## 2 Materials and Methods
### Material Definition
\begin{array}{c}\Big{(}\frac{\partial \overline{p}}{\delta\overline{I}_{1}}+\overline{I}_{1}\frac{\partial \overline{p}}{\delta\overline{I}_{2}}\Big{)}\textbf{1}-\frac{\partial \overline{p}}{\delta\overline{I}_{2}}\overline{\textbf{C}}+\overline{I}_{4} \frac{\partial\overline{p}}{\delta\overline{I}_{4}}\textbf{m}_{0}\otimes \textbf{m}_{0}+\overline{I}_{6}\frac{\partial\overline{p}}{\delta\overline{I}_{ 6}}\textbf{n}_{0}\otimes\textbf{n}_{0}\\ -\frac{1}{3}\left(\frac{\partial\overline{p}}{\delta\overline{I}_{1}}\overline {I}_{1}+2\frac{\partial\overline{p}}{\delta\overline{I}_{2}}\overline{I}_{2}+ \frac{\partial\overline{p}}{\delta\overline{I}_{4}}\overline{I}_{4}+\frac{ \partial\overline{p}}{\delta\overline{I}_{6}}\overline{I}_{6}\right) \overline{\textbf{C}}^{-1}\end{array} \Bigg{]} \tag{14}\]From the second Piola-Kirchhoff tensor, the Cauchy Stress Tensor was evaluated by considering the following relation between the tensor components:  
\[\mathbf{\sigma}=J^{-1}\mathbf{F}\mathbf{S}\mathbf{F}^{T} \tag{15}\]  
That applied to Equation (14) can be expressed as the following equation:  
\[\mathbf{\sigma} = p\mathbbm{1}+\frac{2}{J}\left[ \begin{array}{cc}\left(\frac{ \partial\overline{\Psi}}{\delta T_{1}}+T_{1}\frac{\partial\overline{\Psi}}{ \delta T_{2}}\right)\overline{\mathbf{b}}-\frac{\partial\overline{\Psi}}{ \delta T_{2}}\overline{\mathbf{b}}^{2}+T_{4}\frac{\partial\overline{\Psi}}{ \delta T_{4}}\overline{\mathbf{m}}\otimes\mathbf{m}+T_{6}\frac{\partial \overline{\Psi}}{\delta T_{6}}\mathbf{n}\otimes\mathbf{n}\\ -\frac{1}{3}\left(\frac{\partial\overline{\Psi}}{\delta T_{1}}T_{1}+2\frac{ \partial\overline{\Psi}}{\delta T_{2}}T_{2}+\frac{\partial\overline{\Psi}}{ \delta T_{4}}\overline{\mathbf{I}}_{4}+\frac{\partial\overline{\Psi}}{\delta T _{6}}T_{6}\right)\mathbf{1}\end{array} \right] \tag{16}\]  
In these equations, \(p\) is the hydrostatic pressure, \(\mathbf{1}\) is the identity tensor, \(\overline{\mathbf{b}}\) is the modified Left Cauchy strain tensor defined as \(\overline{\mathbf{b}}=\overline{\mathbf{F}}\mathbf{F}^{T}\) and \(\mathbf{m}\) and \(\mathbf{n}\) are the fibers' orientation vectors in the deformed state. Collagen fibers distribution was considered by the variation in the \(k_{1}\) parameter. In this paper, corneal surface was divided into octants, and concentric zones with diameters from 0 to 10 mm in steps of 2 mm were considered. Furthermore, a transition zone between 10 and 11 mm was modeled until the limbus zone considered from 11 mm to the external diameter. Fibers were oriented in the vertical and horizontal directions with the exception of the limbus, where fibers were oriented circumferentially. The materials' properties are found in Table 1 where the parameters are obtained from the values obtained by [21; 36]. The value of \(k_{1}\) is varied depending on the zone to consider the fiber density variation [21]. The variation of \(k_{1}\) is based on the work of [PERSON], [PERSON], [PERSON] [24]. In their work, 66% of fibers are concentrated in horizontal and vertical octants, whereas 34% is concentrated in the oblique octants. The fiber's variation was considered taking into account a reduced value of \(k_{1}\) in the central oblique zones. A transition zone was considered between zone with minimum and maximum values of \(k_{1}\).