# Micromechanical Analysis in Applications of Active Mono-Slip and Continuum Dislocations in the MDCM
## 2 Continuum Dislocation Formulation
### Thermodynamic Principles
#### 2.3.1 Zero Energy Dissipation Case
In 1-D dislocation theory, the analysis of the previous formulations suggests that the minimizer can be found from the following:  
\[\mathrm{Y}=\left\{ \begin{array}{l}\mathrm{Y}_{1}x_{2}\;for\;x_{2}\in(0, \ell),\mathrm{Y}_{m}\;for\;x_{2}\in(\ell,\xi-\ell),\\ \mathrm{Y}_{1}(\xi-x_{2})\;for\;x_{2}\in(\xi-\ell,\xi)\end{array} \right\} \tag{30}\]  
where \(\mathrm{Y}_{m}\) is a constant value, and \(\ell\) is an unknown parametric length, with \(0\leq\ell\leq\xi/2\) and \(\mathrm{Y}_{1}(\ell)=\mathrm{Y}_{m}\). The total energy functional is  
\[E=2\Big{\langle}\frac{1}{2}\mathrm{Y}_{1}^{2}\sin^{2}2\varphi- \frac{1}{2}k\mathrm{Y}_{1}^{2}\sin^{2}2\varphi+k|\mathrm{Y}_{2}||\sin\varphi| +\frac{1}{2}\mathrm{Y}_{1}^{2}\sin^{2}2\varphi\Big{\rangle}+\frac{1}{2} \mathrm{Y}_{1}^{2}\sin^{2}2\varphi(h-2l)-\frac{1}{2}k\mathrm{Y}_{1}^{2}\sin^{ 2}2\varphi(h-2l)\] \[+\frac{1}{2}h\Big{[}k\mathrm{Y}^{ax\varphi}\sin^{2}2\varphi+ \Gamma^{2}-2\Gamma\mathrm{Y}^{ax\varphi}\cos 2\varphi\;+(\mathrm{Y}^{ax\varphi})^{2} \cos^{2}2\varphi\Big{]} \tag{31}\]  
Where \(\langle\bullet\rangle=\int\limits_{0}^{l}\bullet dx_{2}\)  
From the above analysis, we obtained the optimal method to begin averaging the appropriate variable of plastic distortion, aiming to define other variables to examine the evolution of dislocations and other plasticity-related factors. More importantly, we discovered that the average plastic distortion equation strongly suggests the micro-levelkinematics in the averaging formulas, which should be a physically realistic representation of the incorporated length scale. \[\mathrm{Y}^{nxe}=\frac{1}{\xi}2\left(\int\limits_{0}^{\ell}\mathrm{Y}_{1}dx_{2}+ (\xi-2\ell)\mathrm{Y}_{m}\right) \tag{32}\]  
To generalize the concept, we can vary the energy function Equation (31) \(\mathrm{Y}_{1}\). Then, we can perform integration by parts:  
\[\mathrm{Y}^{nxe}\cos^{2}2\varphi+k\mathrm{Y}^{nxe}\sin^{2}2\varphi+\mathrm{Y}_ {l}\sin^{2}2\varphi-k\mathrm{Y}_{l}\sin^{2}2\varphi=k\mathrm{Y}_{1}^{\prime \prime}\sin^{2}\varphi+\Gamma\cos 2\varphi \tag{33a}\]  
To simplify our modeling, it is better to subject \(\mathrm{Y}_{1}(y)\) to boundary conditions:  
\[\mathrm{Y}_{1}(0)=0,\mathrm{Y}_{1}(\xi)=\mathrm{Y}_{m} \tag{33b}\]  
Next, (31) is manipulated using variation with respect to \(\ell\), which establishes an additional boundary condition. \[y=\ell,\mathrm{Y}_{1,2}(\ell)=0 \tag{33c}\]  
We can also use the variation formulation again on the energy functional Equation (33) with respect to \(\mathrm{Y}_{m}\), followed by integration by parts to simplify it. \[2k\big{|}sign\mathrm{Y}_{1}^{\prime}\big{|}\sin\varphi\big{|}+\mathrm{Y}^{nxe }\cos^{2}2\varphi+\kappa\mathrm{Y}^{nxe}\sin^{2}2\varphi=\ \Gamma\cos 2\varphi-\mathrm{Y}_{m}\sin^{2}2 \varphi+k\mathrm{Y}_{m}\sin^{2}2\varphi(h-2l) \tag{34}\]  
The generation of the described evolution equation is based mainly on an energy minimization formula in the zero-energy dissipation case. The microstructural evolution equations introduce two modeling functions: the average distortion and the average parametric length of the material for a dislocation density. First, by assigning symbols to several components of the equation, we can reduce Equation (33) as follows:  
\[\left\{ \begin{array}{l}\omega_{1}=k\sin^{2}\varphi,\\ \omega_{2}=(1-k)\sin^{2}2\varphi,\\ \omega_{3}=-\mathrm{Y}^{nxe}\big{(}\cos^{2}2\varphi+\kappa\sin^{2}2\varphi \big{)}+\Gamma\cos 2\varphi\end{array}