a function of time and \(x_{2}\) in its evolution state, provided \(\mathrm{Y}(0,x_{2})=0\) and \(\varphi<\pi/4\). Since the irreversible distortion, Y, is primarily zero, we observe from (38) that \(\mathrm{Y}=0\) if, and only if, \(\Gamma<\Gamma_{cr}\). Therefore, the yield stress condition for small \(\mathrm{Y}(t,x)\) and \(\Gamma<\Gamma_{cr}\) becomes  
\[-\mathrm{Y}\sin^{2}2\varphi+\kappa\mathrm{Y}\sin^{2}2\varphi-\cos^{2}2\varphi \mathrm{Y}^{\alpha\
u}+k\mathrm{Y}^{\alpha\
u}\sin^{2}2\varphi+k\mathrm{Y}^{ \alpha\
u}\sin^{2}\varphi+\Gamma\cos 2\varphi=\Gamma_{cr}\cos 2\varphi \tag{38}\]  
The deviation of \(\Gamma(t)\) from the critical shear streamline (38) is introduced to obtain  
\[\Gamma_{f}=\Gamma-\Gamma_{cr}\Rightarrow\Gamma=\Gamma_{f}+\Gamma_{cr}\text{ and }\Gamma_{f}=\mathrm{Y}/\mathrm{Y}_{1}\text{:}\]  
\[-\mathrm{Y}\sin^{2}2\varphi+\kappa\mathrm{Y}\sin^{2}2\varphi-\cos^{2}2\varphi \mathrm{Y}^{\alpha\
u}=-k\mathrm{Y}^{\alpha\
u}\sin^{2}2\varphi-k\mathrm{Y}^{ \alpha\
u}\sin^{2}\varphi-\Gamma_{f}\cos 2\varphi \tag{39}\]  
The analogous problem of dissipated energy in the constrained shear resistance suggests that the result of Equation (37) should be symmetric. Hence, \(\mathrm{Y}_{1}(x_{2})=\mathrm{Y}_{1}(\xi-x_{2})\)_for_\(x_{2}\in(\xi/2,\xi)\). The function \(\mathrm{Y}_{1}(x_{2})\) can be obtained from Equation (38). Similarly, we can use the boundary condition from Equation (33a). In the first case, the boundary region is far from the dislocations because of the stated displacement. In the next case, the plastic distortion is guided by continuity and symmetry. Based on this, \(\mathrm{Y}_{1}\) is evaluated as  
\[\mathrm{Y}_{1}=\mathrm{Y}_{1p}(1-\cos\mathrm{h}(\eta\mathrm{x}_{2})*(1+\mathrm{ Tanh}(\eta\xi/2)*\mathrm{Tanh}(\eta\mathrm{x}_{2}))),0\leq x_{2}\leq\xi/2 \tag{40a}\]  
Figure 3: The evolution of plastic distortion (\(\gamma\)).