A Generic Finite Element Implementation of Arterial Wall Constitutive Laws

Hyper-elastic formulations are usually employed to represent the constitutive response of biological tissues. Particularly these formulations are often encountered in models assessing the arterial wall behavior. A great number of distinct hyper-elastic laws can be found in the specialized literature devised to represent, at different extents, the mechanical aspects of the arterial tissues. However, no one can be, a priori, considered more suitable than the others. The choice depends on the specific applications and on the type of mechanical response the corresponding models are intended to account for. Consequently, it is convenient to take at hand the possibility of implementing and rapid prototyping different constitutive laws in an easy and reliable manner. Therefore, in this work we describe how to implement a generic Finite Element framework capable to accommodate practically any hyper-elastic material law. This is carried out using a spatial variational formulation for the momentum equation which is linearized by means of a Newton-Raphson scheme. The iterative algorithm is such that for a given load, the equilibrium is reached in the deformed spatial configuration. The main feature of our approach is based on the evaluation of the second order stress tensor and of the fourth order constitutive tangent tensor using finite differences. That is, given a strain energy potential we compute, by means of a second order finite difference centered scheme, the first (stress) and second (tangent matrix) derivatives. In this way, a generic computational implementation in the context of Finite Elements is achieved, making possible to change the material behavior just changing the procedure that evaluates the elastic function and reusing the entire numerical element structure. The developments are carried out for cuasi-incompressible materials, and some implementation issues are presented and discussed. The method is validated for two common constitutive laws of the arterial tissues including the mechanical response of the arterial wall considered as a fiber-reinforced multilayer material.