## 2 Methodology
### Numerical Model
The invariants of the right Cauchy-Green deformation tensor are as follows:  
\[I_{1}\ =\ \mathrm{tr}\mathbf{C}I_{2}\ =\ \frac{1}{2}\Big{[}\big{(}\mathrm{tr} \mathbf{C}\big{)}^{2}-\mathrm{tr}\mathbf{C}^{2}\Big{]}\qquad I_{3}\ =\ \mathrm{det}\mathbf{C} \tag{5}\]  
Therefore, the strain energy density function is dependent on the invariants [20], and the Cauchy stress equation is expanded as follows:  
\[\sigma\ =\ 2J^{-1}\bigg{(}\mathbf{F}\frac{\partial\psi}{\partial I_{1}} \frac{\partial I_{1}}{\partial\mathbf{C}}\mathbf{F}^{T}+\mathbf{F}\frac{ \partial\psi}{\partial I_{2}}\frac{\partial I_{2}}{\partial\mathbf{C}} \mathbf{F}^{T}+\mathbf{F}\frac{\partial\psi}{\partial I_{3}}\frac{\partial I _{3}}{\partial\mathbf{C}}\mathbf{F}^{T}\bigg{)} \tag{6}\]  
By obtaining the derivatives of the invariants with respect to the right Cauchy-Green deformation tensor [24] and knowing that \(\mathbf{B}=\mathbf{F}\mathbf{F}^{\mathrm{T}}\), Equation (6) can be expressed as follows:  
\[\sigma\ =\ 2J^{-1}\Big{[}\psi_{1}\mathbf{B}+\psi_{2}\Big{(}I_{1}\mathbf{B}- \mathbf{B}^{2}\Big{)}+I_{3}\psi_{3}\mathbf{I}\Big{]} \tag{7}\]  
where \(\psi=\partial\psi/\partial I_{i}\). In an incompressible isotropic material, \(I_{3}=\) det \(\mathbf{F}=1\), and the Cauchy stress tensor is modified as follows [24]:  
\[\sigma\ =\ -p\mathbf{I}+2\mathbf{F}\frac{\partial\psi}{\partial\mathbf{C}} \mathbf{F}^{T} \tag{8}\]  
where \(p\) is a scalar identified as hydrostatic pressure. Therefore, the Cauchy stress tensor of an incompressible material associated with \(\mathbf{B}\) is expressed as follows:  
\[\sigma\ =\ -p\mathbf{I}+2\psi_{1}\mathbf{B}+2\psi_{2}\Big{(}I_{1}\mathbf{B}- \mathbf{B}^{2}\Big{)} \tag{9}\]  
**Anisotropic hyper-elastic materials. ** In the organization of human soft tissues, the existence of collagen fibers [25] causes the material to have one or more preferred directions (\(\mathbf{M}\)). In this case, the strain energy density is a function of both the right Cauchy-Green deformation tensor and the preferred direction. Two more dependent pseudo-invariants for these materials are defined as follows [19]:  
\[S_{ij}\ =\ \frac{\partial\psi}{\partial\epsilon_{ij}} \tag{1}\]0  
where, for example, \(\mathbf{CM}\) expresses the action of the second-order tensor \(\mathbf{C}\) on the vector \(\mathbf{M}\). In an incompressible material reinforced by one family of fibers, the strain energy density function is dependent on \(I_{1}\), \(I_{2}\), \(I_{4}\), and \(I_{5}\). In this particular case, the Cauchy stress has two additional terms that indicate the effects of anisotropy. Therefore, the Cauchy stress tensor could be represented as follows [20]:  
\[S_{ij}\ =\ \frac{\partial\psi}{\partial\epsilon_{ij}} \tag{1}\]1  
where \(\otimes\) denotes the dyadic product of two vectors and \(\mathbf{m}\) = FM is the deformed form of the vector \(\mathbf{M}\) in the present configuration. In some tissues, such as arterial walls, two families of fibers with different directions can be found within the tissue. Therefore, \(\mathbf{M}^{\prime}\) can be considered as the unit vector in the direction of the second family of fibers. In addition, three more invariants were considered as follows [20]:  
\[S_{ij}\ =\ \frac{\partial\psi}{\partial\epsilon_{ij}} \tag{1}\]2  
In this case, the Cauchy stress is represented as follows [20]:  
\[ \begin{array}{l}\sigma\ =\ -p\mathbf{I}+2\psi_{1}\mathbf{B}+2\psi_{2} \big{(}I_{1}\mathbf{B}-\mathbf{B}^{2}\big{)}+2\psi_{4}\mathbf{m}\otimes \mathbf{m}+2\psi_{5}[\mathbf{m}\otimes\mathbf{B}\mathbf{m}+\mathbf{B}\mathbf{ m}\otimes\mathbf{m}]\\ +2\psi_{6}\mathbf{m}^{{}^{\prime}}\otimes\mathbf{m}^{{}^{\prime}}+2\psi_{7} \Big{[}\mathbf{m}^{{}^{\prime}}\otimes\mathbf{B}\mathbf{m}^{{}^{\prime}}+ \mathbf{B}\mathbf{m}^{{}^{\prime}}\otimes\mathbf{m}^{{}^{\prime}}\Big{]}+2 \psi_{8}\Big{(}\mathbf{M}\otimes\mathbf{M}^{{}^{\prime}}\Big{)}\Big{(} \mathbf{m}\otimes\mathbf{m}^{{}^{\prime}}+\mathbf{m}^{{}^{\prime}}\otimes \mathbf{m}\Big{)}\end{array}