*2.2. Mechanical Properties of Aorta*

2.2.1. Ogden Model for Description of Isotropic Hyperelastic Behavior of Aorta

In the Ogden material model (Equation (6)), the strain energy density is expressed in terms of the principal stretches as:

$$W(\lambda\_{1\prime}, \lambda\_{2\prime}, \lambda\_3, \lambda\_5) = \sum\_{\mathbf{p}=1}^{N} \frac{\mu\_{\mathbf{p}}}{\alpha\_{\mathbf{p}}} \left(\lambda\_1^{\alpha\_{\mathbf{p}}} + \lambda\_2^{\alpha\_{\mathbf{p}}} + \lambda\_3^{\alpha\_{\mathbf{p}}} - 3\right),\tag{6}$$

where N, µp, and α<sup>p</sup> are material constants. Under the assumption of incompressibility, one can rewrite this as

$$\mathcal{W}(\lambda\_{1'}, \lambda\_2 \mid) = \sum\_{\mathsf{p}=1}^{\mathsf{N}} \frac{\mu\_{\mathsf{p}}}{\alpha\_{\mathsf{p}}} \left(\lambda\_1^{\alpha\_{\mathsf{p}}} + \lambda\_2^{\alpha\_{\mathsf{p}}} + \lambda\_1^{-\alpha\_{\mathsf{p}}} \lambda\_2^{-\alpha\_{\mathsf{p}}} - 3\right). \tag{7}$$

In general, the shear modulus is calculated as follows:

$$2\mu = \sum\_{\mathbf{p}=1}^{N} \mu\_{\mathbf{p}} \alpha\_{\mathbf{p}\cdots} \tag{8}$$

where N = 3; by fitting the material parameters, the material behavior of rubbers can be described very accurately. For particular values of material constants, the Ogden (Equation (7)) model will reduce to either the neo-Hookean solid (N = 1, α = 2) or the Mooney–Rivlin material (N = 2, α<sup>1</sup> = 2, α<sup>2</sup> = −2 with the constraint condition λ<sup>1</sup> λ2λ<sup>3</sup> = 1).

2.2.2. Holzapfel–Gasser–Ogden Model for Description of Anisotropic Hyperelastic Behavior of Aorta

The simplified form of the strain energy potential is based on that proposed by Holzapfel, Gasser, and Ogden [18] for modeling arterial layers with distributed collagen fiber orientations:

$$\mathcal{W} = \mathbb{C}\_{10}(\mathbb{I}\_1 - \mathfrak{Z}) + \frac{\mathbf{k}\_1}{2\mathbf{k}\_2} \sum\_{\alpha=1}^{\mathcal{N}} \left\{ \exp\left[\mathbf{k}\_2 \overline{\mathbf{E}}\_{\alpha} ^2\right] - 1 \right\},\tag{9}$$

with

$$\overline{\mathbf{E}}\_{\mathfrak{A}} = \kappa (\overline{\mathbf{I}}\_1 - \mathfrak{A}) + (1 + \mathfrak{A}\kappa)(\overline{\mathbf{I}}\_{4(\alpha \cdot \mathfrak{a})} - \mathbf{1}),\tag{10}$$

where W is the strain energy per unit of reference volume; C10, κ, k1, and k<sup>2</sup> are temperaturedependent material parameters; N is the number of families of fibers (N ≤ 3); I<sup>1</sup> is the first deviatoric strain invariant; and I4(αα) are pseudo-invariants of C and Eα.

The model (Equations (9)–(10)) assumes that the directions of the collagen fibers within each family are dispersed (with rotational symmetry) about a mean preferred direction. The parameter κ (0 ≤ κ ≤ 1/3) describes the level of dispersion in the fiber directions. If ρ(Θ) is the orientation density function that characterizes the distribution (it represents the normalized number of fibers with orientations in the range [Θ, Θ + dΘ] with respect to the mean direction), the parameter κ is defined as

$$\kappa = \frac{1}{4} \int\_0^\pi \rho(\Theta) \sin^3 \Theta d\Theta. \tag{11}$$

It is also assumed that all families of fibers have the same mechanical properties and the same dispersion (Equation (11)). When κ = 0, the fibers are perfectly aligned (no dispersion). When κ = 1/3, the fibers are randomly distributed and the material becomes isotropic; this corresponds to a spherical orientation density function.

In this study, isotropic (Equations (7)–(8)) and anisotropic (Equations (9)–(11)) Holzapfel– Gasser–Ogden models are applied to describe the hyperelastic properties of the aorta.
