Nonlinear elasticity, anisotropy, material stability and residual stresses in soft tissue

In this chapter the basic equations of nonlinear elasticity theory needed for the analysis of the elastic behaviour of soft tissues are summarized. Particular attention is paid to characterizing the material symmetries associated with the anisotropy that arises in soft tissue from its fibrous constituents (collagens) that endow the material with preferred directions. The importance of the issue of convexity in the construction of constitutive laws (strain-energy functions) for soft tissues is emphasized with reference to material stability The problem of extension and inflation of a thick-walled circular cylindrical tube is used throughout as an example that is closely associated with arterial wall mechanics. This is discussed first for isotropic materials, then for cylindrically orthotropic materials. Since residual stresses have a very important role in, in particular, arterial wall mechanics these are examined in some detail. Specifically, for the tube extension/inflation problem the residual stresses arising from the assumption that the circumferential stress is uniform under typical physiological conditions are calculated for a representative constitutive law and compared with those calculated using the 'opening angle' method