## 2 Constitutive Model
### Anisotropic Hyper-Elastic Materials  
The form of the strain energy potential for modeling arterial layers with distributed collagen fiber orientations is based on that proposed by [PERSON] et al. [22; 23; 24]:  
\[U=C_{10}\Big{(}\overline{I}_{1}-3\Big{)}+\frac{1}{D}\Bigg{(}\frac{\left(r^{el }\right)^{2}-1}{2}-\ln J^{el}\Bigg{)}+\frac{k_{1}}{2k_{2}}\sum\limits_{\alpha=1 }^{N}\Big{\{}\exp\Big{[}k_{2}\big{(}\overline{E}_{\alpha}\big{)}^{2}\Big{]}-1 \Big{\}} \tag{14}\]  
\[\overline{E}_{\alpha}\kappa\Big{(}\overline{I}_{1}-3\Big{)}+(1-3\kappa)\Big{(} \overline{I}_{4(\alpha\alpha)}-1\Big{)} \tag{15}\]  
where \(U\) is the strain energy per unit of the reference volume; \(C_{10}\), \(D\), \(k_{1}\), \(k_{2}\), and \(\kappa\) are temperature-dependent material parameters; \(N\) is the number of families of fibers (\(N\leq 3\)); \(\overline{I}_{1}\) is the first invariant of \(\overline{\mathbf{C}}\); \(J^{el}\) is the elastic volume ratio; and \(\overline{I}_{4(\alpha\alpha)}\) are the pseudo-invariants of \(\overline{\mathbf{C}}\) and \(\mathbf{A}_{\alpha}\)This model presumes that the orientation of the collagen fibers in each family is distributed rotationally symmetrically with respect to the mean preferred orientation. The parameter, \(\kappa\) (\(0\leq\kappa\leq 1/3\)), expresses the level of dispersion in the fiber directions. If \(\rho(\Theta)\) is the orientation density function that identifies the distribution (it implies the normalized number of fibers with orientations in the range \([\Theta,\ \Theta+d\Theta]\) associated with the mean orientation) [24], then the parameter, \(\kappa\), is defined as follows:  
\[\kappa=\frac{1}{4}\int_{0}^{\pi}\rho(\Theta)\sin^{3}\Theta d\Theta. \tag{16}\]  
It is also presumed that all the fiber groups exhibit the same mechanical properties and the same dispersion. When \(\kappa=0\), the fibers are completely aligned (no dispersion). When \(\kappa=1/3\), the fibers are randomly distributed, and the material becomes isotropic. This corresponds to the spherical orientation density function. The strain-like quantity, \(\overline{E}_{\alpha}\), identifies the deformation of the fiber group with the mean direction, \(\mathbf{A}_{\alpha}\). \(\overline{E}_{\alpha}=\overline{I}_{4(\alpha\alpha)}-1\) for completely aligned fibers (\(\kappa=0\)), and \(\overline{E}_{\alpha}=\left(\overline{I}_{1}-3\right)/3\) for randomly-distributed fibers (\(\kappa=1/3\)).  
The first two terms in the strain energy function express the distortional and volumetric contributions of the non-collogenous isotropic ground material. The third term expresses the contributions from other families of collagen fibers, considering the effects of dispersion. The basic presumption of the model is that collagen fibers can support only tension because they would buckle under compressive loads. Hence, the anisotropic contribution in the strain energy function only appears when the strain of the fibers is positive, or equivalently, when \(\overline{E}_{\alpha}>0\). This condition is enforced by the term \(\left\langle\overline{E}_{\alpha}\right\rangle\), where operator \(\left\langle\cdot\right\rangle\) represents the Macaulay bracket and is defined as \(\left\langle x\right\rangle=1/2(|x|+x)\)[25].  
The initial deviatoric elasticity tensor, \(\overline{\mathbf{D}}_{0}\), and bulk modulus, \(K_{0}\), are given by:  
\[\overline{\mathbf{D}}_{0}=4C_{10}\mathfrak{I}+2(1-3\kappa)^{2}k_{1}\sum_{ \alpha=1}^{N}H\!\left(\overline{E}_{\alpha}\right)\!\mathbf{A}_{\alpha} \mathbf{A}_{\alpha}\mathbf{A}_{\alpha} \tag{17}\]  
\[K_{0}=\frac{2}{D} \tag{18}\]  
where \(\mathfrak{I}\) is the fourth-order unit tensor and \(H(x)\) is the Heaviside unit step function.