A new strain energy function for the hyperelastic modelling of ligaments and tendons based on fascicle microstructure

Tom Shearer

doi:10.1016/j.jbiomech.2014.11.031

Abstract

A new strain energy function for the hyperelastic modelling of ligaments and tendons based on the geometrical arrangement of their fibrils is derived. The distribution of the crimp angles of the fibrils is used to determine the stress–strain response of a single fascicle, and this stress–strain response is used to determine the form of the strain energy function, the parameters of which can all potentially be directly measured via experiments – unlike those of commonly used strain energy functions such as the Holzapfel–Gasser–Ogden (HGO) model, whose parameters are phenomenological. We compare the new model with the HGO model and show that the new model gives a better match to existing stress–strain data for human patellar tendon than the HGO model, with the average relative error in matching this data when using the new model being 0.053 (compared with 0.57 when using the HGO model), and the average absolute error when using the new model being 0.12MPa (compared with 0.31MPa when using the HGO model).

Highlights

Ligaments and tendons are fundamental structures in the musculoskeletal systems of vertebrates
We develop a model for ligaments and tendons, but which will potentially be adaptable to other fibrous soft tissues
We derive a new strain energy function (SEF) for the hyperelastic modelling of ligaments and tendons. The form of this SEF is determined from the stress–strain response of fascicles as a function of their microstructure, using the method described by Kastelic et al (1980)

Summary

Introduction

Ligaments and tendons are fundamental structures in the musculoskeletal systems of vertebrates. I1 1⁄4 tr C; and I4 1⁄4 M Á ðCMÞ; ð2Þ where C 1⁄4 FTF is the right Cauchy–Green tensor, where F is the deformation gradient tensor (Ogden, 1997), and M is a unit vector pointing in the direction of the tissue's fibres before any deformation has taken place, c, k1 and k2 are material parameters, and the above expression is only valid when I4 Z 1 (when I4 o 1, W 1⁄4 c=2ðI1 À 3Þ) This SEF was proposed as a constitutive model for arteries and is commonly used, along with its variants (Holzapfel and Ogden, 2010) to model a wide variety of biological materials. Note that Kastelic et al (1980) included a “crimp blunting factor”, b, in their model, which resulted in the above expressions for εpðρÞ and εn being multiplied by ð1 ÀbÞ; since we are interested

The effect of fibril crimp angle distribution

Derivation of the strain energy function

Comparison with the HGO model

Findings

Discussion