Role of anisotropic invariants in numerically modeling soft biological tissues as transversely isotropic hyperelastic materials: A comparative study

Abstract

Fiber-reinforced biological soft tissues are often modeled as anisotropic hyperelastic materials. Four strain invariants can be used to define a strain energy function for soft tissues reinforced with a single fiber family: two isotropic invariants I1,I2 and two anisotropic invariants I4,I5. Invariant I5 is often omitted in the strain energy functions to simplify the problem mathematically. In this study, the implications of using only I4 or the use of both anisotropic invariants I4,I5 in the numerical modeling of soft tissues were analyzed. A simple modification to the Holzapfel–Gasser–Ogden (HGO) model is proposed by adding a term that contains invariant I5. Material parameters were calculated by fitting the models with experimental data of uniaxial traction in the tibialis anterior tendon tissue of rats. General analytical solutions for the simple load scenarios were obtained. Such solutions are taken as a reference point to measure the precision of the numerical results obtained in the finite element simulations. The main differences between the models were observed in the shear behavior. The proposed model predicts three different shear responses (two responses with fiber reinforcement and one isotropic response), while the HGO model predicts two equal isotropic responses and only one with fiber reinforcement. Two sets of simple shear experimental data on a fiber-reinforced elastomer material were used to verify the shear stress prediction of the models. The experimental data show that the three shear behaviors are different; thus, the HGO model offers a limited description of the shear behavior. Furthermore, analytical solutions and experimental data suggest that invariant I5 is related to the elastic energy of the fibers when the material is subjected to shear in a direction parallel to the fibers. The models for non-homogeneous deformations were also compared. For this comparison, an irregular geometry was implemented in ABAQUS, and the traction and shear conditions were simulated by changing the fiber inclination. The most critical difference was found when the shear was parallel to the direction of the fiber and the minor difference when the fiber was at 45° to the load force.