# 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 this work, the energy dissipation rate increases when minimizing the volume fraction of the fiber and its modulus. The approach that minimizes the volume fraction increases the dissipation energy to the maximum level. For complete slip, the increase in dissipation energy is directly proportional to the change in shear stress. The formulation for active mono-slip is based on the flow rule; thus, the dissipation potential is described as  
\[\mathbb{Q}=K\Big{|}\dot{\mathrm{Y}}\Big{|} \tag{23}\]  
In the zero-resistance case, the formulation of plastic strain requires the simplification of the overall energy. Subsequently, \(\mathrm{U}\) will be graphically convex with respect to \(\mathrm{Y}\) and \(\mathrm{Y}_{\mathcal{Z}}\) in the variational formulation, which should provide accurate results [33]. Formulations  
To simplify the energy functional equation, some dimensionless quantities are introduced. Continuum dislocation formulations in a single-slip mechanism require a crucial formula that includes the balance and the evolution. Both have a common type of material, geometry, and parametric length. We aimed to develop an analytical model that shows distinctive features of the theory, independent of a specific selection of material parameters. Similarly, we performed a dimensionless analysis on the two components to explore how definite integrations of the parameters affect the outcome predicted by this method. The dimensionless variables include the elevation of the UC, which can be considered comparable to a domain elevation described by \(\overline{x}_{2}\) and is denoted by \(w\). The newly introduced variable \(\overline{x}_{2}\) varies in the range \(\left(0,\overline{\xi}\right)\). By  
dislocation should be small, so we can remove the last term of the equation. Moreover, the boundary layer width approaches zero as the plastic shear strain tends toward a threshold value. This provides the key to obtaining the value of the threshold through the minimizing sequence of Equation (27):  
\[\mathrm{Y}=\left\{ \begin{array}{l}\frac{\mathrm{Y}_{m}x_{2}}{\epsilon}\;\;for\; x_{2}\in(0,\epsilon),\mathrm{Y}_{m}\;for\;x_{2}\in(\epsilon,\xi-\epsilon),\\ \frac{\mathrm{Y}_{m}}{\epsilon}(\xi-x_{2})\;for\;x_{2}\in(\xi-\epsilon,\xi) \end{array} \right\} \tag{27}\]  
where \(\mathrm{Y}_{m}\) is a constant value, and \(\epsilon\) is a new infinite length that moves toward zero as the plastic shear strain tends toward a threshold value. Substituting (27) into the energy functional (25) by removing the last term and ignoring small terms of order \(\epsilon\) and higher results in the following expression of the energy functional:  
\[E[\mathrm{Y}_{m}]=\frac{1}{2}\xi\Big{[}\Gamma^{2}-2\Gamma\mathrm{Y}_{m}\cos 2 \varphi+\mathrm{Y}_{m}^{2}\cos^{2}2\varphi+\mathrm{Y}^{2}\sin^{2}2\varphi \Big{]}+2k|\mathrm{Y}||\sin\varphi| \tag{28}\]  
Simple analysis shows that to achieve a minimum value in (28), the minimized plastic distortion is different from zero if, and only if,  
\[\Gamma>\Gamma_{en}=2k|\sin\varphi|\Big{(}\xi\Big{|}1-2\sin^{2}2\varphi\Big{|} \Big{)}^{-1} \tag{29a}\]  
Dislocations are not nucleated when deformation achieves. \(\mathrm{Y}_{m}=0\). Recall that the sign of \(\mathrm{Y}_{m}\) relies on the inclination of \(\varphi\). Therefore, the angle \(\varphi\) is positive if \(0^{0}\leq\varphi\leq\pi/4\) and negative if \(\pi/4\leq\varphi\leq\pi/2\). With the size effect of the original length expressed in terms of \(\xi\), the energy threshold value can be formulated as  
\[\Gamma_{en}=2k|\sin\varphi|\Big{(}\xi b\rho_{s}\Big{|}1-2\sin^{2}2\varphi\Big{|} \Big{)}^{-1} \tag{29b}\]  
The threshold value deviates from the well-known Hall-Petch relationship. The term can be explained by the following condition: the deviation in (26) is through slip fields, which does not allow dislocations to penetrate grain boundaries. Boundary conditions in the grain boundaries cause \(\mathrm{Y}^{\prime}\) to change its direction in the range \((0,\xi)\).