Influence of the material nearly incompressibility on the numerical prediction of maximum wall stresses in Abdominal Aortic Aneurysm
Résumé
Accurate and reliable numerical stress analysis of Abdominal Aortic Aneurysm (AAA) requires using appropriate constitutive laws for the AAA material. Recently planar biaxial tests, carried out on normal and pathologic human abdominal aortic tissues (Vande Geest J. of Biomech., 2006), clearly showed that the (degeneration) formation of the (aorta) aneurysm leads to an increase in mechanical anisotropy (and also an increase in the circumferential stiffness for) of the AAA tissue as compared to the normal aortic tissue. To take into account such features, hyperelastic anisotropic constitutive models were then proposed using a Green-strain approach (Vande Geest et al, J. of Biomech., 2006) or an invariant approach (Rodriguez et al, J. of Biomech. Eng. 2008, Basciano et al, J. of Biomech. Eng. 2009). In both cases, the strain energy is formulated assuming that the arterial material is incompressible. However in practice, a nearly incompressible approach, i.e. a mixed formulation pressure-displacement, is usually adopted to perform finite element stress analysis of Abdominal Aortic Aneurysm (AAA). Consequently, the strain energy function is split into an isochoric part and a volumetric part U. This volumetric function U depends on the jacobian J and on the initial bulk modulus K of the material. The incompressibility of the material is nearly reached for large values of K which is used as a user-specified penalty parameter. In the present work, the influence of the initial bulk modulus K on both the magnitude and the distribution of anisotropic wall stresses in AAA is emphasized. For that purpose, an analytical analysis of the influence of K on the mechanical response of anisotropic models is first achieved in the case of an equibiaxial test. This analysis is performed using invariant-based anisotropic hyperelastic models only, but the conclusions would be the same for strain elongation formulations. Three strain energies based on polynomial or exponential functions are considered and a classical quadratic volumetric function U is used. Through a parametric study, it is shown that for strongly non-linear anisotropic models, large values of K are sufficient to ensure the incompressibility condition, i.e. to estimate wall stresses with a given precision, in a restricted range of deformations only. Consequently, numerical simulations outside this range may lead to large errors on the estimation of wall stresses. To illustrate such result, the anisotropic model proposed by Rodriguez et al (J. of Biomech. Eng. 2008) was implemented in a finite element code and numerical simulations on idealised AAA geometries were then performed. From these simulations, it is shown that even in simple AAA geometries, too small value of K may lead to important error on both the magnitude of the maximum first principal stress and also on its location.