# Numerical Modeling of Plasticity in Metal Matrix Fiber Composites
## 3 Numerical Homogenization in an Orthogonally Reinforced Composite  
Let us consider a boron-aluminum composite reinforced with continuous fibers in two mutually perpendicular directions. Due to the regularity of the composite structure, it is sufficient to perform a micromechanical analysis of the periodically repeating representative volume element (RVE). The geometrical model of RVE is shown in Figure 1a. The relative (related to the length of the edge parallel to the x-axis) dimensions of the sides of the parallelepiped are 1 \(\times\) 1 \(\times\) 2, and the radius of the fiber is 0.6. The experimental basis for determining the effective characteristics of an equivalent homogeneous material is the mechanical properties of boron fibers and an aluminum matrix from [34]. The elastic modulus and Poisson's ratio of boron are 420 GPa and 0.3, and those for aluminum are 7.1 GPa and 0.34. To approximate the aluminum deformation diagram, a bilinear function was adopted with a yield strength of 262 MPa and a tangential modulus of 8.72 GPa. All stages of numerical homogenization were performed using a structural analysis of RVEs with the finite element method in the ANSYS 2021 R1 software package. To create a finite element model (Figure 1b), the higher-order 3D 20-node SOLID186 element was used. The geometrical model of the RVEs consists of two volumes belonging to the fibers and one to the matrix. When creating the finite element model, the corresponding material properties were set in the mesh attributes for these volumes. On the boundaries between the volumes, the conditions of continuity of nodal displacements were fulfilled. In \"Mesh Tool\", the smart sizing option was used with \"Fine 1\", and the parameters of the \"Mesher Options\" were set by the program choice option. The number of elements between the components of the structure was distributed as follows: fiber--2592; matrix--16,837 elements. All of the results presented in this work were obtained using a finite element model consisting of 19,429 elements and 36,138 nodes. The homogenization of the elastic properties of the composite. For the composite structure under consideration, the equivalent homogeneous material is orthotropic. In this case, it is convenient to combine the coordinate planes of the Cartesian coordinate system with the symmetry planes of the composite. Using Voigt notation, the stress-strain relationship is represented in matrix form:  
\[\left[ \begin{array}{c}\sigma_{x}\\ \sigma_{y}\\ \sigma_{z}\\ \tau_{xy}\\ \tau_{yz}\\ \tau_{xz}\end{array} \right]=\left( \begin{array}{cccccc}a_{11}&a_{12}&a_{13 }&0&0&0\\ a_{21}&a_{22}&a_{23}&0&0&0\\ a_{31}&a_{32}&a_{33}&0&0&0\\ 0&0&0&a_{44}&0&0\\ 0&0&0&0&a_{55}&0\\ 0&0&0&0&a_{66}\end{array} \right)\times\left[ \begin{array}{c}\varepsilon_{x} \\ \varepsilon_{y}\\ \varepsilon_{z}\\ \gamma_{xy}\\ \gamma_{yz}\\ \gamma_{xz}\end{array} \right] \tag{10}\]  
The mechanical properties of the composite in the directions of the OX and OY axes are the same; therefore, the matrix components satisfy the following equalities: \(a_{11}=a_{22}\), \(a_{13}=a_{23}\), \(a_{55}=a_{66}\). To determine the elastic characteristics of the composite, a special methodology for a numerical simulation of basic experiments was used [37]. In each numerical experiment, a certain type of homogeneous stress state of an equivalent orthotropic material is modeled. For this purpose, a micromechanical analysis of the RVEs is performed under a specially designed system of boundary conditions on the surfaces of the RVEs (Table 1). These boundary conditions are used only for the homogenization procedure, the purpose of which is to calculate the effective properties of an equivalent homogeneous material. These boundary  
Figure 1: Representative volume element. (**a**) Geometrical model; (**b**) finite element model. conditions ensure that the stress-strain state inside the RVEs fully corresponds to the state that occurs in each RVE during the homogeneous deformation of the unbounded volume of the composite.