An Improved Non-Affine Arruda-Boyce Type Constitutive Model for Collagen Networks

Abstract

This work investigates, by means of computational modeling, the mechanical properties of soft collagen tissues on the basis of elasticity theory. Bio-polymer networks are structurally disordered and thus compelled to deform non-affine. To capture that in our computational modeling, we supplement the well-known affine Arruda-Boyce model with positional disorder and compute the resultant changes in mechanical response. We characterize this mechanical behavior as a response to various homogeneous deformations in 3D networks, assuming different constitutive behavior for the individual fibers (in the small deformations linear regime, hookean springs under the entropic elasticity assumption, and in the nonlinear regime freely-jointed and worm-like chains), as well as different coordination numbers (4, 6 or 8 chains connecting at each cross linking point) of the resulting fiber networks. Furthermore we compare the moduli of the simulated networks with their affine deformed counterparts. Previous work has clearly demonstrated that non-affine deformation modes in elastic (bio)polymer networks greatly affect their mechanics. As the original Arruda-Boyce model can be represented with a particular form of strain-energy function that is micro-mechanically motivated, incorporation of the non-affinity yields amended predictions of the macroscopic mechanical behavior of soft fibrous networks, based on an improved representation of microscopic network structure and deformations. We show that shear and bulk moduli in the Arruda-Boyce model can be as off as 30% when compared with the shear and bulk moduli in the non-affine model. This entire evaluation of the ways non-affinity enhances the well known Arruda-Boyce model sets the groundwork for developing accurate constitutive relations for fibrous biological materials, for use in finite element analysis of soft tissues.