A Fiber Distributed Model of Biological Tissues

Abstract

Several micro-structured biological tissues are characterized by anisotropy. The dependence of stiffness and strength of the material on the direction is determined mainly by the presence of cable-like micro- and nano-structures made of collagen. Recent findings concerning the arrangement of the structural collagen in biological tissues suggest that, although the functionality would require a prevailing orientation of the fibers, the organization of the organ introduces instead a certain degree of dispersion. In this regard, we propose a material model alternative to the one based on generalized structure tensors, proposed by Gasser et al. (2006). In the present model the strain energy function is assumed to be dependent on the mean value and on the variance of the pseudo-invariant I4 of the fiber distribution. We consider the stress response under standard uniaxial shear and biaxial loading conditions of the proposed model. Finally, we derive an approximated explicit expression of the covariance tensor for the second Piola-Kirchhoff stress and verify such expression via numerical integration.