## 2 Materials and Methods
### Mean Field Homogenization Model  
The Digimat (namely, the MF and MX modules) was used to create the material model. The main advantage of this program is the use of the mean field homogenization approach to determine the characteristics of the composite material. The properties of each phase (PA6 matrix and glass fibers), mass distribution, and mechanical characteristics of the matrix in each load direction were used as input data. The fibers and matrix were homogenized in Digimat-MF using a revised formulation second-order Mori-Tanaka homogenization scheme to calculate the mechanical properties [10].  
It is assumed that the composite stress and strain depend on the stress and strain of the matrix and fiber, which are proportional to their volume fractions:  
\[\epsilon=\Big{(}1-\phi^{f}\Big{)}\epsilon_{m}+\phi^{f}\epsilon^{f},\;\sigma= \Big{(}1-\phi^{f}\Big{)}\sigma_{m}+\phi^{f}\sigma^{f},\]  
where \(\phi^{f}\) and \(\phi^{m}\) are fiber and matrix volume fractions, \(\phi^{m}=1-\phi^{f}\) and \(\phi^{f}=\frac{\rho_{c}}{\rho_{f}}w_{f}\), \(\rho_{c}\) and \(\rho_{f}\) are composite and fiber densities, and \(w_{f}\) is the fiber mass fraction.  
The Mori-Tanaka homogenization model was used to account for composite properties at high fiber concentrations [10]. Useful closed-form expressions for the Eshelby tensor for anisotropic matrix have been given by [PERSON] and [PERSON] [52]. The effective stiffness tensor of the composite is given by  
\[\mathbf{C}=\mathbf{C}^{m}+\phi^{f}\Big{(}\mathbf{C}^{f}-\mathbf{C}^{m}\Big{)}\mathbf{A}^{Eshelby} \Big{(}\phi^{m}\mathbf{I}+\phi^{f}\mathbf{A}^{Eshelby}\Big{)}^{-1},\]  
where \(\mathbf{A}^{Eshelby}\) is the strain-concentration tensor [53].  
Orientation averaging is used to account for the distribution of fiber orientations. [PERSON] and [PERSON] proposed the following formula [11]:  
\[\Big{\langle}C_{ijkl}\Big{\rangle}=B_{1}a_{ijkl}+B_{2}(a_{ij}\delta_{kl}+a_{ kl}\delta_{ij})+B_{3}(a_{ik}\delta_{jl}+a_{il}\delta_{jk}+a_{jl}\delta_{ik}+a_{jk} \delta_{il})+B_{4}\big{(}\delta_{ij}\delta_{kl})+B_{5}\Big{(}\delta_{ik}\delta _{jl}+\delta_{il}\delta_{jk}\Big{)},\]  
where \(a_{ijkl}\) and \(a_{ij}\) are the components of fourth-order and second-order fiber orientation tensors, \(\delta_{ij}\) is the components of second-order unit tensor, and the coefficients \(B\) are related to the components of the stiffness matrix of the transversely isotropic unidirectional composite [54].  
Figure 4: Sample on an electron microscope.  
The plasticity model [55] describes the stress-strain state of the matrix and is based on an equivalent stress \(\sigma_{eq}\):  
\[\sigma_{eq}=\sigma_{Y}+R\big{(}\varepsilon_{p}\big{)},\]  
where \(\sigma_{Y}\) is the initial yield stress; \(R\big{(}\varepsilon_{p}\big{)}=k\varepsilon_{p}+R_{\infty}\big{[}1-e^{-m \varepsilon_{p}}\big{]}\) is an exponential and linear hardening law [56]; and \(\varepsilon_{p}\) is the accumulated plastic strain.  
[PERSON]'s ratio of the matrix, in the plastic range, is predicted through the Lame parameters \(E\) and \(G\) using spectral decomposition [57], where the shear modulus \(G\) is as follows:  
\[G=G_{e}\left(1-\frac{3G_{e}}{3G_{e}+\frac{dR\big{(}\varepsilon_{p}\big{)}}{dx _{p}}}\right)\]  
Because the equation for calculating fiber orientation contains a fourth-order tensor, it was reduced to a second-order tensor using the orthotropic closure approximation presented by [PERSON] [58]. The values of the orientation tensor components are also set in Digimat, and the fiber direction from 0 to 90\({}^{\circ}\) was divided into 20 equal angle increments, with a tolerance on trace of orientation tensor of 0.01. The values of the fiber orientation tensor used for model building can be written in the following form, \(a_{ij}\), which is equivalent to the multilayer approach calculated in [44]:  
\[a_{ij}=\begin{bmatrix}0.763796&-0.00093&-0.00099\\ &0.180333&-0.00038\\ &&0.055871\end{bmatrix}\]