Specialized Strain Energy Functions for Modeling the Contribution of the Collagen Network (Waniso) to the Deformation of Soft Tissues

A popular framework in continuum mechanics modeling of soft tissues is the use of an additive split of the total strain energy function (W) into the contribution of the isotropic matrix (W-iso) and the anisotropic collagen fiber networks (W-aniso): W = W-iso + W-aniso. This paper presents specialized strain energy functions for the W-aniso part of this additive split, in the form of W-aniso(I-4) or W-aniso(I-4, I-6) for one or two fiber families, respectively, accounting for the deformation and contribution of the collagen fibers' 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 of paving the way for predicting the mechanical behavior of the collagen network from pre-established structural parameters. These features render the proposed models as attractive choices for application in continuum-based modeling of collagenous soft tissues.