A constitutive model for fibrous tissues with cross-linked collagen fibers including dispersion — With an analysis of the Poynting effect

The present work deals with cross-links in collagenous tissues and proposes a planar continuum model for large strains to mechanically characterize, for example, the well-known stiffening effect associated with the presence of these cross-links. A key novelty of the paper is the consideration of dispersed fibers connected by randomly distributed cross-links. The model is essentially based on two mechanisms: (i) a fiber dispersion-induced cross-link dispersion and (ii) a fiber-independent cross-link dispersion connecting two arbitrary parallel fibers of the sample space. Within the framework of the generalized structure tensors, we derive appropriate invariants that enter the stored-energy function, taking into account the stiffness and relative orientation of the cross-links and the cross-link-fiber interaction. To illustrate the power of the proposed model, we first consider uniaxial tension and simple shear. We investigate the influence of different cross-link configurations on the stress response. In particular, we are interested in simple shear and the sign of the normal stress perpendicular to the shear planes, which is referred to as the Poynting effect. We show that the cross-links have a significant impact on the normal stress considered. This may provide a deeper insight into the microstructural mechanisms in semi-flexible biopolymer gels that are responsible for the tendency, in simple shear, of the top and bottom faces to approach each other. Finally, we investigate the Poynting effect that occurs in a circular hollow cylinder under pure torsion, i.e. how the axial normal force correlates with the cross-linked fibrous microstructure of the specimen.