# Numerical Modeling of Plasticity in Metal Matrix Fiber Composites
## 3 Numerical Homogenization in an Orthogonally Reinforced Composite
For example, in a numerical experiment that simulates homogeneous stretching of an equivalent orthotropic material only in the x-axis direction, the boundary conditions are given in the first row of Table 1. These conditions both specify such a change in the shape of the RVEs that the original rectangular parallelepiped remains rectangular after deformation, but only the edge parallel to the x-axis is elongated. In each numerical experiment, the boundary conditions set one non-zero average value (\(10^{-3}\)) of the strain tensor. Based on the results of the micromechanical analysis, all of the characteristics of the stress state, including all of the components of the average stress tensor, are determined. Consecutive processing of the results of numerical modeling of basic experiments makes it possible to determine all components of the matrix (10), and after calculating the inverse matrix, the technical elastic constants of the composite can be found (Table 2). Details of this process are given in Appendix A. The numerical prediction of the yield criterion parameters. For the considered orthotropic material in the coordinate system coinciding with the symmetry planes of mechanical properties, criterion (3) contains six independent constants and can be written in the following form:  
\[C_{11}(\sigma_{y}^{2}+\sigma_{x}^{2})+C_{33}\sigma_{z}^{2}+C_{12}\sigma_{x} \sigma_{y}+C_{13}(\sigma_{x}\sigma_{z}+\sigma_{y}\sigma_{z})+C_{44}\tau_{xy}^{ 2}+C_{55}(\tau_{yz}^{2}+\tau_{xz}^{2})=1 \tag{11}\]  
To determine the parameters of criterion (11), it is necessary to know the yield strength under uniaxial tensile stresses in two directions, \(\sigma_{x}^{*}\) and \(\sigma_{z}^{*}\), shear in two planes, \(\tau_{xy}^{*}\) and  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Basic Experiment**} & \multicolumn{6}{c}{**Equations of Surfaces**} \\ \cline{2-7}  & \(\mathbf{x=0}\) & \(\mathbf{y=0}\) & \(\mathbf{z=0}\) & \(\mathbf{x=1}\) & \(\mathbf{y=1}\) & \(\mathbf{z=2}\) \\ \hline \multirow{3}{*}{Stretching along axis 0X} & \multirow{3}{*}{_sim_} & \multirow{3}{*}{_sim_} & & \(u_{x}=10^{-3}\) & \(u_{y}=0\) & \(u_{z}=0\) \\  & & & & \(\tau_{xy}=0\) & \(\tau_{xy}=0\) & \(\tau_{xy}=0\) \\ \hline \multirow{3}{*}{Stretching along axis 0Z} & \multirow{3}{*}{_sim_} & \multirow{3}{*}{_sim_} & & \(u_{x}=0\) & \(u_{y}=0\) & \(u_{z}=2\times 10^{-3}\) \\  & & & \(\tau_{xy}=0\) & \(\tau_{xy}=0\) & \(\tau_{xy}=0\) & \(\tau_{xy}=0\) \\  & & & & \(\tau_{xz}=0\) & \(\tau_{xy}=0\) & \(\tau_{xz}=0\) \\ \hline \multirow{3}{*}{Shear in plane XY} & \(u_{y}=0\) & \(u_{x}=0\) & \(u_{z}=0\) & \(u_{y}=10^{-3}\) & \(u_{x}=0\) & \(u_{z}=0\) \\  & \(\sigma_{x}=0\) & \(\sigma_{y}=0\) & \(\tau_{xz}=0\) & \(\sigma_{x}=0\) & \(\sigma_{y}=0\) & \(\tau_{xz}=0\) \\  & \(\tau_{xz}=0\) & \(\tau_{\(\tau_{xz}^{*}\), and under simultaneous tension in two directions with equal stresses, \(\sigma_{x+y}^{*}\) and \(\sigma_{x+z}^{*}\). Parameters \(C_{ij}\) are related to yield strengths by the following relations:  
\[ \begin{split} C_{11}&=\frac{1}{\left(\sigma_{x}^{*} \right)^{2}}\ \ C_{33}=\frac{1}{\left(\sigma_{z}^{*}\right)^{2}}\\ C_{44}&=\frac{1}{\left(\tau_{xy}^{*}\right)^{2}} \ \ C_{55}=\frac{1}{\left(\tau_{xz}^{*}\right)^{2}}\\ C_{12}&=\frac{1}{\left(\sigma_{x+y}^{*}\right)^{2}} -\left(C_{11}+C_{22}\right)\ \ \ \ \ C_{13}=\frac{1}{\left(\sigma_{x+z}^{*}\right)^{2}}-\left(C_{11}+C_{33} \right)\end{split}  \tag{12}\]  
In the framework of the proposed approach, the determination of yield strengths is performed by a micromechanical analysis of RVEs. To this end, the stress states under the elastic deformation of the composite under uniaxial tensile, shear, and biaxial stretching were numerically modeled. For these basic experiments, average strain values were determined using the technical elastic constants given in Table 2. The procedure for the formation of boundary conditions for finite element modeling of the basic experiments was developed in [31].