# Numerical Modeling of Plasticity in Metal Matrix Fiber Composites
## 2 Government Equation for Equivalent Homogeneous Orthotropic Materials  
Modern technologies to produce fibrous metal matrix composites make it possible to create regular structures in which a periodically repeating representative volume element (RVE) can be distinguished. The characteristic dimensions of such software are comparable to the radius of the fibers, which is many orders of magnitude smaller than the dimensions of real structural elements made of composite materials. The concept of homogenization consists of replacing the real structure of the composite with an equivalent homogeneous material, and its state is characterized by the values of the stress and strain tensor components averaged over the volume of RVE [34]:  
\[\sigma_{ij}=\frac{1}{\upsilon}\iint\limits_{v}\overline{\psi}_{ij}dv\ \ \ \ \ \varepsilon_{ij}=\frac{1}{\upsilon}\iint\limits_{v}\overline{\varepsilon}_{ij}dv \tag{1}\]  
Here, \(\overline{\sigma}_{ij}\) represent the components of the Cauchy stress tensor, \(\overline{\varepsilon}_{ij}\) represent the components of the linear strain tensor inside the RVE, \(\sigma_{ij}\) represent the average stresses, and \(\varepsilon_{ij}\) represent the average strain values for an equivalent homogeneous material. The physical relations of the elastic-plastic deformation of composites are formulated relative to the average characteristics for an equivalent homogeneous material. Before the first plastic strains appear, Hooke's law is used for an anisotropic body [35]:  
\[\sigma_{ij}=A_{ijkl}\varepsilon_{kl}\ (i,\ j,\ k,\ l\ =\ 1,\ 2,\ 3) \tag{2}\]  
where \(A_{ijkl}\) represent components of the elastic tensor of equivalent homogeneous material. Yield criterion. When a certain limiting state is reached, the first plastic deformations appear in the composite. For an equivalent orthotropic material, the criterion for the onset of plasticity is formulated in the form of an invariant quadratic form with respect to the average stresses [31]:  
\[B_{ijkl}\ \sigma_{ij}\sigma_{kl}\ =\ 1 \tag{3}\]  
Due to the symmetry of the stress tensor, the components of the fourth-rank tensor must satisfy the conditions \(B_{ijkl}=B_{ijkl}=B_{klij}\). Independent components \(B_{ijkl}\) determine the elastic limits for various types of complex stress states. The number of such independent components depends on the symmetry type of the equivalent homogeneous material. In particular, for an orthotropic material, criterion (3) contains nine components. For homogeneous orthotropic materials, [PERSON]'s hypothesis of plastic incompressibility is widely used. This makes it possible to exclude the first invariant of the stress tensor from consideration and reduce the number of independent components to six in the yield criterion. However, [PERSON]'s hypothesis is not applicable to composite materials, even if all of their components are incompressible. A theoretical explanation of this effect is given in [31] based on the results of a micromechanical analysis of an orthogonally reinforced composite. Plasticity model. To create physical relations of the plastic deformation of metal matrix composites, a quadratic yield function and its associated flow rule are introduced. In the initial unreformed plastic state, the yield function corresponds to the plasticity criterion (3). To reflect the anisotropy of hardening, a kinematic scheme of plasticity surface variation is used. In the proposed version of the plasticity function, it is possible to take into account different hardening rates when stretching in different directions and shearing in different planes. Within the framework of the homogenization concept, the plasticity function is formulated with respect to the tensors of average stresses and plastic strains:  
\[2F\ \equiv B_{ijkl}\ \left(\sigma_{ij}-\rho_{ij}\ )(\sigma_{kl}-\rho_{kl}\ )-1=0\ \rho_{ij}=D_{ijmn}\ \varepsilon_{mn}^{p} \tag{4}\]  
Here, \(\rho_{ij}\) represent so-called back stresses, which determine the displacement of the plasticity surface as a result of hardening.