# Numerical Modeling of Plasticity in Metal Matrix Fiber Composites
## 2 Government Equation for Equivalent Homogeneous Orthotropic Materials
In contrast to traditional variants of the plasticity theory [35], the tensor of the fourth rank \(D_{ijmn}\) is introduced to account for the difference in the rate of hardening during different types of deformation. The components of this tensor are determined by the tangents of angles for the diagrams of stress-plastic straindependences. For the development of new plastic deformations, two conditions of active loading must be fulfilled [36]:  
\[F\ =0,\ \frac{\partial F}{\partial\sigma_{ij}}\ d\sigma_{ij}>0 \tag{5}\]  
Then, the components of the tensor of plastic strain increments are determined by the associated flow law with accuracy up to the multiplier \(d\lambda\):  
\[de_{ij}^{p}=d\lambda\frac{\partial F}{\partial\sigma_{ij}}=d\lambda\cdot B_{ ijkl}\ (\sigma_{kl}-\rho_{kl}\ ) \tag{6}\]  
These relations determine the direction of the vector of plastic strain increments perpendicular to the plasticity surface. Under active loading, stress increments cause new plastic deformations, but condition \(F=0\) is satisfied, and the total differential of the plasticity function is equal to zero:  
\[\frac{\partial F}{\partial\sigma_{ij}}d\sigma_{ij}+\frac{\partial F}{\partial e _{ij}^{p}}d\varepsilon_{ij}^{p}\ =\ 0 \tag{7}\]  
This condition, together with Formula (2), makes it possible to determine the scalar multiplier \(d\lambda\) for known increments of stresses:  
\[d\lambda=\frac{B_{ijkl}\ (\sigma_{kl}-\rho_{kl})\ d\sigma_{ij}}{B_{ijkl}\ ( \sigma_{kl}-\rho_{kl})\ B_{mnst}\ (\sigma_{st}-\rho_{st})\ C_{mnij}} \tag{8}\]  
Formula (8) sets the norm of the vector of plastic strain increments in explicit dependence on stress increments. If the stresses do not satisfy yield criterion F \(<0\) or the stress increments satisfy the inequality \(\frac{\partial F}{\partial\sigma_{ij}}\ d\sigma_{ij}<0\), then \(d\lambda=0\). When the active loading conditions (5) are met, the components of the plastic deformation increment tensor are determined by taking into account the accumulated plastic deformations. \[de_{ij}^{p}=B_{ijkl}\ (\sigma_{kl}-\rho_{kl}\ )\frac{B_{ijkl}\ (\sigma_{kl}-\rho_{kl}) \ d\sigma_{ij}}{B_{ijkl}\ (\sigma_{kl}-\rho_{kl})\ B_{mnst}\ (\sigma_{st}-\rho_{st})\ D_{mnij}} \tag{9}\]  
The presented plasticity model of an equivalent homogeneous material contains a number of material parameters. These include elastic constants \(A_{ijkl}\) in Hooke's law (2) and tensor components \(B_{ijkl}\) in the plasticity criterion (3) and plasticity function (4). In addition, the plasticity function contains components of the tensor \(D_{mnij}\) to account for differences in the rate of hardening under different types of stress states. The classical method of determining material parameters in a material model involves performing physical experiments. Even for homogeneous anisotropic materials, this method requires a large number of experiments and the fabrication of special samples. In the case of composites, the situation is also complicated by the variety of structures and types of reinforcement. A more rational approach for composites is a combined approach in which the physical properties of the components are determined experimentally, and homogenization is performed theoretically. In the present work, such an approach is implemented for an orthogonally reinforced metal matrix composite using a finite element homogenization procedure.