The relevance of transverse deformation effects in modeling soft biological tissues

Hyperelastic constitutive models for anisotropic biological materials are frequently based on orthotropic incompressible stored energy functions. The material parameters of these models are then obtained through an optimization procedure as to fit some stress-strain experimental data. For example, in arterial wall mechanics the material data usually employed for the Holzapfel-Gasser-Ogden and the Gasser-Ogden-Holzapfel models are two uniaxial tension curves from circumferential and axial specimens. The transverse strains from these specimens are frequently not taken into consideration. In this paper we analyze the evolution of those strains, showing that an unrealistic behaviour may be predicted. We then show how transverse strains may be prescribed using our What-You-Prescribe-Is-What-You-Get (WYPIWYG) model in a very intuitive way, still capturing the longitudinal stress-strain behavior in an exact manner without employing any constitutive parameter. This is possible because, in contrast to what it is usually done, we exactly solve the equilibrium and compatibility equations without imposing the shape of the stored energy function. Furthermore, we show that the small strains formulation is naturally recovered and that the physical insight from the infinitesimal theory is preserved. In fact, for incompressible materials, the present approach can be considered as a natural extension of the infinitesimal continuum elastic framework to large strains. This new physical insight clearly shows that if some subclasses of orthotropic incompressible material models are determined with just two uniaxial curves, then the transverse behavior should be contrasted with additional experimental observations.