Efficient computing of the viscoelastic response of helical tendon subunits

BACKGROUND AND OBJECTIVE: Tendons are hierarchical structures, with a viscoelastic, time-dependent mechanical response upon axial loading. Their mechanical behavior strongly depends on the inner composition and on the material attributes of their subunits. With tens of thousands of damaged tendons each year all over the world, an increasing need for robust simulation tools arises, which can assist tendon diagnosis and restoration praxis in the selection of appropriate treatments and substitute materials. METHODS: In this paper, we elaborate a numerical model for the computation of the viscoelastic response of tendon fascicles to tensile loads. The model can inherently describe the composite inner tendon structure, accounting for its helical geometric arrangement and for different relaxation behaviors among its constituents. The model's weak form and finite element implementation are derived making use of the generalized Maxwell-Wiechert model, which inherently accounts for a complex multi-parameter relaxation behavior. RESULTS: The model results have been validated with respect to existing homogenization results, with the proposed framework to reproduce homogenization results with an accuracy of 5% over the entire relaxation process. The simulation scheme has been further used to study the viscoelastic behavior of tendon fascicles with different inner structural compositions, namely with different physiologically relevant fiber contents, viscosity values and helical angles, extending the applicability of existent schemes. What is more, its robust formulation has been shown to outperform other simulation methods. CONCLUSIONS: Overall, the proposed framework has been shown to constitute a robust simulation tool for the computation of the relaxation behavior of tendon fascicles. It has provided a means to compute the time-evolution of the axial and of the torsional modulus of tendon subunits. The latter has been shown to be significant for typical fascicle structures, with its viscoelastic behavior to well compare with the one recorded for the axial modulus of the tendon's subunit.