Anisotropy and nonlinear elasticity in arterial wall mechanics

In this chapter we provide a theoretical framework based on the nonlinear theory of elasticity that can be used as the background against which the mechanical properties of soft biological tissue can be analyzed by comparing theory with experimental data. Of particular concern will be the elastic properties of arterial wall tissue. The results of mechanical testing are important for the characterization of the material properties through appropriate consitutive laws that are essential for the simulation of, for example, clinical procedures that involve deformation of the tissue. An important constituent of soft tissue is its fibrous structure, particularly the arrangement of collagen fibres. These fibres have a strong influence on the mechanical properties of the tissue and, in particular, they endow the material with anisotropic properties. This anisotropy features in our analysis and is accounted for by one or more families of preferred directions within the constitutive description. Particular attention is focused on biaxial deformations for both transversely isotropic materials and a general class of anisotropic materials. Specific constitutive laws are discussed for models based on invariants, incorporating the influence of fibre orientation and dispersion, and on constitutive formulations based directly on the Green strain tensor. The theory is illustrated by application to a prototype boundary-value problem, namely the extension and inflation of a circular cylindrical tube, representing an artery. Some attention is focused on questions related to the convexity and strong ellipticity of the constitutive laws since these issues are important for the appropriateness of the models from the mathematical point of view, for questions of material and structural stability and for stability of numerical schemes, including finite element computations.