A finite-strain micromechanical model for the hyperelasticity of tendons and ligaments with crimped fibers

Abstract

The hyperelasticity of soft biological tissues such as tendons and ligaments is closely related to their microstructures consisting of wavy fibers embedded in a soft matrix. Large deformation of such composites involves both microstructural evolution and material nonlinearity, posing a challenge of theoretical analysis. In this paper, we propose a finite-strain micromechanical method to predict the hyperelastic constitutive relation of tendons and ligaments reinforced with crimped fibers. The Mori–Tanaka method is extended to account for the interaction of wavy fibers embedded in a soft matrix under large deformation. It is found that the tension–shear coupling mechanism induced by fiber–matrix interaction is crucial for the nonlinear behavior of tendons and ligaments. The hyperelasticity of such composites relies strongly on the initial crimp shape, Young's modulus, and volume fraction of fibers. This model is validated by finite element simulations of a unit cell model with crimped fibers. Furthermore, an explicit expression based on dimensional analysis is derived for the hyperelastic constitutive relation in terms of microstructural and material parameters, and it fits well with experimental data of tendons at different ages. This study can not only deepen the understanding of the mechanical behaviors of such soft tissues of tendons and ligaments, but also help the optimal design of biomimetic composites with superior elasticity.