Third- and fourth-order elasticity of biological soft tissues

TL;DR

This paper extends the third- and fourth-order elasticity theory to biological soft tissues with fiber anisotropy, enabling better modeling of their nonlinear elastic behavior and aiding in the determination of nonlinear constants through wave propagation analysis.

Contribution

It develops a fourth-order strain-energy expansion for anisotropic soft tissues with aligned fibers, advancing the theoretical framework beyond isotropic models.

Findings

Derived fourth-order elasticity model for fiber-reinforced tissues

Applied model to acoustoelasticity for nonlinear constant estimation

Enhanced understanding of soft tissue nonlinear elastic properties

Abstract

In the theory of weakly non-linear elasticity, Hamilton et al. [J. Acoust. Soc. Am. \textbf{116} (2004) 41] identified $W = \mu I_2 + (A/3)I_3 + D I_2^2$ as the fourth-order expansion of the strain-energy density for incompressible isotropic solids. Subsequently, much effort focused on theoretical and experimental developments linked to this expression in order to inform the modeling of gels and soft biological tissues. However, while many soft tissues can be treated as incompressible, they are not in general isotropic, and their anisotropy is associated with the presence of oriented collagen fiber bundles. Here the expansion of $W$ is carried up to fourth-order in the case where there exists one family of parallel fibers in the tissue. The results are then applied to acoustoelasticity, with a view to determining the second- and third-order nonlinear constants by employing small-amplitude transverse waves propagating in a deformed soft tissue.