# Numerical Modeling of Plasticity in Metal Matrix Fiber Composites
## 5 Conclusions  
In this paper, a new generalized quadratic yield function is proposed to describe plastic anisotropy in metal matrix composites. The peculiarity of the reflection of kinematic hardening is that the back stresses are specified using a special fourth-rank tensor. The components of this tensor make it possible to take into account the difference in the dependence of back stresses on plastic strains for different tensile directions and different shear planes. The proposed dependence for the back stress tensor satisfies the fundamental invariance requirement and can be used in an arbitrary system of orthogonal coordinates. The initial determination of the components of this tensor is naturally performed in the system related to the symmetry planes of the composite structure.  
As an alternative to determining all material parameters of the mathematical model using physical experiments, this paper uses a numerical homogenization procedure. The mechanical characteristics of the matrix and fibers, as well as the geometric structure of the reinforcement, are taken as input data. The micromechanical analysis of RVEs was performed using the finite element method in the ANSYS software package. A series of numerical experiments were performed to predict the elastic properties of the composite, parameters of the yield criterion, and modeling of tensile and shear processes. The components of the additive stress tensor were determined as a result of processing the calculated strain diagrams. This made it possible to calculate all material parameters necessary for a structural analysis in finite element program complexes.  
The methodology outlined in this paper is applicable to composites with a proper orthogonal reinforcement structure. In such cases, symmetry conditions can be used for a micromechanical analysis of the representative volume. For other variants of periodic structures, the periodicity boundary conditions should be applied. For stochastic reinforcement with short fibers, the proposed approach is also applicable, but then the representativevolume should include a sufficiently large number of fibers. A promising direction for further research is to take into account the non-linear nature of matrix hardening, which is characteristic of new aluminum-based alloys. Possible clarification of the interaction scheme between the components of the composite may be the purpose of future works.  
The created methodology allows for a theoretical prediction of the development of plastic deformations in metal matrix composites and for a computer analysis of the limit state of structural elements made of such materials to be carried out. The possibility to carry out multivariate calculations at the design stage provides a significant economic effect when choosing the optimal structure of the composite.  
Conceptualization, G.L. and M.T.; methodology, G.L. and M.T.; software M.T.; validation, G.L. and M.T.; formal analysis, G.L. and M.T.; investigation, G.L. and M.T.; resources, G.L. and M.T.; data curation, G.L. and M.T.; writing-- G.L. and M.T.; writing--review and editing, G.L. and M.T.; visualization, G.L. and M.T.; supervision, G.L.; project administration, G.L.; funding acquisition, M.T. All authors have read and agreed to the published version of the manuscript.  
This research received no external funding.  
Not applicable.  
The original contributions presented in the study are included in the article, further inquiries can be directed to the corresponding author.  
The authors declare no conflicts of interest.