## 2 Materials and Methods
### Material Definition
\[\overline{I}_{1}=tr\ (\overline{\textbf{C}}) \tag{3}\]  
\[\overline{I}_{2}=\big{(}\frac{1}{2}\big{)}\Big{[}\big{(}tr\ (\overline{\textbf{C}}) \big{)}^{2}-tr(\overline{\textbf{C}}^{2})\Big{]} \tag{4}\]  
The pseudo-invariants, associated with fibers' families, can be expressed in terms of initial orientations and the modified Cauchy-Green deformation tensor according to the expressions contained in Equations (5)-(10)  
\[\overline{I}_{4}=\textbf{m}_{0}\overline{\textbf{C}}\textbf{m}_{0} \tag{5}\]  
\[\overline{I}_{5}=\textbf{m}_{0}\overline{\textbf{C}}^{2}\ \textbf{m}_{0} \tag{6}\]  
\[\overline{I}_{6}=\textbf{n}_{0}\overline{\textbf{C}}\textbf{n}_{0} \tag{7}\]  
\[\overline{I}_{7}=\textbf{n}_{0}\overline{\textbf{C}}^{2}\textbf{n}_{0} \tag{8}\]  
\[\overline{I}_{8}=\textbf{m}_{0}\overline{\textbf{C}}\textbf{n}_{0} \tag{9}\]  
\[\overline{I}_{9}=\left(\textbf{m}_{0}\textbf{n}_{0}\right)^{2} \tag{10}\]  
\(\overline{I}_{4}\) and \(\overline{I}_{6}\) represent the squared stretch of fibers. \(\overline{I}_{5}\) and \(\overline{I}_{7}\) are associated with transversal behavior. \(\overline{I}_{8}\) considers the interaction between two fibers' families. \(\overline{I}_{9}\) is a constant that depends on the material's configuration. The volumetric component was evaluated according to Equation (11) where the compressibility factor \(d\) was calculated considering a Bulk factor \(K=5.5\) MPa and taken into account that \(d=\)\(\frac{2}{K}\) as was considered in previous work [36]:  
\[\psi=\psi_{\mathrm{volumetric}}\;(J)+\;\overline{\psi}_{\mathrm{Isochoric}}( \overline{I}_{1},\overline{I}_{2},\overline{I}_{4},\overline{I}_{5},\overline{ I}_{6},\overline{I}_{7},\overline{I}_{8},\overline{I}_{9}) \tag{2}\]0  
The isochoric isotropic component was modeled with a 2-parametric Mooney-Rivlin, as seen in Equation (12), where \(a_{1}\) and \(a_{2}\) are the experimentally obtained materials. \[\psi=\psi_{\mathrm{volumetric}}\;(J)+\;\overline{\psi}_{\mathrm{Isochoric}}( \overline{I}_{1},\overline{I}_{2},\overline{I}_{4},\overline{I}_{5},\overline{ I}_{6},\overline{I}_{7},\overline{I}_{8},\overline{I}_{9}) \tag{2}\]1  
This work did not bear in mind the interaction between fiber families, and the same shear modulus was contemplated in the direction of the fiber families and orthogonal to this direction. According to this hypothesis [1], the influence of pseudo-invariants \(\overline{I}_{5}\), \(\overline{I}_{7}\) and \(\overline{I}_{8}\) was not considered. \(\overline{I}_{9}\) was constant during material deformation but was not included in the material deformation description. The isochoric anisotropic component was considered by the Holzapfel model [37], in which only pseudo invariants \(\overline{I}_{4}\) and \(\overline{I}_{6}\) were taken into account according to Equation (13). \[\psi=\psi_{\mathrm{volumetric}}\;(J)+\;\overline{\psi}_{\mathrm{Isochoric}}( \overline{I}_{1},\overline{I}_{2},\overline{I}_{4},\overline{I}_{5},\overline{ I}_{6},\overline{I}_{7},\overline{I}_{8},\overline{I}_{9}) \tag{2}\]2  
In this equation, constants \(k_{1}\) and \(k_{2}\) are related to fiber behavior. The former considers stiffness at low extension, whereas the latter shows the influence at high strain levels. According to the hypothesis, and considering the Clausius-Planck inequality, the second Piola-Kirchhoff tensor can be derived according to Equation (14). \[\textbf{S}=Jp\textbf{C}^{-1}+2\Bigg{[}