2.2.1. Passive Material Behavior

The anisotropic and nonlinear passive mechanical behavior of arterial tissue is often represented by a Gasser-Ogden-Holzapfel (Gasser et al., 2006) hyperelastic formulation. The deviatoric strain energy function is decomposed in an isotropic Neo-Hookean part, representing the elastin fibers in the tissue, and an exponential, anisotropic part, representing two collagen fiber families running in two symmetric directions. Assuming a fully incompressible material and ignoring the volumetric contribution, the strain energy function of the elastin and collagen contribution is respectively written as

where C₁₀ and k₁ represent the stiffness of elastin and collagen. k₂ determines the exponential collagen behavior and κ quantifies the fiber dispersion. I¯1elas and I¯1coll are the first invariants or traces of the deviatoric right Cauchy-Green stretch tensors tr(C-elas)=tr(J-2/3FelasTFelas) and tr(C-coll)=tr(J-2/3FcollTFcoll), where F^elas and F^coll are the deformation gradients of elastin and collagen respectively and J is the Jacobian of the deformation gradient F. More information on these different deformation gradients follows in section 2.2.4. I¯4coll and I¯6coll are the fourth and sixth invariants of C-coll and M_i, representing the stretch along the preferred fiber direction, written as

with M_i the undeformed fiber vector defined by the fiber angle α_i with respect to the circumferential direction. Therefore, Mi=[0cosαisinαi]T, assuming that the radial direction is the first direction, the circumferential direction the second and the axial the third.