# Numerical Modeling of Plasticity in Metal Matrix Fiber Composites
## 4 Results and Discussion  
The results of finite element modeling of tension and shear processes are shown in Figures 2 and 3. Distributions of equivalent (von Mises) stresses and plastic strains are given for the final steps of the deformation processes.  
The average values of coordinate stresses in all numerical experiments were determined in the postprocessor of the ANSYS program complex. The processing of results for the numerical solution of elastic-plastic problems is presented in Figure 4, which shows the calculated strain diagrams for stretching and shear.  
Figure 2: Distributions of equivalent stresses: (**a**) uniaxial stretching along the 0X axis; (**b**) uniaxial stretching along the 0Z axis; (**c**) shear in the XY plane; (**d**) shear in the XZ plane.  
The used form of plasticity function (4) in cases of uniaxial tension or pure shear leads to a stress-plastic strain diagram with linear hardening. In particular, for tension along the 0X axis, the hardening section is given by the following equation:  
\[\sigma_{x}-d_{1111}\epsilon_{x}^{p}=\sigma_{x}^{*} \tag{13}\]  
The above is calculated while taking into account that elastic deformation leads to a bilinear deformation diagram, and the approximation of the calculated diagram (Figure 2a) allows one to determine the parameter \(d_{1111}\). As a result of processing all calculated deformation diagrams, all components of the hardening tensor were obtained, which are given in Table 5.  
Finite element modeling of plastic deformation processes of an orthogonally reinforced composite predicts significant anisotropy of translational hardening. Plasticity theories since [PERSON]'s work [36], in which the back stresses \(\rho_{ij}\) are assumed to be equal to the plastic strains multiplied by a constant or by a scalar strain function, do not reflect this feature of composite materials.  
As a visual illustration of the shift in the plasticity surface, Figure 5 shows its sections by planes corresponding to two non-zero coordinate stresses, \(\sigma_{x}-\sigma_{z}\) and \(\sigma_{x}-\tau_{xy}\), at different values of the back stress, \(\rho_{ij}\) (0--line 1; 10\({}^{7}\) Pa--line 2; 1.5 \(\times\) 10\({}^{7}\) Pa--line 3).  
Figure 3: Distributions of equivalent plastic strains: (**a**) uniaxial stretching along the 0X axis; (**b**) uniaxial stretching along the 0Z axis; (**c**) shear in the XY pane; (**d**) shear in the XZ plane.  
\begin{table}
\begin{tabular}{c c c c} \hline \(d_{1111}=d_{2222}\) & \(d_{3333}\) & \(d_{4444}\) & \(d_{5555}=d_{6666}\) \\ \hline
894 & 18.3 & 5.72 & 6.53 \\ \hline \end{tabular}
\end{table}
Table 5: The components of the hardening tensor (GPa).  
The anisotropy of the plastic properties of a boron-aluminum composite is manifested not only in the difference in yield strengths in different directions, but also in the plasticity surface shift rate. As can be seen from the data in Table 5, the ratio of the increment of back stresses to the increments of plastic deformations differs significantly depending on the type of deformed state.  
Figure 4: Diagrams for (**a**) uniaxial stretching along the 0X and 0Z axes; (**b**) shear in the XY and XZ planes.