# Micromechanical Analysis in Applications of Active Mono-Slip and Continuum Dislocations in the MDCM
## 2 Continuum Dislocation Formulation
### Thermodynamic Principles  
To formulate the MDCS in Figure 1, we can apply free energy theory to investigate the UCs with the required accuracy. This postulation suggests that we can acquire identical nominal material properties since an ideal unrestrained and free MDCS uses identical microstructures for the loaded and constrained unit cells. Hereafter, we can perform the micromechanical calculations since a composite material is composed of a vast number of UCs. For a reversible and tough material, the total amount of energy is the same as the time derivative of free energy plus the power stored in all of the UCs, which is  
\[\prod\limits=\frac{d}{dt}\int\limits_{R}\Theta\big{(}\varepsilon_{ij},\text{Y}_ {ij},\alpha_{ij}\big{)}dx+\Phi\leq 0 \tag{15}\]  
The required extent of power in any solid material can be measured from the scheme of energy. For this study, the power is formulated as  
\[\Phi=\int\limits_{\partial Y}\Big{(}\sigma_{ij}n_{j}\dot{u}_{i}+\sigma_{ijk}n _{k}\dot{\text{Y}}_{ij}\Big{)}da \tag{16}\]  
where \(\Phi\) is the accumulated power, and \(\sigma\) represents stress. We infer that the stresses expressed in terms of higher-order calculus are included under this theory because the gradient of irreversible distortion is based on the density of the stored energy. Thus, we can transform the surface integral into a volume integral by [PERSON]'s principle, and the mathematical requirement can be satisfied for an arbitrary R. At this juncture, the combination of the first and second principles of thermodynamics can be defined by  
\[\zeta_{ij}\dot{u}_{ij}+\lambda_{ij}\text{Y}_{ij}+\zeta_{ijk}\dot{\text{Y}}_{ ij,k}\geq 0 \tag{17}\]  
This confirms that \(\zeta_{ij}\) and \(\zeta_{ijk}\) denote the normal and higher-order stresses, respectively, which cause energy dissipation through the heating of the crystal. Tensor \(\lambda_{ij}\) describes a non-uniform flow of heat; it assumes the same meaning as in the frictional stress formula. Thus, \(\lambda_{ij}\) and \(\zeta_{ijk}\) denote the heat caused by homogeneous and inhomogeneous irreversible distortions, respectively. \[\mathbb{Q}=\mathbb{Q}\Big{(}\dot{u}_{ij},\dot{\text{Y}}_{ij},\dot{\text{Y}}_{ ij,k}\Big{)} \tag{18}\]  
The tensors described as \(\zeta_{ij},\ \lambda_{ij}\) and \(\zeta_{ijk}\) for irreversible progressions are related to \(\dot{u}_{ij},\ \text{Y}_{ij}\) and \(\dot{\text{Y}}_{ij,k}\) and described by the next formulation. \[\zeta_{ij}=\mathbb{Q}_{\dot{u}_{ij}},\lambda_{ij}=\mathbb{Q}_{\dot{Y}_{ij}}, \zeta_{ijk}=\mathbb{Q}_{\dot{Y}_{ij,k}} \tag{19}\]  
Consequently, each model of continuum dislocations can be formulated by the two energy functions, namely, free energy and potentials of dissipation. In the isothermal formulation, we require the free energy density to be contingent on the elastic state \(\varepsilon_{ij}^{e}\). The free energy in terms of \(\varepsilon_{ij}\), \(\text{Y}_{ij}\) & \(\alpha_{ij}\) is given as follows:  
\[\Theta\big{(}\varepsilon_{ij},\text{Y}_{ij},\alpha_{ij}\big{)}=\frac{1}{2} \Re_{ijkl}\varepsilon_{ij}^{e}\varepsilon_{kl}^{e}+\Theta_{m}\big{(}\alpha_{ ij}\big{)} \tag{20}\]  
where \(\Theta_{m}\big{(}\alpha_{ij}\big{)}\) denotes the energy of the dislocation network of the system. When \(\rho\) increases, there must be some threshold value, \(\rho_{s}\), after which new dislocation structures grow, including sub-grains and cell structures or new grain boundaries. Hence, additional parameters must come into play. If we preclude the networked structure of dislocation from surfacing, the dislocation network energy approaches infinity as \(\rho\rightarrow\rho_{s}\)[41]. Then, the dislocation network energy \(\Theta_{m}\big{(}\alpha_{ij}\big{)}\) for the UC distortion in the active mono-slip system can be evaluated as follows:  
\[\Theta_{m}=\text{K}*G_{12}*\ln(1-\rho/\rho_{s})^{-1}\] (21a) where K denotes a material constant, G12 represents the modulus of shear, \[b\] is the Burgers vector, and \[\rho_{s}\] denotes the saturated dislocation density (SDD).