# Micromechanical Analysis in Applications of Active Mono-Slip and Continuum Dislocations in the MDCM
## 3 Numerical Expression and Discussion  
As illustrated in this paper, in the unit cell of the composite material model, the continuum dislocation is mostly formed by complex variables of different parameters, for instance, the dislocation evolution, the material characteristic length \(l\), and the behavior of the fiber/particle and the matrix. Therefore, it becomes a practical challenge to determine the equilibrium location of the dislocations. In this research, only edge dislocations in an active mono-slip system were considered. The slip plane direction is assumed to be in the \(x_{2}\)-direction, and the slip is taken to be in the \(x_{1}\)-direction. When UCs are subjected to simple shear, there should be a dislocation in the veins of the matrix. During this process, the dislocation is trapped by the inclusion, and the number of dislocation densities increases, i.e., it results in hardening. The shear stress due to dislocation is termed micro shear stress, and the shear stress at the hardening point is referred to as critical shear stress. The equilibrium equation from this analysis is given as follows:  
\[\tau_{cr}-\tau-\tau_{micstress}=0 \tag{42}\]  
From the free energy formula \(\Theta\Big{(}\varepsilon_{ij}^{e},\alpha_{ij}\Big{)}\), we can find the standard stress tensor \(\sigma_{ij}\) by using the thermodynamical approach:  
\[\sigma_{ij}=\partial\Theta/\partial\varepsilon_{ij}^{e}=\tau \tag{43}\]  
The micro stress \(\tau_{m}\) is evaluated as follows:  
\[\tau_{m}=e_{ijk}m_{i}\varsigma_{ij}s_{j} \tag{44}\]  
where \(\varsigma_{ij}\) is the distortion stress tensor. The distortion stress tensor is also calculated:  
\[\varsigma_{ij}=\partial\Theta/\partial\alpha_{ij} \tag{45}\]  
Figure 4: Evolution of plastic distortion (\(\gamma\)) in the energy dissipation case. According to the law of thermodynamics, \(\sum\limits_{\kappa}\tau_{cr}\mathrm{Y}\geq 0\) is constant per the inequality; it is also independent of the rate. We formulate the analytical equation as shown below:  
\[\tau_{cr}=\chi sign\mathrm{Y} \tag{46}\]  
where \(\chi>0\) is the resistance of the slip represented by the formula shown in Equation (47):  
\[\chi=h\mathrm{Y} \tag{47}\]  
By using Equations (42)-(44), the standard shear stress is formulated:  
\[\sigma_{12}=\frac{1}{2}G_{12}(u_{1,2}-\mathrm{Y}s_{1}m_{2}-\mathrm{Y}s_{2}m_{ 1}) \tag{48}\]  
We validated the method defined in the aforementioned parts by accounting for the BC formulations demonstrated in Figure 2. The material is assumed to be composed of a double episodic array of UCs with a width of \(2w\) and height of \(2\xi\) as shown in Figure 5 The UC can be defined in a state of typical shear distortion by setting kinematic BCs. Periodic boundaries are necessary along the lateral sides \(x_{1}=\pm w\). The term \(\tau_{ave}\) (average shear stress) is required to withstand the distortions calculated from the shear field \(\sigma_{12}\) of the overall stress \(\sigma\). \[\tau_{ave}=\frac{1}{2w}\langle\sigma_{12}(\pm x_{1},\pm\xi)\rangle_{(-w,w)} \tag{49}\]  
The magnitude \(b\) denotes the Burgers vector of the overall dislocations in the unit cell, which has a value of \(b=2.5\times 10^{-10}m\). In addition, two dislocations with contrasting b annihilate each other when they are dependent on material characteristic length; the critical annihilation length \(l_{e}\) is equal to \(l_{e}=6b\). All inclusions should have an identical value, which is equal to \(\tau_{obs}=5.7\times 10^{-3}G_{12}\), where \(G_{12}\) denotes the elastic modulus of the shear. In the critical length, the stress created by shear distortion on the inclusion can be balanced by the dislocations generated in the slip plane. In the critical length, the value of \(\mathrm{K}=0.000115\). The stress created due to dislocation nucleation corresponds to a nucleation length of \(l_{nuc}=125b\). The height \((\xi/2)\) of the UCs is based on the material characteristic length \((l)\), which equals \(4000\)b. The influence of inclusion size is analyzed by changing the ratio \(\xi/l\)[43].