Specialised strain energy functions for modelling the contribution of the collagen network (Waniso) to the deformation of soft tissues

A popular framework in continuum mechanics modelling of soft tissues is the use of an additive split of the total strain energy function (W) into the contribution of the isotropic matrix (Wiso) and the anisotropic collagen fibres network (Waniso): W=Wiso+Waniso. This paper presents specialised strain energy functions for the Waniso part of this additive split, in the form of Waniso (I4) or Waniso (I4, I6) for one or two fibre families, respectively, accounting for the deformation and contribution of the collagen fibres network. The models have their origins in the statistical mechanics treatment of chains network based on a non-Gaussian, a Gaussian and a modified Gaussian approach. The models are applied to extant experimental stress-stretch data, across multi-scales from a single collagen molecule to the network ensemble, demonstrating an excellent agreement. Due to the direct physical structural basis of the model parameters and therefore their objectivity and uniqueness, these models are proposed as advantageous options next to the existing phenomenological continuum-based strain energy functions in the literature. In addition, and while not exploited in this paper, since the model parameters are inherent structural properties of the collagen molecular chains, they may be established a priori via imaging or molecular techniques. Therefore, the proposed models allow the important possibility of precluding the need for destructive mechanical tests and calibration a posteriori; instead paving the way for predicting the mechanical behaviour of the collagen network from pre-established structural parameters.